A closed-form bound for linear forms in three $p$-adic logarithms of rational numbers

David Niedbala Giraudin

A closed-form bound for linear forms in three $p$ -adic logarithms of rational numbers

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by David Niedbala Giraudin

Version 2 / Nov 02, 2026 / CC BY 4.0

Abstract

We deduce from the

p

-adic kit for three logarithms of the companion note hexagon:2610.00706 a fully explicit bound in closed form. If

\gamma_1,\gamma_2,\gamma_3

are multiplicatively independent rationals, congruent to

1

modulo

p

(modulo

4

if

p=2

), of heights at most

a_j\log p

with

a_j\ge1

, and if

|b_j|\le B

, then

v_p(\gamma_1^{b_1}\gamma_2^{b_2}\gamma_3^{b_3}-1)\le a_1a_2a_3\,\mathcal{B}^2\max\{27\,711,\ C_2(p)\max_j a_j\}

, where

\mathcal{B}=\max\{\log B/\log p,\,10\}

and

C_2(p)=11\,107\,\frac{p}{p-1}\bigl(1+\frac{0.31}{\log p}\bigr)^2

, as soon as

\gamma_1^{b_1}\gamma_2^{b_2}\gamma_3^{b_3}\ne1

. By a change of basis in the multiplicative group, the result applies to arbitrary rationals prime to

p

. On the examples of that note, the bound is

42

to

956

times smaller than that of Yu's explicit theorem, and stays within a factor between

1.2

and

2.5

of the numerical optimisation of the parameters of the kit.

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by independent AI reviews and by exact computation. No professional mathematician has reviewed the note yet.

Tools used

Anthropic: Claude Opus

References

A. Alvarado, A. Koutsianas, B. Malmskog, C. Rasmussen, C. Vincent, M. West, A robust implementation for solving the $S$ -unit equation and several applications, arXiv:1903.00977.arXiv
Y. Bugeaud, M. Laurent, Minoration effective de la distance $p$ -adique entre puissances de nombres algébriques, J. Number Theory 61 (1996), 311--342.
M. Mignotte, P. Voutier, A kit for linear forms in three logarithms, Math. Comp. 93 (2024), 1903--1951; arXiv:2205.08899.arXiv
D. Niedbala Giraudin, A kit for linear forms in three $p$ -adic logarithms, Hexagon preprint hexagon:2610.00706, 2026.hexagon
K. Yu, $p$ -adic logarithmic forms and group varieties III, Forum Math. 19 (2007), 187--280.

Version history

v1 · 6 pagesSubmitted by David Niedbala GiraudinInitial depositSupersededOct 31, 2026
v2 · 6 pagesSubmitted by David Niedbala GiraudinSubmitted as a new versionCurrentNov 03, 2026