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**Zentralblatt MATH**

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

**Zbl.No: ** 111.04701

**Autor: ** Erdös, Pál

**Title: ** On a problem of Sierpinski (In English)

**Source: ** Atti Accad. Naz. Lincei, Rend., Cl. Sci. Fis. Mat. Nat., VIII. Ser. 23, 122-124 (1962).

**Review: ** Let n be a positive integer. Denote by s^{(k)}(n) the sum of the digits of n written in the k-ary system. Let 2 = p_{1} < p_{2} < ··· be the sequence of consecutive primes. In a recent paper, Sierpinski proved that for every k **limsup**_{n = oo} s^{(k)} (p_{n}) = oo, which immediately implies that for infinitely many n s^{(k)} (p_{n+1}) > s^{(k)} (p_{n}). The question with the opposite inequality remained open.

The author settles the question in this note by proving the Theorem: For every k there are infinitely many h for which s^{(k)} (p_{n}) > s^{(k)} (p_{n+1}). The author discusses related unsolved problems.

**Reviewer: ** W.E.Briggs

**Classif.: ** * 11A41 Elemementary prime number theory

11A63 Radix representation

**Index Words: ** number theory

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