2080:
1718:
5932:
2075:{\displaystyle {\begin{aligned}\operatorname {rad} (abc)&=\operatorname {rad} (a)\operatorname {rad} (b)\operatorname {rad} (c)\\&=\operatorname {rad} (1)\operatorname {rad} \left(2^{6n}-1\right)\operatorname {rad} \left(2^{6n}\right)\\&=2\operatorname {rad} \left(2^{6n}-1\right)\\&=2\operatorname {rad} \left(9\cdot {\tfrac {b}{9}}\right)\\&\leqslant 2\cdot 3\cdot {\tfrac {b}{9}}\\&={\tfrac {2}{3}}b\\&<{\tfrac {2}{3}}c.\end{aligned}}}
125:
5609:. This is not only because of their length and the difficulty of understanding them, but also because at least one specific point in the argument has been identified as a gap by some other experts. Although a few mathematicians have vouched for the correctness of the proof and have attempted to communicate their understanding via workshops on IUTT, they have failed to convince the number theory community at large.
136:
2456:
5553:
5363:
5624:
for discussions with
Mochizuki. While they did not resolve the differences, they brought them into clearer focus. Scholze and Stix wrote a report asserting and explaining an error in the logic of the proof and claiming that the resulting gap was "so severe that ... small modifications will not rescue
4931:
3598:
3158:
2741:
of the set of all qualities (defined above) is 1, which implies the much weaker assertion that there is a finite upper bound for qualities. The conjecture that 2 is such an upper bound suffices for a very short proof of Fermat's Last
Theorem for
2238:
3461:
4694:
4796:
3801:
5379:
5189:
1252:
1062:
772:
3363:
2221:
1709:
4813:
5143:
683:
564:
1586:
947:
6264:"Finiteness Theorems for Dynamical Systems", Lucien Szpiro, talk at Conference on L-functions and Automorphic Forms (on the occasion of Dorian Goldfeld's 60th Birthday), Columbia University, May 2007. See
2550:
3472:
2630:
conjecture has a large number of consequences. These include both known results (some of which have been proven separately only since the conjecture has been stated) and conjectures for which it gives a
5055:
2243:
1723:
847:
3039:
602:
4720:
2451:{\displaystyle {\begin{aligned}b&=2^{p(p-1)n}-1\\&=\left(2^{p(p-1)}\right)^{n}-1\\&=\left(2^{p(p-1)}-1\right)(\cdots )\\&=p^{2}\cdot r(\cdots ).\end{aligned}}}
457:
3374:
5636:, the institute's journal. Mochizuki is chief editor of the journal but recused himself from the review of the paper. The announcement was received with skepticism by
4621:
2766:
2727:
4725:
3708:
261:
5548:{\displaystyle c>k\exp \left(4{\sqrt {\frac {3\log k}{\log \log k}}}\left(1+{\frac {\log \log \log k}{2\log \log k}}+{\frac {C_{2}}{\log \log k}}\right)\right)}
5358:{\displaystyle c<k\exp \left(4{\sqrt {\frac {3\log k}{\log \log k}}}\left(1+{\frac {\log \log \log k}{2\log \log k}}+{\frac {C_{1}}{\log \log k}}\right)\right)}
3865:
conjecture, it is hoped that patterns in the triples discovered by this project will lead to insights about the conjecture and about number theory more generally.
4965:
293:
201:
483:
424:
402:
315:
223:
1145:
994:
691:
7024:
A panorama in number theory or The view from Baker's garden. Based on a conference in honor of Alan Baker's 60th birthday, ZĂŒrich, Switzerland, 1999
6711:
Number theory. Diophantine, computational and algebraic aspects. Proceedings of the international conference, Eger, Hungary, July 29-August 2, 1996
6163:
3289:
2105:
1599:
4926:{\displaystyle c<\kappa \operatorname {rad} (abc){\frac {{\Big (}\log {\big (}\operatorname {rad} (abc){\big )}{\Big )}^{\omega }}{\omega !}}}
6349:
7812:
5781:"Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta-functions, notes on the work of Shinichi Mochizuki"
5072:
609:
491:
5629:
2700:
7879:
7608:
7566:
7470:
6536:
3234:
1500:
6327:
892:
6407:
6380:
3593:{\displaystyle c<\exp {\left(K_{3}\operatorname {rad} (abc)^{\frac {1}{3}}\left(\log(\operatorname {rad} (abc)\right)^{3}\right)}}
6288:
Mochizuki, Shinichi (4 March 2021). "Inter-universal TeichmĂŒller Theory IV: Log-Volume
Computations and Set-Theoretic Foundations".
2496:
5659:
4994:
7057:
3252:
5625:
the proof strategy"; Mochizuki claimed that they misunderstood vital aspects of the theory and made invalid simplifications.
7740:
7331:
7273:
7071:
6799:
5598:
368:
claimed to have a proof in 2012, but the conjecture is still regarded as unproven by the mainstream mathematical community.
6597:
the ... discussions ... constitute the first detailed, ... substantive discussions concerning negative positions ... IUTch.
70:
5650:
as "unlikely to move many researchers over to
Mochizuki's camp". In March 2021, Mochizuki's proof was published in RIMS.
7717:
5816:
7233:
7214:
7035:
6971:
6882:
6718:
6677:
2738:
6462:
6176:
5906:
1325:> 1 such as in the second example are rather special, they consist of numbers divisible by high powers of small
6583:"Report on Discussions, Held during the Period March 15 â 20, 2018, Concerning Inter-Universal TeichmĂŒller Theory"
3153:{\displaystyle c_{f}=\prod _{{\text{prime }}p}x_{i}\left(1-{\frac {\omega \,\!_{f}(p)}{p^{2+q_{p}}}}\right).}
2772:
2690:
103:
6650:"Comments on the manuscript (2018-08 version) by Scholze-Stix concerning Inter-Universal TeichmĂŒller Theory"
6115:
809:
7757:
6568:
Web-page by
Mochizuki describing discussions and linking consequent publications and supplementary material
6556:
5605:
conjecture. The papers have not been widely accepted by the mathematical community as providing a proof of
7884:
570:
6206:
7027:
6791:
6736:
6702:
5856:
Castelvecchi, Davide (9 April 2020). "Mathematical proof that rocked number theory will be published".
4699:
4565:
2465:
7839:
7617:
3456:{\displaystyle c<\exp {\left(K_{2}\operatorname {rad} (abc)^{{\frac {2}{3}}+\varepsilon }\right)}}
2643:
431:
3245:
2672:
88:
7422:
7189:
7126:
7063:
5690:, any common factor of two of the values is necessarily shared by the third. Thus, coprimality of
2776:
2696:
98:
6113:
Pasten, Hector (2017), "Definability of
Frobenius orbits and a result on rational distance sets",
318:. A number of famous conjectures and theorems in number theory would follow immediately from the
7874:
4689:{\displaystyle {\big (}\varepsilon ^{-\omega }\operatorname {rad} (abc){\big )}^{1+\varepsilon }}
7167:
7889:
7243:
Langevin, M. (1993). "Cas d'égalité pour le théorÚme de Mason et applications de la conjecture
7184:
4791:{\displaystyle \varepsilon ={\frac {\omega }{\log {\big (}\operatorname {rad} (abc){\big )}}}.}
3796:{\displaystyle c>\operatorname {rad} (abc)\exp {\left(k{\sqrt {\log c}}/\log \log c\right)}}
3671:
There are also theoretical results that provide a lower bound on the best possible form of the
3649:
3622:
7854:
6959:
6874:
7760:
7604:
7575:
7562:
7526:
7462:
7454:
7224:
Lando, Sergei K.; Zvonkin, Alexander K. (2004). "Graphs on
Surfaces and Their Applications".
2860:
2745:
2706:
2665:
2654:
377:
113:
93:
7807:
7118:
6866:
6613:
6235:
1435:> 1.0001, etc. In particular, if the conjecture is true, then there must exist a triple (
7869:
7708:
7671:
7390:
7135:
6833:
6649:
6496:
6136:
6060:
5865:
3281:
2977:
2783:
331:
234:
7821:
7445:
7341:
7300:
7045:
6917:
6809:
6728:
6614:"Comments on the manuscript by Scholze-Stix concerning Inter-Universal TeichmĂŒller Theory"
6404:
6387:
341:
conjecture originated as the outcome of attempts by
Oesterlé and Masser to understand the
16:
The product of distinct prime factors of a,b,c, where c is a+b, is rarely much less than c
8:
7409:
4942:
272:
180:
7727:. Springer Proceedings in Mathematics & Statistics. Vol. 98. pp. 211â230.
7458:
7370:
7139:
7019:
6837:
6778:
Bombieri, Enrico (1994). "Roth's theorem and the abc-conjecture" (Preprint). ETH ZĂŒrich.
6582:
6500:
6166:, Siegelâs theorem and the abc conjecture, Riv. Mat. Univ. Parma (7) 3, 2004, S. 323â332
5869:
1262:(4, 127, 131) = log(131) / log(rad(4·127·131)) = log(131) / log(2·127·131) = 0.46820...
7592:
7550:
7487:
7175:
7151:
7015:
7003:
6981:
6849:
6766:
6673:
6645:
6620:
6609:
6578:
6305:
6156:
6140:
6012:
5889:
5590:
3833:
in the
Netherlands, together with the Dutch Kennislink science institute, launched the
468:
409:
387:
365:
323:
300:
208:
7366:
7085:
6270:
1247:{\displaystyle q(a,b,c)={\frac {\log(c)}{\log {\big (}{\textrm {rad}}(abc){\big )}}}.}
1057:{\displaystyle c<K_{\varepsilon }\cdot \operatorname {rad} (abc)^{1+\varepsilon }.}
166:
129:
44:
7778:
7736:
7596:
7554:
7491:
7436:
7413:
7378:
7327:
7288:
7229:
7210:
7067:
7031:
6967:
6878:
6867:
6795:
6770:
6714:
6624:
6514:
5893:
5881:
5597:
conjecture. He released a series of four preprints developing a new theory he called
3830:
2959:
2639:
2632:
1068:
342:
108:
7119:"ABC implies no "Siegel zeros" for L-functions of characters with negative exponent"
4939:
an absolute constant. After some computational experiments he found that a value of
7842:
7781:
7728:
7694:
7685:
7657:
7648:
7626:
7584:
7542:
7512:
7479:
7441:
7431:
7337:
7296:
7163:
7155:
7143:
7114:
7100:
7081:
7041:
6999:
6995:
6941:
6913:
6841:
6805:
6756:
6724:
6504:
6485:"The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof"
6309:
6297:
6144:
6124:
5873:
5824:
5792:
5703:
5645:
5621:
4982:
4615:
2647:
381:
227:
174:
7630:
7304:
6740:
6428:
5586:
proposed a solution in 2007, but it was found to be incorrect shortly afterwards.
7732:
7704:
7667:
7530:
7386:
7206:
6783:
6541:
6532:
6411:
6132:
3238:
3164:
2804:
2679:
1268:(3, 125, 128) = log(128) / log(rad(3·125·128)) = log(128) / log(30) = 1.426565...
83:
7517:
7500:
767:{\displaystyle {\text{rad}}(1000000)={\text{rad}}(2^{6}\cdot 5^{6})=2\cdot 5=10}
7816:
7397:
7198:
7053:
6323:
5877:
5641:
3838:
2838:
346:
7848:
7105:
6946:
6929:
6128:
5829:
5797:
5780:
7863:
7802:
7382:
7292:
6986:
6817:
6372:
6354:
6184:
5637:
5632:
where
Mochizuki works announced that his claimed proof would be published in
5613:
5583:
2658:
162:
31:
7796:
3358:{\displaystyle c<\exp {\left(K_{1}\operatorname {rad} (abc)^{15}\right)}}
2216:{\displaystyle a=1,\quad b=2^{p(p-1)n}-1,\quad c=2^{p(p-1)n},\qquad n>1.}
1704:{\displaystyle b=2^{6n}-1=64^{n}-1=(64-1)(\cdots )=9\cdot 7\cdot (\cdots ).}
7834:
7699:
7680:
7662:
7639:
7260:
6955:
6518:
6458:
6424:
6301:
5885:
5564:
2953:
all the polynominals (x^n-1)/(x-1) have an infinity of square-free values.
2879:= 1), and Pillai's conjecture (1931) concerning the number of solutions of
2816:
1326:
461:
357:
conjecture was shown to be equivalent to the modified Szpiro's conjecture.
265:
170:
140:
49:
7483:
7147:
6761:
2562:) is given below; the highest quality, 1.6299, was found by Eric Reyssat (
7845:
wiki page linking to various sources of commentary on Mochizuki's papers.
7825:
7770:
6925:
3273:
3231:
conjecture would imply that there are only finitely many counterexamples.
2853:
2808:
858:
7007:
6960:"On Ternary Equations of Fermat Type and Relations with Elliptic Curves"
4555:
3861:. Although no finite set of examples or counterexamples can resolve the
1423:) > 1, the conjecture predicts that only finitely many of those have
7588:
7546:
6865:-conjecture". In Bambah, R. P.; Dumir, V. C.; Hans-Gill, R. J. (eds.).
6853:
6376:
6266:
6063:, Frits Beukers, ABC-DAY, Leiden, Utrecht University, 9 September 2005.
5617:
5138:{\displaystyle O{\big (}\operatorname {rad} (abc)\Theta (abc){\big )},}
2827:
158:
678:{\displaystyle {\text{rad}}(18)={\text{rad}}(2\cdot 3^{2})=2\cdot 3=6}
559:{\displaystyle {\text{rad}}(16)={\text{rad}}(2^{4})={\text{rad}}(2)=2}
264:. The conjecture essentially states that the product of the distinct
7786:
7226:
Encyclopaedia of Mathematical Sciences: Lower-Dimensional Topology II
6160:
349:, which involves more geometric structures in its statement than the
7830:
7018:(2002). "Modular forms, elliptic curves and the abc-conjecture". In
6845:
6509:
6484:
2491:
And now with a similar calculation as above, the following results:
7764:
4372:
3834:
2842:
6350:"Baffling ABC maths proof now has impenetrable 300-page 'summary'"
1581:{\displaystyle a=1,\quad b=2^{6n}-1,\quad c=2^{6n},\qquad n>1.}
364:
conjecture have been made, but none have gained broad acceptance.
124:
7681:"The ABC conjecture implies Vojta's height inequality for curves"
2092:
to have larger square factors, the ratio between the radical and
790:
6290:
Publications of the Research Institute for Mathematical Sciences
5634:
Publications of the Research Institute for Mathematical Sciences
942:{\displaystyle c>\operatorname {rad} (abc)^{1+\varepsilon }.}
6537:"Titans of Mathematics Clash Over Epic Proof of ABC Conjecture"
1109:
A fourth equivalent formulation of the conjecture involves the
135:
5956:
3167:, a generalization of Fermat's Last Theorem proposing that if
2545:{\displaystyle \operatorname {rad} (abc)<{\tfrac {2}{p}}c.}
6246:
6214:
5593:
has claimed a proof of Szpiro's conjecture and therefore the
7718:"Lecture on the abc Conjecture and Some of Its Consequences"
1383:
Whereas it is known that there are infinitely many triples (
7503:[On the distribution of the kernel of an integer].
5718:. So in this case, it does not matter which concept we use.
1475:> 0 is necessary as there exist infinitely many triples
6042:
5050:{\displaystyle K^{\Omega (abc)}\operatorname {rad} (abc),}
3253:
Siegel's theorem about integral points on algebraic curves
3248:
on dense sets of Euclidean points with rational distances.
2894:
As equivalent, the GranvilleâLangevin conjecture, that if
7808:
http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf
5922:
5920:
5918:
1466:
854:
deals with the exceptions. Specifically, it states that:
7831:
Philosophy behind Mochizukiâs work on the ABC conjecture
7052:
6892:
DÄ
browski, Andrzej (1996). "On the diophantine equation
2693:
on the separation between squares and cubes of integers.
7776:
3284:. Specifically, the following bounds have been proven:
2991:
has only finitely many solutions for any given integer
2558:(triples with a particularly small radical relative to
7773:: Easy to follow, detailed explanation by Brian Hayes.
7452:
7265:
Proceedings of the Symposium on Analytic Number Theory
5915:
5738:
5736:
5734:
5162:
4491:
Jerzy Browkin, Juliusz Brzezinski, Abderrahmane Nitaj
2525:
2051:
2026:
2004:
1965:
330:
conjecture as "The most important unsolved problem in
6873:. Trends in Mathematics. Basel: BirkhÀuser. pp.
6094:
6030:
5980:
5382:
5192:
5075:
4997:
4945:
4816:
4728:
4702:
4624:
4556:
Refined forms, generalizations and related statements
3711:
3475:
3377:
3292:
3241:
of a non-torsion rational point of an elliptic curve.
3042:
2823:
conjecture as formulated above for rational integers.
2748:
2709:
2499:
2241:
2108:
1721:
1602:
1503:
1148:
997:
895:
812:
694:
612:
573:
494:
471:
434:
412:
390:
303:
275:
237:
211:
183:
7499:
Robert, Olivier; Tenenbaum, GĂ©rald (November 2013).
5968:
5628:
On April 3, 2020, two mathematicians from the Kyoto
1713:
Using this fact, the following calculation is made:
6820:; BrzeziĆski, Juliusz (1994). "Some remarks on the
5760:
5731:
7678:
7263:(1985). "Open problems". In Chen, W. W. L. (ed.).
7062:. Princeton: Princeton University Press. pp.
6463:"Notes on the Oxford IUT workshop by Brian Conrad"
5962:
5944:
5817:"Proof claimed for deep connection between primes"
5547:
5357:
5137:
5049:
4959:
4925:
4790:
4714:
4688:
3841:system, which aims to discover additional triples
3795:
3592:
3455:
3357:
3152:
2760:
2729:, from an effective form of a weak version of the
2721:
2544:
2450:
2215:
2074:
1703:
1580:
1246:
1056:
941:
841:
766:
677:
596:
558:
477:
451:
418:
396:
376:Before stating the conjecture, the notion of the
309:
287:
255:
217:
195:
7637:
7056:; Barrow-Green, June; Leader, Imre, eds. (2008).
6816:
5992:
5748:
5560:
4901:
4852:
3818:
3100:
2096:can be made arbitrarily small. Specifically, let
7861:
6328:"The ABC conjecture has (still) not been proved"
4600:is the total number of distinct primes dividing
2675:allowing for a finite number of counterexamples.
7525:
7498:
7404:. Princeton University Press. pp. 361â362.
7162:
6010:
5166:
3679:showed that there are infinitely many triples (
3676:
3366:
7113:
6782:
6252:
6048:
7533:(1986). "On the Oesterlé-Masser conjecture".
7377:, SĂ©minaire Bourbaki exp 694 (161): 165â186,
7371:"Nouvelles approches du "théorÚme" de Fermat"
5851:
5849:
5847:
5127:
5081:
4893:
4865:
4777:
4749:
4669:
4627:
1447:) that achieves the maximal possible quality
1233:
1204:
7223:
6790:. New Mathematical Monographs. Vol. 4.
6482:
6367:
6365:
6281:
5855:
5165:proposed a more precise inequality based on
3276:by a near-linear function of the radical of
2563:
2555:
7715:
7471:Bulletin of the London Mathematical Society
7349:Nitaj, Abderrahmane (1996). "La conjecture
5926:
5601:(IUTT), which is then applied to prove the
3664:. The bounds apply to any triple for which
2464:divides 2 â 1. This follows from
322:conjecture or its versions. Mathematician
7093:International Mathematics Research Notices
6934:International Mathematics Research Notices
6011:Granville, Andrew; Tucker, Thomas (2002).
5844:
5063:) is the total number of prime factors of
2779:concerning powers that are sums of powers.
2703:. However it follows easily, at least for
2701:a famously difficult proof by Andrew Wiles
1336:, there exist only finitely many triples (
864:, there exist only finitely many triples (
7698:
7661:
7603:
7561:
7516:
7501:"Sur la répartition du noyau d'un entier"
7435:
7408:
7396:
7281:Far East Journal of Mathematical Sciences
7249:Comptes rendus de l'Académie des sciences
7188:
7104:
7080:
6945:
6891:
6760:
6608:
6531:
6508:
6362:
6287:
6100:
6036:
5986:
5828:
5796:
4618:noticed that the minimum of the function
4564:conjecture is an integer analogue of the
3601:
3464:
3098:
2969:
7365:
7324:Advanced number theory with applications
7242:
7014:
6980:
6966:. New York: Springer. pp. 527â548.
6777:
6322:
5974:
5938:
5766:
5742:
5660:List of unsolved problems in mathematics
5177:). They conjectured there is a constant
3824:
173:in 1985. It is stated in terms of three
134:
123:
7400:(2008). "Computational Number Theory".
6964:Modular Forms and Fermat's Last Theorem
6860:
6483:Castelvecchi, Davide (8 October 2015).
6423:
6371:
6088:
5778:
4971:. This version is called the "explicit
3829:In 2006, the Mathematics Department of
2898:is a square-free binary form of degree
7862:
7402:The Princeton Companion to Mathematics
7321:
7271:
7259:
7059:The Princeton Companion to Mathematics
6924:
6713:. Berlin: de Gruyter. pp. 37â44.
6709:-conjecture". In GyĆry, KĂĄlmĂĄn (ed.).
6457:
6347:
6112:
6076:
6072:
5950:
5810:
5808:
5754:
5163:Robert, Stewart & Tenenbaum (2014)
4981:also describes related conjectures of
3259:
3035:> 0 a positive constant defined as:
2863:concerning the number of solutions of
1467:Examples of triples with small radical
842:{\displaystyle c<{\text{rad}}(abc)}
7777:
7348:
6735:
6701:
6678:"Mochizuki's proof of ABC conjecture"
6672:
6644:
6577:
6348:Revell, Timothy (September 7, 2017).
6199:
6169:
6054:
5998:
4978:
4801:
4572:
2918:) such that for all coprime integers
2875:(Tijdeman's theorem answers the case
2678:The existence of infinitely many non-
2480:. Raising both sides to the power of
2225:Now it may be plausibly claimed that
978:) of coprime positive integers, with
876:) of coprime positive integers, with
7679:Van Frankenhuijsen, Machiel (2002).
7463:"A refinement of the abc conjecture"
7086:"ABC Allows Us to Count Squarefrees"
6954:
6265:
5814:
2621:
1395:) of coprime positive integers with
1348:) of coprime positive integers with
1285:) of coprime positive integers with
1086:) of coprime positive integers with
7638:van Frankenhuysen, Machiel (2000).
7197:
6984:(1996). "Beyond the last theorem".
6749:Publicationes Mathematicae Debrecen
6705:(1998). "Logarithmic forms and the
6557:"March 2018 Discussions on IUTeich"
5805:
2962:, which would yield a bound of rad(
597:{\displaystyle {\text{rad}}(17)=17}
13:
7880:Unsolved problems in number theory
7228:. Vol. 141. Springer-Verlag.
7203:Unsolved Problems in Number Theory
5599:inter-universal TeichmĂŒller theory
5367:holds whereas there is a constant
5155:divisible only by primes dividing
5151:) is the number of integers up to
5107:
5003:
4471:Jerzy Browkin, Juliusz Brzezinski
165:that arose out of a discussion of
14:
7901:
7751:
5907:Further comment by P. Scholze at
5815:Ball, Peter (10 September 2012).
5578:
4804:to propose a sharper form of the
4715:{\displaystyle \varepsilon >0}
4549:
2811:, given a uniform version of the
2472: > 2, 2 =
2460:The last step uses the fact that
460:, is the product of the distinct
296:is usually not much smaller than
5779:Fesenko, Ivan (September 2015).
5644:, as well as being described by
4985:that would give upper bounds on
4375:had found 23.8 million triples.
3227:have a common prime factor. The
2476: + 1 for some integer
7725:Mathematics in the 21st Century
7414:"Wieferich's criterion and the
6906:Nieuw Archief voor Wiskunde, IV
6788:Heights in Diophantine Geometry
6666:
6638:
6602:
6571:
6549:
6525:
6476:
6451:
6417:
6381:"Why abc is still a conjecture"
6341:
6316:
6258:
6228:
6150:
6106:
6082:
6066:
6004:
5785:European Journal of Mathematics
5561:Browkin & BrzeziĆski (1994)
4579:conjecture one can replace rad(
2203:
2165:
2121:
2100:> 2 be a prime and consider
1568:
1545:
1516:
1332:For every positive real number
955:For every positive real number
452:{\displaystyle {\text{rad}}(n)}
371:
7274:"A note on the ABC-conjecture"
7000:10.1080/10724117.1996.11974985
6271:"Proof of the abc Conjecture?"
5900:
5772:
5672:
5122:
5110:
5104:
5092:
5041:
5029:
5018:
5006:
4888:
4876:
4844:
4832:
4772:
4760:
4663:
4651:
3736:
3724:
3644:is a constant that depends on
3570:
3558:
3549:
3523:
3510:
3425:
3412:
3340:
3327:
3113:
3107:
2657:(already proven in general by
2518:
2506:
2484:then shows that 2 =
2438:
2432:
2403:
2397:
2381:
2369:
2325:
2313:
2276:
2264:
2192:
2180:
2148:
2136:
1821:
1815:
1796:
1790:
1781:
1775:
1766:
1760:
1744:
1732:
1695:
1689:
1671:
1665:
1662:
1650:
1228:
1216:
1191:
1185:
1170:
1152:
1036:
1023:
952:An equivalent formulation is:
921:
908:
836:
824:
805:, it turns out that "usually"
743:
717:
706:
700:
654:
635:
624:
618:
585:
579:
547:
541:
530:
517:
506:
500:
446:
440:
360:Various attempts to prove the
1:
7631:10.1215/S0012-7094-01-10815-6
7326:. Boca Raton, FL: CRC Press.
5725:
5167:Robert & Tenenbaum (2013)
4571:A strengthening, proposed by
3869:Distribution of triples with
3677:Stewart & Tijdeman (1986)
2088:with other exponents forcing
1329:. The fourth formulation is:
7733:10.1007/978-3-0348-0859-0_13
7716:Waldschmidt, Michel (2015).
7437:10.1016/0022-314X(88)90019-4
6861:Browkin, Jerzy (2000). "The
6253:Bombieri & Gubler (2006)
6049:Granville & Stark (2000)
3675:conjecture. In particular,
3280:. Bounds are known that are
2958:As equivalent, the modified
2902:> 2, then for every real
2635:. The consequences include:
793:positive integers such that
7:
7855:News about IUT by Mochizuki
7518:10.1016/j.indag.2013.07.007
7322:Mollin, Richard A. (2010).
7267:. London: Imperial College.
5653:
3691:) of coprime integers with
3367:Stewart & Tijdeman 1986
3244:A negative solution to the
3191:are positive integers with
2906:> 2 there is a constant
2084:By replacing the exponent 6
966:such that for all triples (
10:
7906:
7813:The ABC's of Number Theory
7535:Monatshefte fĂŒr Mathematik
7028:Cambridge University Press
6792:Cambridge University Press
6694:
6403:(updated version of their
6183:(in Dutch), archived from
6116:Monatshefte fĂŒr Mathematik
5878:10.1038/d41586-020-00998-2
3026:)/B' is square-free, with
2691:Marshall Hall's conjecture
959:, there exists a constant
380:must be introduced: for a
226:(hence the name) that are
155:OesterlĂ©âMasser conjecture
71:Modified Szpiro conjecture
7618:Duke Mathematical Journal
7505:Indagationes Mathematicae
7166:; Tucker, Thomas (2002).
7106:10.1155/S1073792898000592
6947:10.1155/S1073792891000144
6786:; Gubler, Walter (2006).
6129:10.1007/s00605-016-0973-2
5963:Van Frankenhuijsen (2002)
5830:10.1038/nature.2012.11378
5798:10.1007/s40879-015-0066-0
4495:
4475:
4455:
4435:
4415:
4410:
4405:
4400:
4395:
4390:
4387:
4343:
4317:
4291:
4265:
4239:
4213:
4187:
4161:
4135:
4109:
4083:
4057:
4031:
4005:
3979:
3953:
3927:
3919:
3913:
3907:
3901:
3895:
3889:
3877:
2978:the Diophantine equation
2837:) has only finitely many
2773:FermatâCatalan conjecture
2644:Diophantine approximation
104:FermatâCatalan conjecture
76:
66:
58:
37:
27:
7797:ABC conjecture home page
7423:Journal of Number Theory
7127:Inventiones Mathematicae
5665:
5557:holds infinitely often.
3819:van Frankenhuysen (2000)
3268:conjecture implies that
3251:An effective version of
3237:, a lower bound for the
2976:conjecture implies that
2564:Lando & Zvonkin 2004
2468:, which shows that, for
1067:Equivalently (using the
4381:Highest-quality triples
2761:{\displaystyle n\geq 6}
2722:{\displaystyle n\geq 6}
2556:highest-quality triples
2466:Fermat's little theorem
1321:) < 1. Triples with
1140:), which is defined as
7822:Questions about Number
7700:10.1006/jnth.2001.2769
7663:10.1006/jnth.1999.2484
7640:"A Lower Bound in the
6535:(September 20, 2018).
6236:"100 unbeaten triples"
6207:"Data collected sofar"
5549:
5359:
5139:
5051:
4961:
4927:
4792:
4716:
4690:
4566:MasonâStothers theorem
3809:< 4. The constant
3797:
3650:effectively computable
3625:that do not depend on
3594:
3457:
3359:
3154:
2775:, a generalization of
2762:
2723:
2673:ErdĆsâWoods conjecture
2546:
2452:
2217:
2076:
1705:
1582:
1248:
1058:
943:
843:
768:
679:
598:
560:
479:
453:
420:
398:
311:
289:
257:
219:
197:
143:
132:
7795:Abderrahmane Nitaj's
7761:Distributed computing
7576:Mathematische Annalen
7272:Mollin, R.A. (2009).
7168:"It's As Easy As abc"
7148:10.1007/s002229900036
6930:"ABC implies Mordell"
6762:10.5486/PMD.2004.3348
6461:(December 15, 2015).
6427:(28 September 2016).
6326:(December 17, 2017).
6177:"Synthese resultaten"
6013:"It's As Easy As abc"
5571:conjecture involving
5550:
5360:
5140:
5052:
4962:
4928:
4793:
4717:
4691:
4575:, states that in the
3825:Computational results
3798:
3602:Stewart & Yu 2001
3595:
3465:Stewart & Yu 1991
3458:
3360:
3155:
2777:Fermat's Last Theorem
2763:
2724:
2697:Fermat's Last Theorem
2547:
2488:(...) + 1.
2453:
2218:
2077:
1706:
1583:
1249:
1059:
944:
844:
769:
680:
599:
561:
480:
454:
421:
399:
378:radical of an integer
312:
290:
258:
256:{\displaystyle a+b=c}
220:
198:
138:
127:
99:Fermat's Last Theorem
7410:Silverman, Joseph H.
7117:; Stark, H. (2000).
7030:. pp. 128â147.
6994:(September): 26â34.
6741:"Experiments on the
6302:10.4171/PRIMS/57-1-4
6187:on December 22, 2008
5704:pairwise coprimality
5380:
5190:
5073:
4995:
4943:
4814:
4808:conjecture, namely:
4726:
4700:
4622:
3709:
3473:
3375:
3290:
3040:
2859:A generalization of
2746:
2737:conjecture says the
2707:
2497:
2239:
2106:
1719:
1600:
1501:
1495:). For example, let
1146:
995:
893:
810:
692:
610:
571:
492:
469:
432:
410:
388:
332:Diophantine analysis
301:
273:
235:
209:
181:
7803:ABC Triples webpage
7484:10.1112/blms/bdu069
7455:Stewart, Cameron L.
7140:2000InMat.139..509G
6838:1994MaCom..62..931B
6674:Mochizuki, Shinichi
6646:Mochizuki, Shinichi
6610:Mochizuki, Shinichi
6579:Mochizuki, Shinichi
6501:2015Natur.526..178C
6393:on February 8, 2020
5870:2020Natur.580..177C
5589:Since August 2012,
4967:was admissible for
4960:{\displaystyle 6/5}
4384:
3874:
3260:Theoretical results
2972:has shown that the
2852:has at least three
2566:, p. 137) for
1594:is divisible by 9:
1471:The condition that
1431:> 1.001 or even
857:For every positive
288:{\displaystyle abc}
196:{\displaystyle a,b}
153:(also known as the
24:
7885:1985 introductions
7779:Weisstein, Eric W.
7589:10.1007/BF01445201
7547:10.1007/BF01294603
7176:Notices of the AMS
6410:2020-02-08 at the
6061:The ABC-conjecture
6020:Notices of the AMS
5630:research institute
5591:Shinichi Mochizuki
5567:âa version of the
5545:
5355:
5135:
5047:
4957:
4923:
4788:
4712:
4686:
4378:
3868:
3793:
3590:
3453:
3355:
3246:ErdĆsâUlam problem
3150:
3070:
3010:positive integers
2861:Tijdeman's theorem
2803:) formed with the
2758:
2719:
2666:Vojta's conjecture
2655:Mordell conjecture
2542:
2534:
2448:
2446:
2213:
2072:
2070:
2060:
2035:
2013:
1974:
1701:
1578:
1273:A typical triple (
1244:
1054:
939:
839:
764:
675:
594:
556:
475:
449:
416:
394:
366:Shinichi Mochizuki
307:
285:
253:
215:
193:
144:
133:
114:Tijdeman's theorem
94:Faltings's theorem
89:ErdĆsâUlam problem
19:
7851:Numberphile video
7742:978-3-0348-0858-3
7615:conjecture, II".
7459:Tenenbaum, GĂ©rald
7453:Robert, Olivier;
7333:978-1-4200-8328-6
7183:(10): 1224â1231.
7164:Granville, Andrew
7115:Granville, Andrew
7073:978-0-691-11880-2
7020:WĂŒstholz, Gisbert
6801:978-0-521-71229-3
6495:(7572): 178â181.
6240:Reken mee met ABC
6181:RekenMeeMetABC.nl
5912:math.columbia.edu
5575:> 2 integers.
5533:
5499:
5441:
5440:
5343:
5309:
5251:
5250:
4921:
4783:
4568:for polynomials.
4536:) of the triple (
4515:
4514:
4369:
4368:
3831:Leiden University
3765:
3607:In these bounds,
3534:
3437:
3235:Lang's conjecture
3140:
3064:
3056:
2960:Szpiro conjecture
2926:, the radical of
2689:The weak form of
2648:algebraic numbers
2633:conditional proof
2622:Some consequences
2533:
2059:
2034:
2012:
1973:
1239:
1213:
1128:) of the triple (
1074:For all triples (
1069:little o notation
822:
715:
698:
633:
616:
577:
539:
515:
498:
478:{\displaystyle n}
438:
419:{\displaystyle n}
405:, the radical of
397:{\displaystyle n}
343:Szpiro conjecture
310:{\displaystyle c}
218:{\displaystyle c}
175:positive integers
122:
121:
7897:
7843:Polymath project
7792:
7791:
7782:"abc Conjecture"
7746:
7722:
7712:
7702:
7686:J. Number Theory
7675:
7665:
7649:J. Number Theory
7634:
7611:(2001). "On the
7600:
7569:(1991). "On the
7558:
7522:
7520:
7495:
7478:(6): 1156â1166.
7467:
7449:
7439:
7405:
7393:
7367:Oesterlé, Joseph
7362:
7345:
7318:
7316:
7315:
7309:
7303:. Archived from
7278:
7268:
7256:
7239:
7220:
7194:
7192:
7172:
7159:
7123:
7110:
7108:
7099:(19): 991â1009.
7090:
7077:
7049:
7016:Goldfeld, Dorian
7011:
6982:Goldfeld, Dorian
6977:
6951:
6949:
6921:
6888:
6872:
6857:
6832:(206): 931â939.
6813:
6784:Bombieri, Enrico
6779:
6774:
6764:
6755:(3â4): 253â260.
6732:
6689:
6688:
6686:
6684:
6670:
6664:
6663:
6661:
6659:
6654:
6642:
6636:
6635:
6633:
6631:
6618:
6606:
6600:
6599:
6594:
6592:
6587:
6575:
6569:
6567:
6565:
6563:
6553:
6547:
6546:
6533:Klarreich, Erica
6529:
6523:
6522:
6512:
6480:
6474:
6473:
6471:
6469:
6455:
6449:
6448:
6446:
6444:
6421:
6415:
6402:
6400:
6398:
6392:
6386:. Archived from
6385:
6369:
6360:
6359:
6345:
6339:
6338:
6336:
6334:
6320:
6314:
6313:
6285:
6279:
6277:
6269:(May 26, 2007),
6262:
6256:
6250:
6244:
6243:
6232:
6226:
6225:
6224:
6222:
6213:, archived from
6203:
6197:
6195:
6194:
6192:
6173:
6167:
6154:
6148:
6147:
6110:
6104:
6101:Granville (1998)
6098:
6092:
6086:
6080:
6070:
6064:
6058:
6052:
6046:
6040:
6037:Pomerance (2008)
6034:
6028:
6027:
6026:(10): 1224â1231.
6017:
6008:
6002:
5996:
5990:
5987:Silverman (1988)
5984:
5978:
5972:
5966:
5960:
5954:
5948:
5942:
5936:
5930:
5927:Waldschmidt 2015
5924:
5913:
5904:
5898:
5897:
5853:
5842:
5841:
5839:
5837:
5832:
5812:
5803:
5802:
5800:
5776:
5770:
5764:
5758:
5752:
5746:
5740:
5719:
5676:
5554:
5552:
5551:
5546:
5544:
5540:
5539:
5535:
5534:
5532:
5515:
5514:
5505:
5500:
5498:
5478:
5455:
5442:
5439:
5422:
5408:
5407:
5364:
5362:
5361:
5356:
5354:
5350:
5349:
5345:
5344:
5342:
5325:
5324:
5315:
5310:
5308:
5288:
5265:
5252:
5249:
5232:
5218:
5217:
5144:
5142:
5141:
5136:
5131:
5130:
5085:
5084:
5056:
5054:
5053:
5048:
5022:
5021:
4983:Andrew Granville
4966:
4964:
4963:
4958:
4953:
4932:
4930:
4929:
4924:
4922:
4920:
4912:
4911:
4910:
4905:
4904:
4897:
4896:
4869:
4868:
4856:
4855:
4848:
4797:
4795:
4794:
4789:
4784:
4782:
4781:
4780:
4753:
4752:
4736:
4721:
4719:
4718:
4713:
4695:
4693:
4692:
4687:
4685:
4684:
4673:
4672:
4644:
4643:
4631:
4630:
4616:Andrew Granville
4385:
4383:
4382:
4377:
4371:As of May 2014,
3875:
3867:
3813:was improved to
3802:
3800:
3799:
3794:
3792:
3791:
3787:
3771:
3766:
3755:
3652:way) but not on
3599:
3597:
3596:
3591:
3589:
3588:
3584:
3583:
3582:
3577:
3573:
3536:
3535:
3527:
3503:
3502:
3462:
3460:
3459:
3454:
3452:
3451:
3447:
3446:
3445:
3438:
3430:
3405:
3404:
3364:
3362:
3361:
3356:
3354:
3353:
3349:
3348:
3347:
3320:
3319:
3159:
3157:
3156:
3151:
3146:
3142:
3141:
3139:
3138:
3137:
3136:
3116:
3106:
3105:
3093:
3080:
3079:
3069:
3065:
3062:
3052:
3051:
2970:DÄ
browski (1996)
2767:
2765:
2764:
2759:
2733:conjecture. The
2728:
2726:
2725:
2720:
2680:Wieferich primes
2617:
2616:
2603:
2602:
2599:
2587:
2586:
2583:
2551:
2549:
2548:
2543:
2535:
2526:
2457:
2455:
2454:
2449:
2447:
2425:
2424:
2409:
2396:
2392:
2385:
2384:
2349:
2339:
2338:
2333:
2329:
2328:
2293:
2283:
2282:
2229:is divisible by
2222:
2220:
2219:
2214:
2199:
2198:
2155:
2154:
2081:
2079:
2078:
2073:
2071:
2061:
2052:
2043:
2036:
2027:
2018:
2014:
2005:
1984:
1980:
1976:
1975:
1966:
1937:
1933:
1929:
1922:
1921:
1889:
1885:
1881:
1880:
1858:
1854:
1847:
1846:
1802:
1710:
1708:
1707:
1702:
1640:
1639:
1621:
1620:
1587:
1585:
1584:
1579:
1564:
1563:
1535:
1534:
1253:
1251:
1250:
1245:
1240:
1238:
1237:
1236:
1215:
1214:
1211:
1208:
1207:
1194:
1177:
1063:
1061:
1060:
1055:
1050:
1049:
1013:
1012:
948:
946:
945:
940:
935:
934:
848:
846:
845:
840:
823:
820:
773:
771:
770:
765:
742:
741:
729:
728:
716:
713:
699:
696:
684:
682:
681:
676:
653:
652:
634:
631:
617:
614:
603:
601:
600:
595:
578:
575:
565:
563:
562:
557:
540:
537:
529:
528:
516:
513:
499:
496:
484:
482:
481:
476:
458:
456:
455:
450:
439:
436:
425:
423:
422:
417:
403:
401:
400:
395:
382:positive integer
353:conjecture. The
316:
314:
313:
308:
294:
292:
291:
286:
262:
260:
259:
254:
228:relatively prime
224:
222:
221:
216:
202:
200:
199:
194:
45:Joseph Oesterlé
25:
18:
7905:
7904:
7900:
7899:
7898:
7896:
7895:
7894:
7860:
7859:
7801:Bart de Smit's
7763:project called
7754:
7749:
7743:
7720:
7465:
7398:Pomerance, Carl
7334:
7313:
7311:
7307:
7276:
7236:
7217:
7207:Springer-Verlag
7199:Guy, Richard K.
7170:
7121:
7088:
7074:
7054:Gowers, Timothy
7038:
6974:
6885:
6846:10.2307/2153551
6802:
6721:
6697:
6692:
6682:
6680:
6671:
6667:
6657:
6655:
6652:
6643:
6639:
6629:
6627:
6616:
6607:
6603:
6590:
6588:
6585:
6576:
6572:
6561:
6559:
6555:
6554:
6550:
6542:Quanta Magazine
6530:
6526:
6510:10.1038/526178a
6481:
6477:
6467:
6465:
6456:
6452:
6442:
6440:
6422:
6418:
6412:Wayback Machine
6396:
6394:
6390:
6383:
6370:
6363:
6346:
6342:
6332:
6330:
6324:Calegari, Frank
6321:
6317:
6286:
6282:
6263:
6259:
6251:
6247:
6234:
6233:
6229:
6220:
6218:
6217:on May 15, 2014
6205:
6204:
6200:
6190:
6188:
6175:
6174:
6170:
6155:
6151:
6111:
6107:
6099:
6095:
6087:
6083:
6071:
6067:
6059:
6055:
6047:
6043:
6035:
6031:
6015:
6009:
6005:
5997:
5993:
5985:
5981:
5975:Langevin (1993)
5973:
5969:
5961:
5957:
5949:
5945:
5939:Bombieri (1994)
5937:
5933:
5925:
5916:
5905:
5901:
5854:
5845:
5835:
5833:
5813:
5806:
5777:
5773:
5765:
5761:
5753:
5749:
5741:
5732:
5728:
5723:
5722:
5677:
5673:
5668:
5656:
5612:In March 2018,
5581:
5563:formulated the
5555:
5516:
5510:
5506:
5504:
5479:
5456:
5454:
5447:
5443:
5423:
5409:
5406:
5402:
5398:
5381:
5378:
5377:
5373:
5365:
5326:
5320:
5316:
5314:
5289:
5266:
5264:
5257:
5253:
5233:
5219:
5216:
5212:
5208:
5191:
5188:
5187:
5183:
5145:
5126:
5125:
5080:
5079:
5074:
5071:
5070:
5057:
5002:
4998:
4996:
4993:
4992:
4949:
4944:
4941:
4940:
4933:
4913:
4906:
4900:
4899:
4898:
4892:
4891:
4864:
4863:
4851:
4850:
4849:
4847:
4815:
4812:
4811:
4776:
4775:
4748:
4747:
4740:
4735:
4727:
4724:
4723:
4701:
4698:
4697:
4674:
4668:
4667:
4666:
4636:
4632:
4626:
4625:
4623:
4620:
4619:
4594:
4558:
4511:Benne de Weger
4451:Benne de Weger
4380:
4379:
3887:
3882:
3827:
3803:
3767:
3754:
3750:
3746:
3745:
3710:
3707:
3706:
3643:
3620:
3613:
3605:
3578:
3542:
3538:
3537:
3526:
3522:
3498:
3494:
3493:
3489:
3488:
3474:
3471:
3470:
3468:
3429:
3428:
3424:
3400:
3396:
3395:
3391:
3390:
3376:
3373:
3372:
3370:
3343:
3339:
3315:
3311:
3310:
3306:
3305:
3291:
3288:
3287:
3262:
3165:Beal conjecture
3160:
3132:
3128:
3121:
3117:
3101:
3099:
3094:
3092:
3085:
3081:
3075:
3071:
3061:
3060:
3047:
3043:
3041:
3038:
3037:
3034:
3006:
2819:, not just the
2805:Legendre symbol
2801:
2747:
2744:
2743:
2708:
2705:
2704:
2668:in dimension 1.
2664:As equivalent,
2624:
2619:
2614:
2612:
2605:
2600:
2597:
2595:
2589:
2584:
2581:
2579:
2573:
2552:
2524:
2498:
2495:
2494:
2458:
2445:
2444:
2420:
2416:
2407:
2406:
2365:
2361:
2360:
2356:
2347:
2346:
2334:
2309:
2305:
2301:
2300:
2291:
2290:
2260:
2256:
2249:
2242:
2240:
2237:
2236:
2223:
2176:
2172:
2132:
2128:
2107:
2104:
2103:
2082:
2069:
2068:
2050:
2041:
2040:
2025:
2016:
2015:
2003:
1982:
1981:
1964:
1957:
1953:
1935:
1934:
1914:
1910:
1909:
1905:
1887:
1886:
1873:
1869:
1865:
1839:
1835:
1834:
1830:
1800:
1799:
1747:
1722:
1720:
1717:
1716:
1711:
1635:
1631:
1613:
1609:
1601:
1598:
1597:
1588:
1556:
1552:
1527:
1523:
1502:
1499:
1498:
1469:
1381:
1271:
1254:
1232:
1231:
1210:
1209:
1203:
1202:
1195:
1178:
1176:
1147:
1144:
1143:
1107:
1065:
1064:
1039:
1035:
1008:
1004:
996:
993:
992:
964:
950:
949:
924:
920:
894:
891:
890:
819:
811:
808:
807:
737:
733:
724:
720:
712:
695:
693:
690:
689:
648:
644:
630:
613:
611:
608:
607:
574:
572:
569:
568:
536:
524:
520:
512:
495:
493:
490:
489:
486:. For example,
470:
467:
466:
435:
433:
430:
429:
411:
408:
407:
389:
386:
385:
374:
347:elliptic curves
324:Dorian Goldfeld
302:
299:
298:
274:
271:
270:
236:
233:
232:
210:
207:
206:
182:
179:
178:
167:Joseph Oesterlé
130:Joseph Oesterlé
118:
84:Beal conjecture
54:
17:
12:
11:
5:
7903:
7893:
7892:
7887:
7882:
7877:
7875:Abc conjecture
7872:
7858:
7857:
7852:
7849:abc Conjecture
7846:
7840:ABC Conjecture
7837:
7828:
7819:
7817:Noam D. Elkies
7810:
7805:
7799:
7793:
7774:
7768:
7753:
7752:External links
7750:
7748:
7747:
7741:
7713:
7693:(2): 289â302.
7676:
7635:
7625:(1): 169â181.
7605:Stewart, C. L.
7601:
7583:(1): 225â230.
7563:Stewart, C. L.
7559:
7541:(3): 251â257.
7527:Stewart, C. L.
7523:
7511:(4): 802â914.
7496:
7450:
7430:(2): 226â237.
7406:
7394:
7363:
7355:Enseign. Math.
7346:
7332:
7319:
7287:(3): 267â275.
7269:
7257:
7240:
7234:
7221:
7215:
7195:
7190:10.1.1.146.610
7160:
7134:(3): 509â523.
7111:
7078:
7072:
7050:
7036:
7012:
6978:
6972:
6952:
6922:
6889:
6883:
6858:
6824:-conjecture".
6818:Browkin, Jerzy
6814:
6800:
6780:
6775:
6733:
6719:
6698:
6696:
6693:
6691:
6690:
6665:
6637:
6601:
6570:
6548:
6524:
6475:
6450:
6416:
6373:Scholze, Peter
6361:
6340:
6315:
6296:(1): 627â723.
6280:
6275:Not Even Wrong
6257:
6255:, p. 404.
6245:
6227:
6198:
6168:
6164:Andrea Surroca
6149:
6105:
6093:
6081:
6079:, p. 297)
6065:
6053:
6041:
6029:
6003:
5991:
5979:
5967:
5955:
5943:
5931:
5914:
5909:Not Even Wrong
5899:
5843:
5804:
5791:(3): 405â440.
5771:
5759:
5747:
5729:
5727:
5724:
5721:
5720:
5670:
5669:
5667:
5664:
5663:
5662:
5655:
5652:
5642:Edward Frenkel
5580:
5579:Claimed proofs
5577:
5543:
5538:
5531:
5528:
5525:
5522:
5519:
5513:
5509:
5503:
5497:
5494:
5491:
5488:
5485:
5482:
5477:
5474:
5471:
5468:
5465:
5462:
5459:
5453:
5450:
5446:
5438:
5435:
5432:
5429:
5426:
5421:
5418:
5415:
5412:
5405:
5401:
5397:
5394:
5391:
5388:
5385:
5376:
5371:
5353:
5348:
5341:
5338:
5335:
5332:
5329:
5323:
5319:
5313:
5307:
5304:
5301:
5298:
5295:
5292:
5287:
5284:
5281:
5278:
5275:
5272:
5269:
5263:
5260:
5256:
5248:
5245:
5242:
5239:
5236:
5231:
5228:
5225:
5222:
5215:
5211:
5207:
5204:
5201:
5198:
5195:
5186:
5181:
5134:
5129:
5124:
5121:
5118:
5115:
5112:
5109:
5106:
5103:
5100:
5097:
5094:
5091:
5088:
5083:
5078:
5069:
5046:
5043:
5040:
5037:
5034:
5031:
5028:
5025:
5020:
5017:
5014:
5011:
5008:
5005:
5001:
4991:
4956:
4952:
4948:
4919:
4916:
4909:
4903:
4895:
4890:
4887:
4884:
4881:
4878:
4875:
4872:
4867:
4862:
4859:
4854:
4846:
4843:
4840:
4837:
4834:
4831:
4828:
4825:
4822:
4819:
4810:
4800:This inspired
4787:
4779:
4774:
4771:
4768:
4765:
4762:
4759:
4756:
4751:
4746:
4743:
4739:
4734:
4731:
4711:
4708:
4705:
4683:
4680:
4677:
4671:
4665:
4662:
4659:
4656:
4653:
4650:
4647:
4642:
4639:
4635:
4629:
4585:
4557:
4554:
4513:
4512:
4509:
4506:
4503:
4500:
4497:
4493:
4492:
4489:
4486:
4483:
4480:
4477:
4473:
4472:
4469:
4466:
4463:
4460:
4457:
4453:
4452:
4449:
4446:
4443:
4440:
4437:
4433:
4432:
4429:
4426:
4423:
4420:
4417:
4413:
4412:
4411:Discovered by
4409:
4404:
4399:
4394:
4389:
4367:
4366:
4363:
4360:
4357:
4354:
4351:
4348:
4341:
4340:
4337:
4334:
4331:
4328:
4325:
4322:
4315:
4314:
4311:
4308:
4305:
4302:
4299:
4296:
4289:
4288:
4285:
4282:
4279:
4276:
4273:
4270:
4263:
4262:
4259:
4256:
4253:
4250:
4247:
4244:
4237:
4236:
4233:
4230:
4227:
4224:
4221:
4218:
4211:
4210:
4207:
4204:
4201:
4198:
4195:
4192:
4185:
4184:
4181:
4178:
4175:
4172:
4169:
4166:
4159:
4158:
4155:
4152:
4149:
4146:
4143:
4140:
4133:
4132:
4129:
4126:
4123:
4120:
4117:
4114:
4107:
4106:
4103:
4100:
4097:
4094:
4091:
4088:
4081:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4055:
4054:
4051:
4048:
4045:
4042:
4039:
4036:
4029:
4028:
4025:
4022:
4019:
4016:
4013:
4010:
4003:
4002:
3999:
3996:
3993:
3990:
3987:
3984:
3977:
3976:
3973:
3970:
3967:
3964:
3961:
3958:
3951:
3950:
3947:
3944:
3941:
3938:
3935:
3932:
3925:
3924:
3918:
3912:
3906:
3900:
3894:
3888:
3883:
3878:
3839:grid computing
3826:
3823:
3790:
3786:
3783:
3780:
3777:
3774:
3770:
3764:
3761:
3758:
3753:
3749:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3705:
3641:
3618:
3611:
3587:
3581:
3576:
3572:
3569:
3566:
3563:
3560:
3557:
3554:
3551:
3548:
3545:
3541:
3533:
3530:
3525:
3521:
3518:
3515:
3512:
3509:
3506:
3501:
3497:
3492:
3487:
3484:
3481:
3478:
3469:
3450:
3444:
3441:
3436:
3433:
3427:
3423:
3420:
3417:
3414:
3411:
3408:
3403:
3399:
3394:
3389:
3386:
3383:
3380:
3371:
3352:
3346:
3342:
3338:
3335:
3332:
3329:
3326:
3323:
3318:
3314:
3309:
3304:
3301:
3298:
3295:
3286:
3261:
3258:
3257:
3256:
3249:
3242:
3232:
3161:
3149:
3145:
3135:
3131:
3127:
3124:
3120:
3115:
3112:
3109:
3104:
3097:
3091:
3088:
3084:
3078:
3074:
3068:
3059:
3055:
3050:
3046:
3036:
3030:
3002:
2996:
2967:
2955:
2954:
2951:
2892:
2857:
2839:perfect powers
2824:
2815:conjecture in
2799:
2780:
2769:
2757:
2754:
2751:
2718:
2715:
2712:
2694:
2687:
2682:in every base
2676:
2669:
2662:
2651:
2640:Roth's theorem
2623:
2620:
2606:
2590:
2574:
2568:
2554:A list of the
2541:
2538:
2532:
2529:
2523:
2520:
2517:
2514:
2511:
2508:
2505:
2502:
2493:
2443:
2440:
2437:
2434:
2431:
2428:
2423:
2419:
2415:
2412:
2410:
2408:
2405:
2402:
2399:
2395:
2391:
2388:
2383:
2380:
2377:
2374:
2371:
2368:
2364:
2359:
2355:
2352:
2350:
2348:
2345:
2342:
2337:
2332:
2327:
2324:
2321:
2318:
2315:
2312:
2308:
2304:
2299:
2296:
2294:
2292:
2289:
2286:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2259:
2255:
2252:
2250:
2248:
2245:
2244:
2235:
2212:
2209:
2206:
2202:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2175:
2171:
2168:
2164:
2161:
2158:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2131:
2127:
2124:
2120:
2117:
2114:
2111:
2102:
2067:
2064:
2058:
2055:
2049:
2046:
2044:
2042:
2039:
2033:
2030:
2024:
2021:
2019:
2017:
2011:
2008:
2002:
1999:
1996:
1993:
1990:
1987:
1985:
1983:
1979:
1972:
1969:
1963:
1960:
1956:
1952:
1949:
1946:
1943:
1940:
1938:
1936:
1932:
1928:
1925:
1920:
1917:
1913:
1908:
1904:
1901:
1898:
1895:
1892:
1890:
1888:
1884:
1879:
1876:
1872:
1868:
1864:
1861:
1857:
1853:
1850:
1845:
1842:
1838:
1833:
1829:
1826:
1823:
1820:
1817:
1814:
1811:
1808:
1805:
1803:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1748:
1746:
1743:
1740:
1737:
1734:
1731:
1728:
1725:
1724:
1715:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1638:
1634:
1630:
1627:
1624:
1619:
1616:
1612:
1608:
1605:
1596:
1577:
1574:
1571:
1567:
1562:
1559:
1555:
1551:
1548:
1544:
1541:
1538:
1533:
1530:
1526:
1522:
1519:
1515:
1512:
1509:
1506:
1497:
1468:
1465:
1331:
1270:
1269:
1258:
1243:
1235:
1230:
1227:
1224:
1221:
1218:
1206:
1201:
1198:
1193:
1190:
1187:
1184:
1181:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1142:
1102:) is at least
1073:
1053:
1048:
1045:
1042:
1038:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1011:
1007:
1003:
1000:
991:
962:
954:
938:
933:
930:
927:
923:
919:
916:
913:
910:
907:
904:
901:
898:
889:
856:
852:abc conjecture
838:
835:
832:
829:
826:
818:
815:
763:
760:
757:
754:
751:
748:
745:
740:
736:
732:
727:
723:
719:
711:
708:
705:
702:
674:
671:
668:
665:
662:
659:
656:
651:
647:
643:
640:
637:
629:
626:
623:
620:
593:
590:
587:
584:
581:
555:
552:
549:
546:
543:
535:
532:
527:
523:
519:
511:
508:
505:
502:
474:
448:
445:
442:
415:
393:
373:
370:
326:described the
306:
284:
281:
278:
252:
249:
246:
243:
240:
214:
192:
189:
186:
139:Mathematician
128:Mathematician
120:
119:
117:
116:
111:
109:Roth's theorem
106:
101:
96:
91:
86:
80:
78:
74:
73:
68:
64:
63:
60:
59:Conjectured in
56:
55:
53:
52:
47:
41:
39:
38:Conjectured by
35:
34:
29:
15:
9:
6:
4:
3:
2:
7902:
7891:
7890:Number theory
7888:
7886:
7883:
7881:
7878:
7876:
7873:
7871:
7868:
7867:
7865:
7856:
7853:
7850:
7847:
7844:
7841:
7838:
7836:
7832:
7829:
7827:
7823:
7820:
7818:
7814:
7811:
7809:
7806:
7804:
7800:
7798:
7794:
7789:
7788:
7783:
7780:
7775:
7772:
7769:
7766:
7762:
7759:
7756:
7755:
7744:
7738:
7734:
7730:
7726:
7719:
7714:
7710:
7706:
7701:
7696:
7692:
7688:
7687:
7682:
7677:
7673:
7669:
7664:
7659:
7655:
7651:
7650:
7645:
7643:
7636:
7632:
7628:
7624:
7620:
7619:
7614:
7610:
7606:
7602:
7598:
7594:
7590:
7586:
7582:
7578:
7577:
7573:conjecture".
7572:
7568:
7564:
7560:
7556:
7552:
7548:
7544:
7540:
7536:
7532:
7528:
7524:
7519:
7514:
7510:
7507:(in French).
7506:
7502:
7497:
7493:
7489:
7485:
7481:
7477:
7473:
7472:
7464:
7460:
7456:
7451:
7447:
7443:
7438:
7433:
7429:
7425:
7424:
7419:
7417:
7411:
7407:
7403:
7399:
7395:
7392:
7388:
7384:
7380:
7376:
7372:
7368:
7364:
7360:
7357:(in French).
7356:
7352:
7347:
7343:
7339:
7335:
7329:
7325:
7320:
7310:on 2016-03-04
7306:
7302:
7298:
7294:
7290:
7286:
7282:
7275:
7270:
7266:
7262:
7261:Masser, D. W.
7258:
7255:(5): 441â444.
7254:
7251:(in French).
7250:
7246:
7241:
7237:
7235:3-540-00203-0
7231:
7227:
7222:
7218:
7216:0-387-20860-7
7212:
7208:
7204:
7200:
7196:
7191:
7186:
7182:
7178:
7177:
7169:
7165:
7161:
7157:
7153:
7149:
7145:
7141:
7137:
7133:
7129:
7128:
7120:
7116:
7112:
7107:
7102:
7098:
7094:
7087:
7083:
7082:Granville, A.
7079:
7075:
7069:
7065:
7061:
7060:
7055:
7051:
7047:
7043:
7039:
7037:0-521-80799-9
7033:
7029:
7026:. Cambridge:
7025:
7021:
7017:
7013:
7009:
7005:
7001:
6997:
6993:
6989:
6988:
6987:Math Horizons
6983:
6979:
6975:
6973:0-387-94609-8
6969:
6965:
6961:
6957:
6956:Frey, Gerhard
6953:
6948:
6943:
6940:(7): 99â109.
6939:
6935:
6931:
6927:
6926:Elkies, N. D.
6923:
6919:
6915:
6911:
6907:
6903:
6899:
6895:
6890:
6886:
6884:3-7643-6259-6
6880:
6876:
6871:
6870:
6869:Number Theory
6864:
6859:
6855:
6851:
6847:
6843:
6839:
6835:
6831:
6827:
6823:
6819:
6815:
6811:
6807:
6803:
6797:
6793:
6789:
6785:
6781:
6776:
6772:
6768:
6763:
6758:
6754:
6750:
6746:
6744:
6738:
6734:
6730:
6726:
6722:
6720:3-11-015364-5
6716:
6712:
6708:
6704:
6700:
6699:
6679:
6675:
6669:
6651:
6647:
6641:
6626:
6622:
6615:
6612:(July 2018).
6611:
6605:
6598:
6584:
6580:
6574:
6558:
6552:
6544:
6543:
6538:
6534:
6528:
6520:
6516:
6511:
6506:
6502:
6498:
6494:
6490:
6486:
6479:
6464:
6460:
6459:Conrad, Brian
6454:
6438:
6434:
6430:
6426:
6425:Fesenko, Ivan
6420:
6413:
6409:
6406:
6397:September 23,
6389:
6382:
6378:
6374:
6368:
6366:
6357:
6356:
6355:New Scientist
6351:
6344:
6329:
6325:
6319:
6311:
6307:
6303:
6299:
6295:
6291:
6284:
6276:
6272:
6268:
6261:
6254:
6249:
6242:. 2010-11-07.
6241:
6237:
6231:
6216:
6212:
6208:
6202:
6186:
6182:
6178:
6172:
6165:
6162:
6158:
6153:
6146:
6142:
6138:
6134:
6130:
6126:
6123:(1): 99â126,
6122:
6118:
6117:
6109:
6102:
6097:
6091:, p. 10)
6090:
6089:Browkin (2000
6085:
6078:
6074:
6073:Mollin (2009)
6069:
6062:
6057:
6050:
6045:
6038:
6033:
6025:
6021:
6014:
6007:
6000:
5995:
5988:
5983:
5976:
5971:
5964:
5959:
5952:
5951:Elkies (1991)
5947:
5940:
5935:
5928:
5923:
5921:
5919:
5911:
5910:
5903:
5895:
5891:
5887:
5883:
5879:
5875:
5871:
5867:
5864:(7802): 177.
5863:
5859:
5852:
5850:
5848:
5831:
5826:
5822:
5818:
5811:
5809:
5799:
5794:
5790:
5786:
5782:
5775:
5768:
5767:Goldfeld 1996
5763:
5756:
5751:
5744:
5743:Oesterlé 1988
5739:
5737:
5735:
5730:
5717:
5713:
5709:
5705:
5701:
5697:
5693:
5689:
5685:
5681:
5675:
5671:
5661:
5658:
5657:
5651:
5649:
5648:
5643:
5639:
5638:Kiran Kedlaya
5635:
5631:
5626:
5623:
5619:
5615:
5614:Peter Scholze
5610:
5608:
5604:
5600:
5596:
5592:
5587:
5585:
5584:Lucien Szpiro
5576:
5574:
5570:
5566:
5562:
5558:
5541:
5536:
5529:
5526:
5523:
5520:
5517:
5511:
5507:
5501:
5495:
5492:
5489:
5486:
5483:
5480:
5475:
5472:
5469:
5466:
5463:
5460:
5457:
5451:
5448:
5444:
5436:
5433:
5430:
5427:
5424:
5419:
5416:
5413:
5410:
5403:
5399:
5395:
5392:
5389:
5386:
5383:
5375:
5370:
5351:
5346:
5339:
5336:
5333:
5330:
5327:
5321:
5317:
5311:
5305:
5302:
5299:
5296:
5293:
5290:
5285:
5282:
5279:
5276:
5273:
5270:
5267:
5261:
5258:
5254:
5246:
5243:
5240:
5237:
5234:
5229:
5226:
5223:
5220:
5213:
5209:
5205:
5202:
5199:
5196:
5193:
5185:
5180:
5176:
5172:
5168:
5164:
5160:
5158:
5154:
5150:
5132:
5119:
5116:
5113:
5101:
5098:
5095:
5089:
5086:
5076:
5068:
5066:
5062:
5044:
5038:
5035:
5032:
5026:
5023:
5015:
5012:
5009:
4999:
4990:
4988:
4984:
4980:
4976:
4975:conjecture".
4974:
4970:
4954:
4950:
4946:
4938:
4917:
4914:
4907:
4885:
4882:
4879:
4873:
4870:
4860:
4857:
4841:
4838:
4835:
4829:
4826:
4823:
4820:
4817:
4809:
4807:
4803:
4798:
4785:
4769:
4766:
4763:
4757:
4754:
4744:
4741:
4737:
4732:
4729:
4709:
4706:
4703:
4681:
4678:
4675:
4660:
4657:
4654:
4648:
4645:
4640:
4637:
4633:
4617:
4613:
4611:
4607:
4603:
4599:
4592:
4588:
4584:
4582:
4578:
4574:
4569:
4567:
4563:
4553:
4551:
4548:) is defined
4547:
4543:
4539:
4535:
4531:
4527:
4523:
4520:
4510:
4507:
4504:
4501:
4498:
4494:
4490:
4487:
4484:
4481:
4478:
4474:
4470:
4467:
4464:
4461:
4458:
4454:
4450:
4447:
4444:
4441:
4438:
4434:
4431:Eric Reyssat
4430:
4427:
4424:
4421:
4418:
4414:
4408:
4403:
4398:
4393:
4386:
4376:
4374:
4364:
4361:
4358:
4355:
4352:
4349:
4346:
4342:
4338:
4335:
4332:
4329:
4326:
4323:
4320:
4316:
4312:
4309:
4306:
4303:
4300:
4297:
4294:
4290:
4286:
4283:
4280:
4277:
4274:
4271:
4268:
4264:
4260:
4257:
4254:
4251:
4248:
4245:
4242:
4238:
4234:
4231:
4228:
4225:
4222:
4219:
4216:
4212:
4208:
4205:
4202:
4199:
4196:
4193:
4190:
4186:
4182:
4179:
4176:
4173:
4170:
4167:
4164:
4160:
4156:
4153:
4150:
4147:
4144:
4141:
4138:
4134:
4130:
4127:
4124:
4121:
4118:
4115:
4112:
4108:
4104:
4101:
4098:
4095:
4092:
4089:
4086:
4082:
4078:
4075:
4072:
4069:
4066:
4063:
4060:
4056:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4030:
4026:
4023:
4020:
4017:
4014:
4011:
4008:
4004:
4000:
3997:
3994:
3991:
3988:
3985:
3982:
3978:
3974:
3971:
3968:
3965:
3962:
3959:
3956:
3952:
3948:
3945:
3942:
3939:
3936:
3933:
3930:
3926:
3922:
3916:
3910:
3904:
3898:
3892:
3886:
3881:
3876:
3872:
3866:
3864:
3860:
3856:
3852:
3848:
3844:
3840:
3836:
3832:
3822:
3820:
3816:
3812:
3808:
3788:
3784:
3781:
3778:
3775:
3772:
3768:
3762:
3759:
3756:
3751:
3747:
3742:
3739:
3733:
3730:
3727:
3721:
3718:
3715:
3712:
3704:
3702:
3698:
3694:
3690:
3686:
3682:
3678:
3674:
3669:
3667:
3663:
3659:
3655:
3651:
3647:
3640:
3636:
3632:
3628:
3624:
3617:
3610:
3603:
3585:
3579:
3574:
3567:
3564:
3561:
3555:
3552:
3546:
3543:
3539:
3531:
3528:
3519:
3516:
3513:
3507:
3504:
3499:
3495:
3490:
3485:
3482:
3479:
3476:
3466:
3448:
3442:
3439:
3434:
3431:
3421:
3418:
3415:
3409:
3406:
3401:
3397:
3392:
3387:
3384:
3381:
3378:
3368:
3350:
3344:
3336:
3333:
3330:
3324:
3321:
3316:
3312:
3307:
3302:
3299:
3296:
3293:
3285:
3283:
3279:
3275:
3274:bounded above
3271:
3267:
3254:
3250:
3247:
3243:
3240:
3236:
3233:
3230:
3226:
3222:
3218:
3215:> 2, then
3214:
3210:
3206:
3202:
3198:
3194:
3190:
3186:
3182:
3178:
3174:
3170:
3166:
3162:
3147:
3143:
3133:
3129:
3125:
3122:
3118:
3110:
3102:
3095:
3089:
3086:
3082:
3076:
3072:
3066:
3057:
3053:
3048:
3044:
3033:
3029:
3025:
3021:
3017:
3013:
3009:
3005:
3001:
2997:
2994:
2990:
2989:
2985:
2981:
2975:
2971:
2968:
2965:
2961:
2957:
2956:
2952:
2949:
2945:
2941:
2937:
2933:
2929:
2925:
2921:
2917:
2913:
2909:
2905:
2901:
2897:
2893:
2890:
2886:
2882:
2878:
2874:
2870:
2866:
2862:
2858:
2855:
2851:
2847:
2844:
2840:
2836:
2832:
2829:
2825:
2822:
2818:
2817:number fields
2814:
2810:
2806:
2802:
2795:
2791:
2788:
2786:
2781:
2778:
2774:
2770:
2755:
2752:
2749:
2740:
2736:
2732:
2716:
2713:
2710:
2702:
2698:
2695:
2692:
2688:
2685:
2681:
2677:
2674:
2670:
2667:
2663:
2660:
2659:Gerd Faltings
2656:
2652:
2649:
2645:
2641:
2638:
2637:
2636:
2634:
2629:
2610:
2593:
2577:
2571:
2567:
2565:
2561:
2557:
2539:
2536:
2530:
2527:
2521:
2515:
2512:
2509:
2503:
2500:
2492:
2489:
2487:
2483:
2479:
2475:
2471:
2467:
2463:
2441:
2435:
2429:
2426:
2421:
2417:
2413:
2411:
2400:
2393:
2389:
2386:
2378:
2375:
2372:
2366:
2362:
2357:
2353:
2351:
2343:
2340:
2335:
2330:
2322:
2319:
2316:
2310:
2306:
2302:
2297:
2295:
2287:
2284:
2279:
2273:
2270:
2267:
2261:
2257:
2253:
2251:
2246:
2234:
2232:
2228:
2210:
2207:
2204:
2200:
2195:
2189:
2186:
2183:
2177:
2173:
2169:
2166:
2162:
2159:
2156:
2151:
2145:
2142:
2139:
2133:
2129:
2125:
2122:
2118:
2115:
2112:
2109:
2101:
2099:
2095:
2091:
2087:
2065:
2062:
2056:
2053:
2047:
2045:
2037:
2031:
2028:
2022:
2020:
2009:
2006:
2000:
1997:
1994:
1991:
1988:
1986:
1977:
1970:
1967:
1961:
1958:
1954:
1950:
1947:
1944:
1941:
1939:
1930:
1926:
1923:
1918:
1915:
1911:
1906:
1902:
1899:
1896:
1893:
1891:
1882:
1877:
1874:
1870:
1866:
1862:
1859:
1855:
1851:
1848:
1843:
1840:
1836:
1831:
1827:
1824:
1818:
1812:
1809:
1806:
1804:
1793:
1787:
1784:
1778:
1772:
1769:
1763:
1757:
1754:
1751:
1749:
1741:
1738:
1735:
1729:
1726:
1714:
1698:
1692:
1686:
1683:
1680:
1677:
1674:
1668:
1659:
1656:
1653:
1647:
1644:
1641:
1636:
1632:
1628:
1625:
1622:
1617:
1614:
1610:
1606:
1603:
1595:
1593:
1575:
1572:
1569:
1565:
1560:
1557:
1553:
1549:
1546:
1542:
1539:
1536:
1531:
1528:
1524:
1520:
1517:
1513:
1510:
1507:
1504:
1496:
1494:
1490:
1486:
1482:
1478:
1474:
1464:
1462:
1458:
1454:
1450:
1446:
1442:
1438:
1434:
1430:
1427:> 1.01 or
1426:
1422:
1418:
1414:
1410:
1406:
1402:
1398:
1394:
1390:
1386:
1379:
1375:
1371:
1367:
1363:
1359:
1355:
1351:
1347:
1343:
1339:
1335:
1330:
1328:
1327:prime numbers
1324:
1320:
1316:
1312:
1308:
1304:
1300:
1296:
1292:
1288:
1284:
1280:
1276:
1267:
1264:
1263:
1261:
1257:
1256:For example:
1241:
1225:
1222:
1219:
1199:
1196:
1188:
1182:
1179:
1173:
1167:
1164:
1161:
1158:
1155:
1149:
1141:
1139:
1135:
1131:
1127:
1123:
1119:
1115:
1112:
1105:
1101:
1097:
1093:
1089:
1085:
1081:
1077:
1072:
1070:
1051:
1046:
1043:
1040:
1032:
1029:
1026:
1020:
1017:
1014:
1009:
1005:
1001:
998:
989:
985:
981:
977:
973:
969:
965:
958:
953:
936:
931:
928:
925:
917:
914:
911:
905:
902:
899:
896:
887:
883:
879:
875:
871:
867:
863:
860:
855:
853:
849:
833:
830:
827:
816:
813:
804:
800:
796:
792:
788:
784:
780:
775:
774:
761:
758:
755:
752:
749:
746:
738:
734:
730:
725:
721:
709:
703:
686:
685:
672:
669:
666:
663:
660:
657:
649:
645:
641:
638:
627:
621:
604:
591:
588:
582:
566:
553:
550:
544:
533:
525:
521:
509:
503:
487:
485:
472:
463:
462:prime factors
459:
443:
426:
413:
404:
391:
383:
379:
369:
367:
363:
358:
356:
352:
348:
344:
340:
335:
333:
329:
325:
321:
317:
304:
295:
282:
279:
276:
267:
266:prime factors
263:
250:
247:
244:
241:
238:
229:
225:
212:
203:
190:
187:
184:
176:
172:
168:
164:
163:number theory
160:
156:
152:
150:
142:
137:
131:
126:
115:
112:
110:
107:
105:
102:
100:
97:
95:
92:
90:
87:
85:
82:
81:
79:
75:
72:
69:
67:Equivalent to
65:
61:
57:
51:
48:
46:
43:
42:
40:
36:
33:
32:Number theory
30:
26:
22:
7835:MathOverflow
7785:
7724:
7690:
7684:
7656:(1): 91â95.
7653:
7647:
7641:
7622:
7616:
7612:
7580:
7574:
7570:
7538:
7534:
7531:Tijdeman, R.
7508:
7504:
7475:
7469:
7427:
7421:
7418:-conjecture"
7415:
7401:
7374:
7361:(1â2): 3â24.
7358:
7354:
7350:
7323:
7312:. Retrieved
7305:the original
7284:
7280:
7264:
7252:
7248:
7244:
7225:
7202:
7180:
7174:
7131:
7125:
7096:
7092:
7058:
7023:
6991:
6985:
6963:
6937:
6933:
6909:
6905:
6901:
6897:
6893:
6868:
6862:
6829:
6825:
6821:
6787:
6752:
6748:
6745:-conjecture"
6742:
6710:
6706:
6681:. Retrieved
6668:
6656:. Retrieved
6640:
6628:. Retrieved
6604:
6596:
6589:. Retrieved
6573:
6560:. Retrieved
6551:
6540:
6527:
6492:
6488:
6478:
6466:. Retrieved
6453:
6441:. Retrieved
6436:
6432:
6419:
6395:. Retrieved
6388:the original
6353:
6343:
6331:. Retrieved
6318:
6293:
6289:
6283:
6274:
6260:
6248:
6239:
6230:
6219:, retrieved
6215:the original
6210:
6201:
6189:, retrieved
6185:the original
6180:
6171:
6161:math/0408168
6152:
6120:
6114:
6108:
6096:
6084:
6077:Mollin (2010
6068:
6056:
6044:
6032:
6023:
6019:
6006:
5999:Nitaj (1996)
5994:
5982:
5970:
5958:
5946:
5934:
5908:
5902:
5861:
5857:
5834:. Retrieved
5820:
5788:
5784:
5774:
5762:
5750:
5715:
5711:
5707:
5699:
5695:
5691:
5687:
5683:
5679:
5674:
5646:
5633:
5627:
5611:
5606:
5602:
5594:
5588:
5582:
5572:
5568:
5565:n conjecture
5559:
5556:
5368:
5366:
5178:
5174:
5170:
5161:
5156:
5152:
5148:
5146:
5064:
5060:
5058:
4989:of the form
4986:
4979:Baker (1998)
4977:
4972:
4968:
4936:
4934:
4805:
4802:Baker (2004)
4799:
4722:occurs when
4614:
4609:
4605:
4601:
4597:
4595:
4590:
4586:
4580:
4576:
4573:Baker (1998)
4570:
4561:
4559:
4545:
4541:
4537:
4533:
4529:
4525:
4521:
4518:
4516:
4406:
4401:
4396:
4391:
4370:
4344:
4318:
4292:
4266:
4240:
4214:
4188:
4162:
4136:
4110:
4084:
4058:
4032:
4006:
3980:
3954:
3928:
3920:
3914:
3908:
3902:
3896:
3890:
3884:
3879:
3870:
3862:
3858:
3854:
3850:
3846:
3842:
3828:
3814:
3810:
3806:
3804:
3700:
3696:
3692:
3688:
3684:
3680:
3672:
3670:
3665:
3661:
3657:
3653:
3645:
3638:
3634:
3630:
3626:
3615:
3608:
3606:
3277:
3269:
3265:
3263:
3228:
3224:
3220:
3216:
3212:
3208:
3204:
3200:
3196:
3192:
3188:
3184:
3180:
3176:
3172:
3168:
3031:
3027:
3023:
3019:
3015:
3011:
3007:
3003:
2999:
2992:
2987:
2983:
2979:
2973:
2963:
2947:
2943:
2939:
2935:
2931:
2927:
2923:
2919:
2915:
2911:
2907:
2903:
2899:
2895:
2888:
2884:
2880:
2876:
2872:
2868:
2864:
2854:simple zeros
2849:
2845:
2834:
2830:
2820:
2812:
2797:
2793:
2789:
2784:
2734:
2730:
2683:
2627:
2625:
2608:
2591:
2575:
2569:
2559:
2553:
2490:
2485:
2481:
2477:
2473:
2469:
2461:
2459:
2230:
2226:
2224:
2097:
2093:
2089:
2085:
2083:
1712:
1591:
1590:The integer
1589:
1492:
1488:
1484:
1480:
1476:
1472:
1470:
1460:
1456:
1452:
1448:
1444:
1440:
1436:
1432:
1428:
1424:
1420:
1416:
1412:
1408:
1404:
1400:
1396:
1392:
1388:
1384:
1382:
1377:
1373:
1369:
1365:
1361:
1357:
1353:
1349:
1345:
1341:
1337:
1333:
1322:
1318:
1314:
1310:
1306:
1302:
1298:
1294:
1290:
1286:
1282:
1278:
1274:
1272:
1265:
1259:
1255:
1137:
1133:
1129:
1125:
1121:
1117:
1113:
1110:
1108:
1103:
1099:
1095:
1091:
1087:
1083:
1079:
1075:
1066:
987:
983:
979:
975:
971:
967:
960:
956:
951:
888:, such that
885:
881:
877:
873:
869:
865:
861:
851:
806:
802:
798:
794:
786:
782:
778:
776:
688:
687:
606:
605:
567:
488:
465:
428:
406:
384:
375:
372:Formulations
361:
359:
354:
350:
338:
336:
327:
319:
297:
269:
231:
230:and satisfy
205:
177:
171:David Masser
154:
148:
147:
145:
141:David Masser
77:Consequences
50:David Masser
20:
7870:Conjectures
7826:Barry Mazur
7771:Easy as ABC
7644:Conjecture"
7066:â362, 681.
6912:: 321â324.
6737:Baker, Alan
6703:Baker, Alan
6591:February 1,
6377:Stix, Jakob
6267:Woit, Peter
5941:, p. .
5755:Masser 1985
3837:project, a
3817:= 6.068 by
3282:exponential
3063:prime
2998:There are ~
2809:Siegel zero
1376:) > 1 +
859:real number
7864:Categories
7609:Yu, Kunrui
7567:Yu, Kunrui
7446:0654.10019
7375:Astérisque
7342:1200.11002
7314:2013-06-14
7301:1241.11034
7205:. Berlin:
7046:1046.11035
6918:0876.11015
6826:Math. Comp
6810:1130.11034
6729:0973.11047
6658:October 2,
6630:October 2,
6562:October 2,
6443:30 October
6405:May report
6191:October 3,
5726:References
5618:Jakob Stix
5374:such that
5184:such that
4517:Note: the
4350:14,482,065
3899:> 1.05
3018:for which
2938:) exceeds
2828:polynomial
2578:= 3·109 =
1407:such that
1360:such that
1297:will have
427:, denoted
159:conjecture
151:conjecture
23:conjecture
7787:MathWorld
7597:123894587
7555:123621917
7492:123460044
7383:0303-1179
7293:0972-0871
7185:CiteSeerX
6771:253834357
6625:174791744
6468:March 18,
6433:Inference
6429:"Fukugen"
6333:March 17,
6221:April 30,
5894:214786566
5527:
5521:
5493:
5487:
5473:
5467:
5461:
5434:
5428:
5417:
5396:
5337:
5331:
5303:
5297:
5283:
5277:
5271:
5244:
5238:
5227:
5206:
5108:Θ
5090:
5027:
5004:Ω
4915:ω
4908:ω
4874:
4861:
4830:
4824:κ
4758:
4745:
4738:ω
4730:ε
4704:ε
4682:ε
4649:
4641:ω
4638:−
4634:ε
4353:2,352,105
4327:1,396,909
4324:7,801,334
4298:4,119,410
4272:2,131,671
4246:1,075,319
3923:> 1.4
3917:> 1.3
3911:> 1.2
3905:> 1.1
3853:with rad(
3782:
3776:
3760:
3743:
3722:
3623:constants
3556:
3547:
3508:
3486:
3443:ε
3410:
3388:
3325:
3303:
3096:ω
3090:−
3058:∏
2807:, has no
2787:-function
2753:≥
2714:≥
2504:
2436:⋯
2427:⋅
2401:⋯
2387:−
2376:−
2341:−
2320:−
2285:−
2271:−
2187:−
2157:−
2143:−
2001:⋅
1995:⋅
1989:⩽
1962:⋅
1951:
1924:−
1903:
1863:
1849:−
1828:
1813:
1788:
1773:
1758:
1730:
1693:⋯
1687:⋅
1681:⋅
1669:⋯
1657:−
1642:−
1623:−
1537:−
1491:> rad(
1301:< rad(
1200:
1183:
1047:ε
1021:
1015:⋅
1010:ε
932:ε
906:
753:⋅
731:⋅
664:⋅
642:⋅
7765:ABC@Home
7758:ABC@home
7461:(2014).
7412:(1988).
7369:(1988),
7201:(2004).
7084:(1998).
7008:25678079
6958:(1997).
6928:(1991).
6739:(2004).
6683:July 13,
6519:26450038
6408:Archived
6211:ABC@Home
5886:32246118
5836:19 March
5702:implies
5654:See also
5620:visited
5147:where Î(
5059:where Ω(
4373:ABC@Home
4347:< 10
4321:< 10
4295:< 10
4269:< 10
4243:< 10
4217:< 10
4191:< 10
4165:< 10
4139:< 10
4113:< 10
4087:< 10
4061:< 10
4035:< 10
4009:< 10
3983:< 10
3957:< 10
3931:< 10
3835:ABC@Home
3805:for all
3668:> 2.
2843:integers
2841:for all
1305:), i.e.
7709:1924103
7672:1755155
7391:0992208
7156:6901166
7136:Bibcode
7022:(ed.).
6854:2153551
6834:Bibcode
6695:Sources
6497:Bibcode
6310:3135393
6145:7805117
6137:3592123
5866:Bibcode
4519:quality
4465:7·29·31
4462:19·1307
4356:449,194
4330:290,965
4304:184,727
4301:812,499
4278:115,041
4275:463,446
4249:258,168
4223:139,762
4220:528,275
4194:252,856
4168:116,978
3893:> 1
3873:> 1
3857:) <
3648:(in an
3272:can be
2942:· max{|
2739:lim sup
2686:> 1.
2594:= 23 =
1111:quality
791:coprime
704:1000000
157:) is a
7739:
7707:
7670:
7595:
7553:
7490:
7444:
7389:
7381:
7340:
7330:
7299:
7291:
7232:
7213:
7187:
7154:
7070:
7044:
7034:
7006:
6970:
6916:
6881:
6877:â106.
6852:
6808:
6798:
6769:
6727:
6717:
6623:
6517:
6489:Nature
6308:
6143:
6135:
5892:
5884:
5858:Nature
5821:Nature
5647:Nature
5173:= rad(
5169:. Let
5067:, and
4596:where
4499:1.5679
4488:2·3·17
4479:1.5808
4459:1.6235
4439:1.6260
4419:1.6299
4359:24,013
4333:17,890
4307:13,118
4252:70,047
4226:41,438
4200:23,773
4197:73,714
4174:13,266
4171:37,612
4145:18,233
4142:51,677
4116:22,316
3637:, and
3467:), and
3239:height
3223:, and
3187:, and
1098:, rad(
850:. The
785:, and
345:about
7721:(PDF)
7593:S2CID
7551:S2CID
7488:S2CID
7466:(PDF)
7308:(PDF)
7277:(PDF)
7171:(PDF)
7152:S2CID
7122:(PDF)
7089:(PDF)
7004:JSTOR
6850:JSTOR
6767:S2CID
6653:(PDF)
6621:S2CID
6617:(PDF)
6586:(PDF)
6391:(PDF)
6384:(PDF)
6306:S2CID
6157:arXiv
6141:S2CID
6016:(PDF)
5890:S2CID
5678:When
5666:Notes
5622:Kyoto
4935:with
4696:over
4583:) by
4550:above
4468:2·3·5
4445:3·5·7
4425:3·109
4388:Rank
4362:1,843
4336:1,530
4310:1,232
4281:9,497
4255:6,665
4229:4,519
4203:3,028
4177:1,947
4151:1,159
4148:7,035
4122:3,693
4119:8,742
4096:1,801
4093:3,869
4090:8,987
4067:1,669
4064:3,499
4038:1,268
3660:, or
3633:, or
1487:with
28:Field
7737:ISBN
7379:ISSN
7328:ISBN
7289:ISSN
7230:ISBN
7211:ISBN
7097:1998
7068:ISBN
7032:ISBN
6968:ISBN
6938:1991
6896:! +
6879:ISBN
6796:ISBN
6715:ISBN
6685:2021
6660:2018
6632:2018
6593:2019
6564:2018
6515:PMID
6470:2018
6445:2021
6399:2018
6335:2018
6223:2014
6193:2012
5882:PMID
5838:2018
5640:and
5616:and
5387:>
5197:<
4821:<
4707:>
4608:and
4589:rad(
4560:The
4485:5·13
4448:2·23
4365:160
4339:143
4313:126
4287:112
3716:>
3703:and
3621:are
3614:and
3480:<
3382:<
3297:<
3264:The
3203:and
3163:The
2982:! +
2946:|, |
2782:The
2771:The
2699:has
2671:The
2653:The
2626:The
2611:) =
2607:rad(
2572:= 2,
2522:<
2208:>
2048:<
1573:>
1002:<
900:>
817:<
789:are
337:The
204:and
169:and
146:The
62:1985
7833:on
7824:by
7815:by
7729:doi
7695:doi
7658:doi
7642:abc
7627:doi
7623:108
7613:abc
7585:doi
7581:291
7571:abc
7543:doi
7539:102
7513:doi
7480:doi
7442:Zbl
7432:doi
7416:abc
7353:".
7351:abc
7338:Zbl
7297:Zbl
7253:317
7247:".
7245:abc
7144:doi
7132:139
7101:doi
7064:361
7042:Zbl
6996:doi
6942:doi
6914:Zbl
6904:".
6863:abc
6842:doi
6822:abc
6806:Zbl
6757:doi
6743:abc
6725:Zbl
6707:abc
6505:doi
6493:526
6439:(3)
6298:doi
6125:doi
6121:182
5874:doi
5862:580
5825:doi
5793:doi
5706:of
5607:abc
5603:abc
5595:abc
5569:abc
5524:log
5518:log
5490:log
5484:log
5470:log
5464:log
5458:log
5431:log
5425:log
5414:log
5393:exp
5334:log
5328:log
5300:log
5294:log
5280:log
5274:log
5268:log
5241:log
5235:log
5224:log
5203:exp
5175:abc
5087:rad
5024:rad
4973:abc
4871:rad
4858:log
4827:rad
4806:abc
4755:rad
4742:log
4646:rad
4591:abc
4581:abc
4577:abc
4562:abc
4508:5·7
4505:2·3
4482:283
4284:998
4261:98
4258:769
4235:84
4232:599
4209:74
4206:455
4183:64
4180:327
4157:51
4154:218
4131:34
4128:144
4125:706
4105:25
4099:384
4079:17
4073:210
4070:856
4053:11
4047:102
4044:379
4041:667
4018:152
4015:240
4012:418
3986:120
3863:abc
3855:abc
3779:log
3773:log
3757:log
3740:exp
3719:rad
3673:abc
3553:rad
3544:log
3505:rad
3483:exp
3407:rad
3385:exp
3322:rad
3300:exp
3278:abc
3266:abc
3229:abc
2974:abc
2964:abc
2950:|}.
2848:if
2821:abc
2813:abc
2735:abc
2731:abc
2646:of
2642:on
2628:abc
2615:042
2609:abc
2601:343
2598:436
2585:341
2582:436
2501:rad
1948:rad
1900:rad
1860:rad
1825:rad
1810:rad
1785:rad
1770:rad
1755:rad
1727:rad
1493:abc
1463:).
1303:abc
1212:rad
1197:log
1180:log
1100:abc
1071:):
1018:rad
903:rad
821:rad
777:If
714:rad
697:rad
632:rad
615:rad
576:rad
538:rad
514:rad
497:rad
464:of
437:rad
362:abc
355:abc
351:abc
339:abc
334:".
328:abc
320:abc
268:of
161:in
149:abc
21:abc
7866::
7784:.
7735:.
7723:.
7705:MR
7703:.
7691:95
7689:.
7683:.
7668:MR
7666:.
7654:82
7652:.
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