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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 781.11008

**Autor: ** Erdös, Paul; Lacampagne, C.B.; Selfridge, J.L.

**Title: ** Estimates of the least prime factor of a binomial coefficient. (In English)

**Source: ** Math. Comput. 61, No.203, 215-224 (1993).

**Review: ** We estimate the least prime factor p of the binomial coefficient {N\choose k} 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 \binom{N}{k} is good. The bound g(k) > ck^{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 {N\choose k} with positive deficiency and k \leq 101 are listed.

**Classif.: ** * 11B65 Binomial coefficients, etc.

11N37 Asymptotic results on arithmetic functions

**Keywords: ** least prime factor; binomial coefficient; deficiency of a good binomial coefficient; positive deficiency

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