Divisibility, periodicity and quadratic lifting in the products ∏(1 + j√−d): the sequences of Moll and Bala
Year: 2026
Type: preprint
Abstract: We study the real and imaginary parts of the products of the numbers one plus j times the square root of minus d, for j from one to n. The case d equal to one arises from V. H. Moll's study of an arctangent sum, and the cases d from two to five are the sequences A363409 to A363416 of the OEIS, in which P. Bala proposed to extend Moll's classification of primes. We prove that every prime divides some term of the imaginary part, and we give a finite criterion that determines exactly the primes dividing no term of the real part; this criterion confirms the initial lists of Moll and Bala and includes the prime 139 in that of A363415. For the odd primes not dividing d modulo which minus d is not a quadratic residue, we prove that divisibility depends only on the index modulo the prime, which excludes linear growth of the valuation, and that the values modulo the prime are periodic with a minimal period given by the multiplicative order of 2. We also describe the passage from roots modulo the prime to roots modulo its square by means of quadratic polynomials and their discriminants, with four roots common to all cases, and we give examples of zero discriminant, with a single lifting class. As consequences we prove Bala's observations on the 7-adic valuation of A363409 and the 5-adic valuation of A363411, and we obtain the periods 171 and 301 of Moll's sequence modulo 19 and 43, together with the congruences by which Moll proved that certain subsequences of the tangent of the arctangent sum take no integer values. A Python program in exact arithmetic, which checks the statements and carries out the finite computations to which some proofs reduce, is included together with its output. This is the English-language version of the paper. A Spanish-language version is available at DOI: 10.5281/zenodo.24017465
Author: Cerdá Bennassar, Miguel
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225384086