Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 669.10011
Autor: Erdös, Paul; Lacampagne, C.B.; Selfridge, J.L.
Title: Prime factors of binomial coefficients and related problems. (In English)
Source: Acta Arith. 49, No.5, 507-523 (1988).
Review: Consider a sequence of k positive integers {ai}ki = 1 with the following properties: (i) ai \leq k for i = 1,...,k; (ii) there is an n such that ai is the quotient when n+i is divided by all its prime factors greater than k (in other words: for each prime p \leq k, the pattern of that prime and its powers in the sequence a1,...,ak is the same as the pattern of that prime and its powers in some sequence of k consecutive integers).
Example: the sequence 1,4,3,2,5 satisfies (i) and (ii) since 91,92,93,94,95 factor into 1· 7· 13, 4· 23, 3· 31, 2· 47, 5· 19.
This long but captivating paper starts with a proof that any sequence which satisfies (i) and (ii) is a permutation of 1,2,...,k. However, only very few of the k! possible permutations can actually occur in such sequences. Six different types of solutions are characterized, and it is proved - the main result of the paper - that these indeed are the only possible patterns which can occur in sequences which obey (i) and (ii). From this, all the solutions for k \leq 27 are derived and listed.
Next, for given k, the number of possible sequences satisfying (i) and (ii) is studied. It is proved that the logarithmic density of those values of k with exactly two solutions is positive. Finally, these results are applied to the problem of estimating the least prime factor of binomial coefficients.
Reviewer: H.J.J.te Riele
Classif.: * 11A41 Elemementary prime number theory
11B39 Special numbers, etc.
05A10 Combinatorial functions
11A05 Multiplicative structure of the integers
Keywords: binomial coefficients; consecutive integers; logarithmic density; least prime factor of binomial coefficients
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