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On a problem in the elementary theory of numbers. (English) Zbl 0010.29401
The following two theorems are proved by elementary methods.
1. If \(a_1,...,a_n\) are different positive integers, and \(n \geq 3 \cdot 2^{k-1}\), then the numbers \(a_i+a_j(i,j=1,2,...,n)\) cannot all be composed only of \(k\) given primes.
2. If \(a_1<...<a_{k+1}\) are positive integers, and \(b > a^{k}_{k+1}\), then the numbers \(a_i+b\) \((i=1,2,...,k+1)\) cannot all be composed of only \(k\) given primes. On p.610, line 8 from below, read \(p^{\alpha_k}_k \text{ for } p^{\alpha_{k-1}}_{k-1}\) on p.611, line 7, read ”that each one” for ”that one”.

MSC:
11B83 Special sequences and polynomials
Keywords:
number theory
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