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Pré-Publication, Document De Travail Année : 2018

Test of Vandiver's conjecture with Gauss sums -- Heuristics

Résumé

The link between Vandiver's conjecture and Gauss sums is well-known since the papers of Iwasawa (1975), Angìès–Nuccio (2010) and has been considered by many authors with various purposes (e.g., Iwasawa theory). In this article, we give again the interpretation of Vandiver's conjecture in terms of the minus part of the torsion group of the Galois group of the maximal abelian p-ramified prop extension of the pth cyclotomic field, that we had published at the Laval University (1984), in relation with the result of Iwasawa. Then we provide a specific use of Gauss sums that allows a new numerical test of Vandiver's conjecture (see Theorem 4.5 using both the set of exponents of p-irregularity and the set of exponents of p-primarity of the Gauss sum associated to a totally split prime number). Then we propose new favorable heuristics for its rightness. We show that a counterexample to Vandiver's conjecture leads to a totally strange phenomenon on the congruential properties of Gauss sums and to an unusual complexity of some classical algorithms. Some tables with PARI programs are given to strength our arguments.
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Dates et versions

hal-01856083 , version 1 (09-08-2018)
hal-01856083 , version 2 (29-08-2018)
hal-01856083 , version 3 (08-12-2018)
hal-01856083 , version 4 (25-06-2019)

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Georges Gras. Test of Vandiver's conjecture with Gauss sums -- Heuristics. 2018. ⟨hal-01856083v1⟩
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