Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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), 215-224
MSC: Primary 11B65; Secondary 11N37
MathSciNet review: 1199990
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We estimate the least prime factor p of the binomial coefficient $ \left( {_k^N} \right)$ for $ k \geq 2$. The conjecture that $ p \leq \max (N/k,29)$ is supported by considerable numerical evidence. Call a binomial coefficient good if $ p > k$. For $ 1 \leq i \leq k$ write $ N - k + i = {a_i}{b_i}$, where $ {b_i}$ contains just those prime factors $ > k$ , and define the deficiency of a good binomial coefficient as the number of i for which $ {b_i} = 1$. Let $ g(k)$ be the least integer $ N > k + 1$ such that $ \left( {_k^N} \right)$ is good. The bound $ g(k) > c{k^2}/\ln k$ is proved. We conjecture that our list of 17 binomial coefficients with deficiency $ > 1$ is complete, and it seems that the number with deficiency 1 is finite. All $ \left( {_k^N} \right)$ with positive deficiency and $ k \leq 101$ are listed.

References [Enhancements On Off] (What's this?)

  • [1] E. F. Ecklund, P. Erdős, and J. L. Selfridge, A new function associated with the prime factors of $ \left( {_k^n} \right)$, Math. Comp. 28 (1974), 647-649. 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), 507-523. MR 967334 (90f:11009)
  • [3] A. E. Ingham, On the differences between consecutive primes, Quart. J. Math. Oxford 8 (1937), 255-266.
  • [4] L. J. Lander and T. R. Parkin, On first appearance of prime differences, Math. Comp. 21 (1967), 483-488. MR 0230677 (37:6237)
  • [5] R. Scheidler and H. C. Williams, A method of tabulating the number-theoretic function $ g(k)$, Math. Comp. 59 (1992), 251-257. MR 1134737 (92k:11146)
  • [6] J. L. Selfridge, Some problems on the prime factors of consecutive integers, Abstract 747-10-9, Notices Amer. Math. Soc. 24 (1977), A456-A457.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11B65, 11N37

Retrieve articles in all journals with MSC: 11B65, 11N37

Additional Information

Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society