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An elementary proof of a congruence by Skula and Granville

Romeo Meštrović (2012)

Archivum Mathematicum

Let p ≥ 5 be a prime, and let q p ( 2 ) : = ( 2 p - 1 - 1 ) / p be the Fermat quotient of p to base 2 . The following curious congruence was conjectured by L. Skula and proved by A. Granville q p ( 2 ) 2 ≡ - ∑ k = 1 p - 1 2 k k 2 ( mod p ) . In this note we establish the above congruence by entirely elementary number theory arguments.

An elliptic curve having large integral points

Yanfeng He, Wenpeng Zhang (2010)

Czechoslovak Mathematical Journal

The main purpose of this paper is to prove that the elliptic curve E : y 2 = x 3 + 27 x - 62 has only the integral points ( x , y ) = ( 2 , 0 ) and ( 28844402 , ± 154914585540 ) , using elementary number theory methods and some known results on quadratic and quartic Diophantine equations.

An equicharacteristic analogue of Hesselholt's conjecture on cohomology of Witt vectors

Amit Hogadi, Supriya Pisolkar (2013)

Acta Arithmetica

Let L/K be a finite Galois extension of complete discrete valued fields of characteristic p. Assume that the induced residue field extension k L / k K is separable. For an integer n ≥ 0, let W n ( L ) denote the ring of Witt vectors of length n with coefficients in L . We show that the proabelian group H 1 ( G , W n ( L ) ) n ∈ ℕ is zero. This is an equicharacteristic analogue of Hesselholt’s conjecture, which was proved before when the discrete valued fields are of mixed characteristic.

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