Fermat's Last Theorem (case 1) and the Wieferich Criterion
نویسنده
چکیده
This note continues work by the Lehmers [3], Gunderson [2], Granville and Monagan [1], and Tanner and Wagstaff [6], producing lower bounds for the prime exponent p in any counterexample to the first case of Fermat's Last Theorem. We improve the estimate of the number of residues r mod p" such that fP = r mod p~ , and thereby improve the lower bound on p to 7.568 x 1017.
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A New Criterion for the First Case of Fermat's Last Theorem
It is shown that if the first case of Fermat's last theorem fails for an odd prime /, then the sums of reciprocals modulo /, s(k, N) = £ 1/7 (kl/N < j < (k + 1)//A0 are congruent to zero mod/ for all integers N and k with I < N < 46 and 0 < k < N 1 . This is equivalent to Bi_{(k/N) B¡_x =0 (mod/), where B„ and B„(x) are the «th Bernoulli number and polynomial, respectively. The work can be cons...
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On primitive roots of 1 mod p k , divisors of p 2 − 1 , Wieferich primes and quadratic analysis mod p Abstract Primitive roots of 1 mod p k (k > 2 and odd prime p) are sought, in cyclic units group G k ≡ A k B k mod p k , coprime to p, of order (p − 1)p k−1. 'Core' subgroup A k has order p − 1 independent of precision k, and 'extension' subgroup B k of all p k−1 residues 1 mod p is generated by...
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IH. S. Vandiver, "Fermat's Last Theorem," Am. Math. Mlonthly, 53, 567-568, 1946. 2 H. S. Vandiver, "The Relation of Some Data Obtained from Rapid Computing Machines to the Theory of Cyclotomic Fields, these PROCEEDINGS, 40, 474-480, 1954. 3H. S. Vandiver, "A Theorem of Kummer's concerning the Second Factor of the Class Number of a Cyclotomic Field," Bull. Am. Math. Soc., 35, 333-335, 1929. 4D. ...
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On primitive roots of 1 mod p k , divisors of p 2 − 1 , Wieferich primes and quadratic analysis mod p Abstract Primitive roots of 1 mod p k (k > 2 and odd prime p) are sought, in cyclic units group G k ≡ A k B k mod p k , coprime to p, of order (p − 1)p k−1. 'Core' subgroup A k has order p − 1 independent of precision k, and 'extension' subgroup B k of all p k−1 residues 1 mod p is generated by...
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On primitive roots of 1 mod p k , divisors of p ± 1, Wieferich primes, and quadratic analysis mod p Abstract Primitive roots of 1 mod p k (k > 2 and odd prime p) are sought, in cyclic units group G k ≡ A k B k mod p k , coprime to p, of order (p − 1)p k−1. 'Core' subgroup A k has order p − 1 independent of precision k, and 'extension' subgroup B k of all p k−1 residues 1 mod p is generated by p...
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