Machine learning the arithmetic of Boyd's Mahler measure conjectures

Boyd conjectured that the Mahler measure of $P_k(x,y)=x+y+\frac{1}{x}+\frac{1}{y}+k$ for $k$ an integer, is given by $r_kL'(E_k,0)$, where $E_k$ is the elliptic curve associated to the zero locus of $P_k$ and $r_k$ is a rational number. We study various arithmetic properties of $r_k$ 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, $r_k$ 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 $p\geq 5$, the probability of $v_p(r_k)=-m$ for $m\geq 1$ 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 $E_k$. Although the neural networks do not predict $r_k$ 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.