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Bonse's inequality

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356: 157: 241: 193: 397: 79: 416: 297: 426: 390: 210: 383: 371: 421: 168: 272:
Bonse, H. (1907). "Über eine bekannte Eigenschaft der Zahl 30 und ihre Verallgemeinerung".
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Inequality relating the primorial to square of the next prime number
152:{\displaystyle p_{n}\#=p_{1}\cdots p_{n}>p_{n+1}^{2}.} 207:
Mathematician Denis Hanson showed an upper bound where
213: 171: 82: 27:, named after H. Bonse, relates the size of a 235: 187: 151: 31:to the smallest prime that does not appear in its 408: 337: 391: 398: 384: 162:(the middle product is short-hand for the 310: 338:Uspensky, J. V.; Heaslet, M. A. (1939). 409: 290: 271: 350: 342:. New York: McGraw Hill. p. 87. 13: 217: 182: 93: 14: 438: 354: 274:Archiv der Mathematik und Physik 298:Canadian Mathematical Bulletin 293:"On the Product of the Primes" 284: 265: 1: 331: 236:{\displaystyle n\#\leq 3^{n}} 417:Theorems about prime numbers 370:. You can help Knowledge by 291:Hanson, Denis (March 1972). 7: 246: 10: 443: 349: 340:Elementary Number Theory 258: 188:{\displaystyle p_{n}\#} 366:-related article is a 312:10.4153/cmb-1972-007-7 237: 189: 153: 238: 190: 154: 73: β‰₯ 4, then 211: 169: 80: 35:. It states that if 427:Number theory stubs 145: 33:prime factorization 233: 185: 149: 125: 25:Bonse's inequality 379: 378: 61:are the smallest 42:, ...,  434: 400: 393: 386: 358: 351: 343: 325: 324: 314: 288: 282: 281: 269: 242: 240: 239: 234: 232: 231: 194: 192: 191: 186: 181: 180: 158: 156: 155: 150: 144: 139: 121: 120: 108: 107: 92: 91: 442: 441: 437: 436: 435: 433: 432: 431: 407: 406: 405: 404: 347: 334: 329: 328: 289: 285: 270: 266: 261: 253:Primorial prime 249: 227: 223: 212: 209: 208: 203: 176: 172: 170: 167: 166: 140: 129: 116: 112: 103: 99: 87: 83: 81: 78: 77: 65: + 1 60: 50: 41: 17: 12: 11: 5: 440: 430: 429: 424: 419: 403: 402: 395: 388: 380: 377: 376: 359: 345: 344: 333: 330: 327: 326: 283: 280:(12): 292–295. 263: 262: 260: 257: 256: 255: 248: 245: 230: 226: 222: 219: 216: 199: 184: 179: 175: 160: 159: 148: 143: 138: 135: 132: 128: 124: 119: 115: 111: 106: 102: 98: 95: 90: 86: 55: 46: 39: 15: 9: 6: 4: 3: 2: 439: 428: 425: 423: 420: 418: 415: 414: 412: 401: 396: 394: 389: 387: 382: 381: 375: 373: 369: 365: 364:number theory 360: 357: 353: 352: 348: 341: 336: 335: 322: 318: 313: 308: 304: 300: 299: 294: 287: 279: 275: 268: 264: 254: 251: 250: 244: 228: 224: 220: 214: 205: 202: 198: 177: 173: 165: 146: 141: 136: 133: 130: 126: 122: 117: 113: 109: 104: 100: 96: 88: 84: 76: 75: 74: 72: 68: 67:prime numbers 64: 58: 54: 49: 45: 38: 34: 30: 26: 22: 21:number theory 422:Inequalities 372:expanding it 361: 346: 339: 305:(1): 33–37. 302: 296: 286: 277: 273: 267: 206: 200: 196: 161: 70: 62: 56: 52: 47: 43: 36: 24: 18: 411:Categories 332:References 321:0008-4395 221:≤ 218:# 183:# 164:primorial 110:⋯ 94:# 29:primorial 247:See also 51:,  319:  362:This 259:Notes 368:stub 317:ISSN 123:> 69:and 307:doi 195:of 19:In 413:: 315:. 303:15 301:. 295:. 276:. 243:. 204:) 59:+1 23:, 399:e 392:t 385:v 374:. 323:. 309:: 278:3 229:n 225:3 215:n 201:n 197:p 178:n 174:p 147:. 142:2 137:1 134:+ 131:n 127:p 118:n 114:p 105:1 101:p 97:= 89:n 85:p 71:n 63:n 57:n 53:p 48:n 44:p 40:1 37:p

Index

number theory
primorial
prime factorization
prime numbers
primorial
Primorial prime
"On the Product of the Primes"
Canadian Mathematical Bulletin
doi
10.4153/cmb-1972-007-7
ISSN
0008-4395
Stub icon
number theory
stub
expanding it
v
t
e
Categories
Theorems about prime numbers
Inequalities
Number theory stubs

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