We show that the abc conjecture implies that the number of terms of any arithmetic progression consisting of almost perfect ”inhomogeneous” powers is bounded, moreover, if the exponents of the powers are all ≥ 4, then the number of such progressions is finite. We derive a similar statement unconditionally, provided that the exponents of the terms in the progression are bounded from above.

Let A, B ⊆ Z be finite, nonempty subsets with minA = minB = 0, and let δ(A,B) = n 1 if A ⊆ B, 0 otherwise. If maxB ≤ maxA ≤ |A|+ |B| − 3 and (1) |A+B| ≤ |A|+ 2|B| − 3− δ(A,B), then we show A + B contains an arithmetic progression with difference 1 and length |A|+ |B| − 1. As a corollary, if (1) holds, max(B) ≤ max(A) and either gcd(A) = 1 or else gcd(A+ B) = 1 and |A+B| ≤ 2|A|+ |B| − 3, then A+...

Abstract. We show that the classical Faulhaber’s theorem on sums of odd powers also holds for an arbitrary arithmetic progression, namely, the odd power sums of any arithmetic progression a + b, a + 2b, . . . , a + nb is a polynomial in na+ n(n+ 1)b/2. The coefficients of these polynomials are given in terms of the Bernoulli polynomials. Following Knuth’s approach by using the central factorial...

According to a theorem of Szemerédi (1975), for every positive integer k and every δ > 0, there exists S = S(k, δ) such that every subset X of {1, 2, . . . , S} of size at least δS contains an arithmetic progression with k terms. Here we generalize the above result to any set of real numbers. For every positive integer k and every c > 0, there exists N = N(k, c) such that every set of n ≥ N poi...

Let d be a squarefree integer. Does there exist four squares in arithmetic progression over Q( √ d )? We shall give a partial answer to this question, depending on the value of d. In the affirmative case, we construct explicit arithmetic progressions consisting of four squares over Q( √ d ).

In a recent paper, Thorne [5] proved the existence of arbitrarily long strings of consecutive primes in arithmetic progressions in the polynomial ring Fq[t]. Here we extend this result to show that given any k there exists a string of k consecutive primes of degree D in arithmetic progression for all sufficiently large D.

We describe eÆcient output-sensitive algorithms to nd the longest arithmetic progression in a given set of numbers.

We report some regular organizations of stability phases discovered among self-sustained oscillations of a biochemical oscillator. The signature of such organizations is a nested arithmetic progression in the number of spikes of consecutive windows of periodic oscillations. In one of them, there is a main progression of windows whose consecutive number of spikes differs by one unit. Such window...

We prove a result which implies that, for any real numbers a and b satisfying 0 ≤ a ≤ b ≤ 1, there exists an infinite sequence of positive integers A with lower density a and upper density b such that the sets A and N \ A contain no infinite arithmetic and geometric progressions. Furthermore, for any m ≥ 2 and any positive numbers a1, . . . , am satisfying a1 + · · · + am = 1, we give an explic...

Abstract We give sharp, in some sense uniform bounds for the number of $$\ell $$ ℓ -th powers and arbitrary among first N terms an arithmetic progression, large enough.