Estimates of the least prime factor of a binomial coefficient
Authors:
P. Erdős, C. B. Lacampagne and J. L. Selfridge
Journal:
Math. Comp. 61 (1993), 215224
MSC:
Primary 11B65; Secondary 11N37
MathSciNet review:
1199990
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Abstract: We estimate the least prime factor p of the binomial coefficient for . The conjecture that is supported by considerable numerical evidence. Call a binomial coefficient good if . For write , where contains just those prime factors , and define the deficiency of a good binomial coefficient as the number of i for which . Let be the least integer such that is good. The bound is proved. We conjecture that our list of 17 binomial coefficients with deficiency is complete, and it seems that the number with deficiency 1 is finite. All with positive deficiency and are listed.
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 E. F. Ecklund, P. Erdős, and J. L. Selfridge, A new function associated with the prime factors of , Math. Comp. 28 (1974), 647649. MR 0337732 (49:2501)
 [2]
 P. Erdős, C. B. Lacampagne, and J. L. Selfridge, Prime factors of binomial coefficients and related problems, Acta Arith. 49 (1988), 507523. MR 967334 (90f:11009)
 [3]
 A. E. Ingham, On the differences between consecutive primes, Quart. J. Math. Oxford 8 (1937), 255266.
 [4]
 L. J. Lander and T. R. Parkin, On first appearance of prime differences, Math. Comp. 21 (1967), 483488. MR 0230677 (37:6237)
 [5]
 R. Scheidler and H. C. Williams, A method of tabulating the numbertheoretic function , Math. Comp. 59 (1992), 251257. MR 1134737 (92k:11146)
 [6]
 J. L. Selfridge, Some problems on the prime factors of consecutive integers, Abstract 747109, Notices Amer. Math. Soc. 24 (1977), A456A457.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199311999906
PII:
S 00255718(1993)11999906
Article copyright:
© Copyright 1993
American Mathematical Society
