arXiv subjects

Preda Mihăilescu

Publications and source records attributed to Preda Mihăilescu.

4 recordsLinked to original sources

arXiv2021

Double exponential lower bounds for possible solutions in the Second Case of the Fermat Last Theorem

In a recent paper, the first author provided some lower bounds to solutions of the equations of Fermat and Catalan, based on local power series developments at the ramified prime of a prime cyclotomic extension. Although both equations have in fact been proved not to have any unknown solutions, these improved bounds are interesting in the context of a new effective abc inequality announced in the paper \cite{MFHMP} based on Mochizuki's \cite{Mo}[IUT-IV, Theorem A]. In this paper we provide a strengthening of the lower bound for FLT2, which is necessary in order to take advantage of the best upper bounds for primes $p$ for which it was verified on a computer that FLT2 has no solutions.

math.NT↗

arXiv2015

The Gross-Kuz'min Connjecture for CM fields

Let $A' = \varprojlim_n A'_n$ be the projective limit of the $p$-parts of the ideal class groups of the $p$ integers in the $\mathbb{Z}_p$-cyclotomic extension $K_{\infty}/K$ of a CM number field $K$. We prove in this paper that the $T$-part $(A')^-(T) = \{ 1 \}$ for CM extensions $K/\mathbb{Q}$. This fact has been conjectured for arbitrary fields $K$ by Kuz'min in 1972 and was proved by Greenberg in 1973, for abelian extensions $K/\mathbb{Q}$. Federer and Gross had shown in 1981 that $(A')^-(T) = \{ 1 \}$ is equivalent to the non-vanishing of the $p$-adic regulator of the $p$-units of $K$.

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arXiv2015

Applications of Baker Theory to the Conjecture of Leopoldt

In this paper we give a short, elementary proof of the following too extreme cases of the Leopoldt conjecture: the case when $\K/\Q$ is a solvable extension and the case when it is a totally real extension in which $p$ splits completely. The first proof uses Baker theory, the second class field theory. The methods used here are a sharpening of the ones presented at the SANT meeting in Göttingen, 2008 and exposed in \cite{Mi2}, \cite{Mi1}.

math.NT↗

arXiv2012

Turning Washington's heuristics in favor of Vandiver's conjecture

A famous conjecture bearing the name of Vandiver states that $p \nmid h_p^+$ in the $p$ - cyclotomic extension of $\Q$. Heuristics arguments of Washington, which have been briefly exposed in Lang (1978), p. 261 and Washington (1996), p. 158 suggest that the Vandiver conjecture should be false if certain conditions of statistical independence are fulfilled. In this note, we assume that Greenberg's conjecture is true for the \nth{p} cyclotomic extensions and prove an elementary consequence of the assumption that Vandiver's conjecture fails for a certain value of $p$: the result indicates that there are deep correlations between this fact and the defect $λ^- > i(p)$, where $i(p)$ is like usual the irregularity index of $p$, i.e. the number of Bernoulli numbers $B_{2k} \equiv 0 \bmod p, 1 < k < (p-1)/2$. As a consequence, this result could turn Washington's heuristic arguments, in a certain sense into an argument in favor of Vandiver's conjecture.

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