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The exponent of pp in xp−1−1x^{p-1}-1: Yamada's constant 283 lowered to 69

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

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

Yamada (J. Number Theory 130, 2010) deduced from the method of interpolation determinants of Bugeaud and Laurent the bound vp(xp−1−1)≤283 (p−1)log⁡y log⁡(xy)/(log⁡p)2+4v_p(x^{p-1}-1)\le 283\,(p-1)\log y\,\log(xy)/(\log p)^2+4, for multiplicatively independent integers x,yx,y coprime to pp. We replace 283 by 69. The new ingredient is an elementary bound, by rearrangement, on the spread of the exponents that occur in the Liouville inequality: it at least halves the height term of Yamada's criterion, involves no congruence class, and allows the parameters to be re-optimised. The proof is complete up to the lemmas of Bugeaud–Laurent (zero lemma, analytic upper bound) and of Laurent–Mignotte–Nesterenko (bound for the polynomial), which are used as they stand. A self-contained verifier checks the numerical inequalities.

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

  1. Y. Bugeaud, M. Laurent, Minoration effective de la distance pp-adique entre puissances de nombres algébriques, J. Number Theory 61 (1996), 311--342.
  2. M. Laurent, M. Mignotte, Y. Nesterenko, Formes linéaires en deux logarithmes et déterminants d'interpolation, J. Number Theory 55 (1995), 285--321.
  3. T. Yamada, A note on the paper by Bugeaud and Laurent ``Minoration effective de la distance pp-adique entre puissances de nombres algébriques'', J. Number Theory 130 (2010), 1889--1897; arXiv:math/0607072.arXiv
  4. D. Niedbala Giraudin, A kit for linear forms in three pp-adic logarithms, note, 22 October 2026, unpublished.
  5. C. L. Stewart, On divisors of Lucas and Lehmer numbers, Acta Math. 211 (2013), 291--314.
  6. K. Yu, pp-adic logarithmic forms and a problem of Erdős, Acta Math. 211 (2013), 315--382.

Version history

  1. v1Submitted by David Niedbala GiraudinInitial depositCurrentOct 31, 2026