2026/08/01 by Alberto Alfarano, Pablo Bianucci, Matilde N. Lal\'ın +1
Mathematics · #math.NT #msc:11R06 #msc:11F67 #msc:11-11 #msc:68T05
36 pages, 19 figures
arxiv created 2026/08/01 · arxiv updated 2026/08/04
Boyd conjectured that the Mahler measure of Pk(x,y)=x+y+(1)/(x)+(1)/(y)+k for k an integer, is given by rkL'(Ek,0), where Ek is the elliptic curve associated to the zero locus of Pk and rk is a rational number. We study various arithmetic properties of rk using a dataset containing the first 250,000 values of k, combining large-scale statistical analysis assisted by Claude with transformer-based experiments carried out using Axolver. We recover Boyd's observation that, apart from a few exceptions, rk is the reciprocal of an integer. The size of this integer is governed by the conductor of the elliptic curve. Moreover, its p-adic valuations display markedly different behavior according to the prime. For p≥ 5, the probability of vp(rk)=-m for m≥ 1 appears to be p-m. For the primes 2 and 3, however, we find additional arithmetic structure involving congruence conditions on k and the primes of bad reduction of Ek. Although the neural networks do not predict rk exactly, they recover significant information about its magnitude and valuations. In particular, the experiments at the prime 2 suggest arithmetic structure beyond the explicit predictor obtained from our statistical analysis.