Thomas Stoll

Octahedrons with Equally Many Lattice Points

Denote by pn(r )t he number of points (x1 ,... , xn) ∈ n satisfying |x1| + ··· + |xn |≤ r. We prove that, for given distinct integers n, m ≥ 2, the equation pn(x )= pm(y) has at most finitely many solutions in integers x, y.

Diophantine equations involving general Meixner and Krawtchouk polynomials

While counting lattice points in octahedra of different dimensions n and m, it is an interesting question to ask, how many octahedra exist containing equally many such points. This gives rise to the Diophantine equation p n (x) = p m (y) in rational integers x, y, where {p k (x)} denote special Meixner polynomials {M (β,c) k (x)} with β = 1, c = −1. In this paper we join the algorithmic criterion of Bilu and Tichy [4] with a famous result of Erdös and Selfridge [6] and prove that the Diophantine equation M (β,c1) n (x) = M (β,c2) m (y) with m, n ≥ 3, β ∈ Z \ {0, −1, −2, − max(n, m) + 1} and c 1, c2 ∈ Q \ {0, 1} only admits a finite number of integral solutions x, y. This generalizes a result given by Bilu and the authors [3]. As an immediate consequence of the investigation an analogous result for general Krawtchouk polynomials {K (p,N) k(x)} is obtained.

Stolarsky’s conjecture and the sum of digits of polynomial values

Let s q (n) denote the sum of the digits in the q-ary expansion of an integer n. In 1978, Stolarsky showed that lim n→∞ inf s 2 (n 2 )/s 2 (n) = 0. He conjectured that, just as for n 2 , this limit infimum should be 0 for higher powers of n. We prove and generalize this conjecture showing that for any polynomial p(x) = a h x h + a h-1 x h-1 + ··· + a 0 ∈ ℤ[x] with h > 2 and a h > 0 and any base q, l im in f s q (p(n)) = 0. n→∞ s q (n) For any e > 0 we give a bound on the minimal n such that the ratio s q (p(n))/ s q (n) < e. Further, we give lower bounds for the number of n < N such that s q (p(n))/s q (n) < e.

THE SUM OF DIGITS OF n AND n 2

Let

Diophantine equations for Morgan-Voyce and other modified orthogonal polynomials

s_q(n)

Reconstruction Problems for Graphs, Krawtchouk Polynomials, and Diophantine Equations

denote the sum of the digits in the

Construction of Some Nonautomatic Sequences by Cellular Automata

q

On digital blocks of polynomial values and extractions in the Rudin–Shapiro sequence

-ary expansion of an integer

Thue–Morse Along Two Polynomial Subsequences

n

On a problem of Chen and Liu concerning the prime power factorization of

. In 2005, Melfi examined the structure of