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 = p1 < p2 < ··· be the sequence of consecutive primes. In a recent paper, Sierpinski proved that for every k limsupn = oo s(k) (pn) = oo, which immediately implies that for infinitely many n s(k) (pn+1) > s(k) (pn). 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) (pn) > s(k) (pn+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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