We consider the minimal number of solutions to (n) = (n + k) for xed values of k. Previous work of A. Schinzel concerning this problem is described, as is the outcome of recent computational searches conducted to extend Schinzel's results. Let k be a xed positive integer. In this note, we consider the minimum number of solutions to the equation (n) = (n + k); (1) where n is a positive integer a...

• questions about prime numbers and their “placement” among all numbers (e.g., the Goldbach conjecture, the twin prime conjecture, the “Schinzel hypothesis” predicting when there are an infinite number of prime number values of a given polynomial, etc.); and also • questions about the behavior of the sets of “perfect powers” under simple arithmetic operations. It is this second type of question...

We generalise a necessary and sufficient condition given by Cohn for all the zeros of a self-inversive polynomial to be on the unit circle. Our theorem implies some sufficient conditions found by Lakatos, Losonczi and Schinzel. We apply our result to the study of a polynomial family closely related to Ramanujan polynomials, recently introduced by Gun, Murty and Rath, and studied by Murty, Smyth...

The starting point of the present paper are the papers of Schinzel [10] and of Conway, Guy, Schneeberger and Sloane [4]. Following recent papers ([1], [4], [6], [7]) a composite n is called a pseudoprime to base b if bn−1 ≡ 1 mod n. This definition does not coincide with the definition given in my book [9], where I defined (i) a pseudoprime as a composite number dividing 2 − 2, (ii) a pseudopri...

Let an (r, s)-formation be a concatenation of s permutations of r distinct letters, and let a block of a sequence be a subsequence of consecutive distinct letters. A k-chain on [1,m] is a sequence of k consecutive, disjoint, nonempty intervals of the form [a0, a1][a1 + 1, a2] . . . [ak−1 + 1, ak] for integers 1 6 a0 6 a1 < . . . < ak 6 m, and an s-tuple is a set of s distinct integers. An s-tup...

This is largely a survey paper in which we discuss new and old problems about the reductions Ep modulo primes p of a fixed elliptic curve E defined over the field of rational numbers. We investigate, in particular, how the “noncyclic” part of the group of points of Ep is distributed, thus making progress toward a conjecture of R. Takeuchi. The new result is Theorem 2 of Section 3. Many interest...

A weighted geometric fitting problem between two corresponding sets of points is to minimize the maximum weighted distance between two corresponding pairs of points by translating and rotating one set to the other set. For this weighted geometric fitting problem, dynamic weighted Voronoi diagrams have been applied to obtain an almost cubic algorithm. In this paper, we show a new reduction of th...