Let G be an abelian group, let S be a sequence of terms s1, s2, . . . , sn ∈ G not all contained in a coset of a proper subgroup of G, and let W be a sequence of n consecutive integers. Let W ̄ S = {w1s1 + . . . + wnsn : wi a term of W, wi 6= wj for i 6= j}, which is a particular kind of weighted restricted sumset. We show that |W ̄S| ≥ min{|G| − 1, n}, that W ̄ S = G if n ≥ |G| + 1, and also c...

Some patterns cannot be avoided ad infinitum. A well-known example of such a pattern is an arithmetic progression in partitions of natural numbers. We observed that in order to avoid arithmetic progressions, other patterns emerge. A visualization is presented that reveals these patterns. We capitalize on the observed patterns by constructing techniques to avoid arithmetic progressions. More for...

In this list there is an arithmetic progression: 1, 25, 49 (common difference 24). If we search further along, another arithmetic progression of squares is found: 289, 625, 961 (common difference 336). Yet another is 529, 1369, 2209 (common difference 840). How can these examples, and all others, be found? In Section 2 we will use plane geometry to describe the 3-term arithmetic progressions of...

In this paper, we look at long arithmetic progressions on conics. By an arithmetic progression on a curve, we mean the existence of rational points on the curve whose x-coordinates are in arithmetic progression. We revisit arithmetic progressions on the unit circle, constructing 3-term progressions of points in the first quadrant containing an arbitrary rational point on the unit circle. We als...

Let p(n) denote the number of partitions of a non-negative integer n. A well-known conjecture asserts that every arithmetic progression contains infinitely many integers M for which p(M) is odd, as well as infinitely many integers N for which p(N) is even (see Subbarao [23]). In this paper we prove that there indeed are infinitely many integers N in every arithmetic progression for which p(N) i...

Multivariate Public Key Cryptosystems (MPKCs) are often touted as future-proofing against Quantum Computers. It also has been known for efficiency compared to “traditional” alternatives. However, this advantage seems to erode with the increase of arithmetic resources in modern CPUs and improved algorithms, especially with respect to Elliptic Curve Cryptography (ECC). In this paper, we show that...

and we consider two equations related by such a change of variables to represent the same curve (equivalently, we will deal with elliptic curves up to so-called Weierstrass changes of variables). Consider P0, . . . , Pn ∈ E(K), with Pi = (xi, yi) such that x0, . . . , xn is an arithmetic progression. We say that P0, . . . , Pn are in x-arithmetic progression (x-a.p.) and also say that E has an ...

The partitions of a positive integer n in which the parts are in arithmetic progression possess interesting combinatorial properties that distinguish them from other classes of partitions. We exhibit the properties by analyzing partitions with respect to a fixed length of the arithmetic progressions. We also address an open question concerning the number of integers k for which there is a k-par...