P-adic Logarithms Research Articles

A note on the paper by Bugeaud and Laurent “Minoration effective de la distance p-adique entre puissances de nombres algébriques”

We shall make a slight improvement to a result of p-adic logarithms, which gives a nontrivial upper bound for the exponent of p dividing the Fermat quotient x p − 1 − 1 .

ON A RESULT OF IWASAWA

We recover a result of Iwasawa on the p-adic logarithm of principal units of ℚp(ζpn+1) by studying the value at s = 1 of p-adic L-functions.

Approximations simultanées de nombres algébriques de Q p par des rationnels

In this paper, we prove that if β1,…, βn are p-adic numbers belonging to an algebraic number field K of degree n + 1 over Q such that 1, β1,…,βn are linearly independent over Z, there exist infinitely many sets of integers (q0,…, qn), with q0 ≠ 0 and with H = H(q0,…, qn). Therefore, these numbers satisfy the p-adic Littlewood conjecture. To obtain this result, we are using, as in the real case by Peck [2], the structure of a group of units of K. The essential argument to obtain the exponent 1/(n-1) (the same as in the real case) is the use of the p-adic logarithm. We also prove that with the same hypothesis, the inequalities have no integer solution (q0,…, qn) with q0 ≠ 0, if ɛ > 0 is small enough.

On multivariatep-adicq-integrals

By using multiple p-adic q-integrals, we define the p-adic q-L-function in n-variables and the q-extension of p-adic log multiple gamma functions. From these definitions, we show that the values of the p-adic q-L-function at positive integers can be expressed in terms of the q-extension of p-adic log multiple gamma functions.

Linear forms in p-adic logarithms and the Diophantine equation formula here

A longstanding conjecture claims that the Diophantine equationformula herehas finitely many solutions and, maybe, only those given byformula hereAmong the known results, let us mention that Ljunggren [9] solved (1) completely when q = 2 and that very recently Bugeaud et al. [3] showed that (1) has no solution when x is a square. For more information and in particular for finiteness type results under some extra hypotheses, we refer the reader to Nagell [11], Shorey & Tijdeman [16, 17] and to the recent survey of Shorey [15]. One of the main tools used in most of the proofs is Baker's theory of linear form in archimedean logarithms of algebraic numbers and especially a dramatic sharpening obtained when the algebraic numbers involved are closed to 1. This was first noticed by Shorey [14], and has been applied in numerous works relating to (1) or to the Diophantine equationformula heresee for instance [1, 5, 10]. The purpose of the present work is to prove a similar sharpening for linear forms in p-adic logarithms and to show how it can be applied in the context of (1). Further, following previous investigations by Sander [12] and Saradha & Shorey [13], we derive from our results an irrationality statement for Mahler's numbers.

$S$-integral solutions to a Weierstrass equation

On détermine les solutions rationnelles de l’équation diophantienne y 2 =x 3 -228x+848 dont les dénominateurs sont des puissances de 2. On applique une idée de Yuri Bilu, qui évite le recours à des équations de Thue et de Thue-Mahler, et qui permet d’obtenir des équations aux (S-) unités à quatre termes dotées de propriétés spéciales, que l’on résout par la théorie des formes linéaires en logarithmes réels et p-adiques.

On the diophantine equation apx + bqy = c + dpzqw

In this paper, the existence of effectively computable bounds on the solutions to the diophantine equation ap x+bq y = c+dp zq w is shown. In this equation p, q are taken to be fixed relatively prime positive integers and a, b, c, d positive integers. The methods involve the application of linear forms in both real and p-adic logarithms. Also, a result on an inequality involving S-integers is used. All constants involved can be explicitly computed in terms of the parameters a, b, c, d, p, q, conceivably allowing one to list all solutions to ( ∗) for any set of parameters. It is also indicated how the bounds in a particular case can be reduced to allow the practical solution of the equation. Finally, the methods are demonstrated by the solving of the equations 2 x + 3 y = 1 + 2 z 3 w and 5 · 2 x + 7 · 3 y = 11 + 2 z 3 w .

Linear forms in p-adic logarithms

Linear forms in p-adic logarithms

The $p$-adic gamma measures

The p-adic log gamma function and its derivatives are used to define distributions and measures on the p-adic units. These measures are then used to interpolate the Leopoldt-Kubota p-adic L-functions on the positive integers.

Interpretation of the p-Adic Log Gamma Function and Euler Constants Using the Bernoulli Measure

Interpretation of the p-Adic Log Gamma Function and Euler Constants Using the Bernoulli Measure

Interpretation of the 𝑝-adic log gamma function and Euler constants using the Bernoulli measure

A regularized version of J. Diamond’sp-adic log gamma function and hisp-adic Euler constants are represented as integrals using B. Mazur’sp-adic Bernoulli measure.

The p-Adic Log Gamma Function and p-Adic Euler Constants

The p-Adic Log Gamma Function and p-Adic Euler Constants