Estimates of the least prime factor of a binomial coefficient

P. Erdős, C. B. Lacampagne, J. L. Selfridge

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5 Citations (Scopus)


We estimate the least prime factor p of the binomial coefficient (£) for k ≥ 2. The conjecture that p ≤ max(N/k, 29) is supported by considerable numerical evidence. Call a binomial coefficient good if p > k. For 1≤ i ≤ k write N - k + i =aibiwhere b¡ contains just those prime factors > k, and define the deficiency of a good binomial coefficient as the number of i for which bi= 1. Let g(k) be the least integer N≥ k + 1 such that (£) is good. The bound g(k) ≥ ck2/lnk 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 (n k) with positive deficiency and k ≤ 101 are listed.

Original languageEnglish
Pages (from-to)215-224
Number of pages10
JournalMathematics of Computation
Issue number203
Publication statusPublished - 1993

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

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