On Simultaneous Digital Expansions of Polynomial Values

Abstract

Abstract. Let sq denote the q-ary sum-of-digits function and let P1(X), P2(X) ∈ Z[X] with P1(N), P2(N) ⊂ N be polynomials of degree h, l ≥ 1, h = l, respectively. In this note we show that (sq(P1(n))/sq(P2(n))) n≥1 is dense in R + . This extends work by Stolarsky (1978) and Hare, Laishram and Stoll (2011).

On a second conjecture of Stolarsky: the sum of digits of polynomial values

“…Stolarsky also showed that the sequence s 2 (n r )/s 2 (n) is unbounded as n → ∞ (this is also true for sq(P (n)) sq(n) , see [6]). In very recent work, the authors [9] show that the ratio…”

Products of integers with few nonzero digits

“…For example, they showed that s(a 2 ) = s(a) = 8 only allows finitely many odd solutions a, whereas s(a 2 ) = s(a) = 12 has infinitely many odd solutions a. Note also, that due to a classical result of Stolarsky [18], we have lim inf a→∞ s(a 2 )/s(a) = 0, so that it may not be too rare to have integers a such that a 2 has a much lower number of nonzero bits than the integer a itself (the results hold true also for higher powers and in a more general context, see [9,11,[14][15][16]). Before going any further, let us first mention several related results for the nonzero bits of powers of integers from the literature that we will translate into our language.…”

Quotients of Palindromic and Antipalindromic Numbers

“…A related conjecture was made by Selfridge and Lacampagne [14, §7]. If we let B = {1, 2, 4, 5,7,11,13,14,16,20,22, . .…”

“…Let s q (n) be the sum of the base-q digits of n. Madritsch and Stoll [16] showed that if P 1 and P 2 are polynomials with integer coefficients, of distinct degrees, such that P 1 (N), P 2 (N) ⊆ N, then the sequence of quotients (s q (P 1 (n))/s q (P 2 (n))) n≥1 is dense in R + .…”

“…By considering the numerator and denominator modulo 3 i , it is easy to see that if N ∈ D, then N ∈ V . Let 5,6,14,15,16,17,18,41,42,43,44,45,46, 47, . .…”