Note on a Cubic Character Sum.
3 as a Ninth Power (mod p).
Forms Representable by an Integral Positive-Definite Binary Quadratic Form.
Note on a result of Barrucand and Cohn.
Explicit forms of Kummer's complementary theorems to his law of quintic reciprocity.
Quadratic forms and a product-to-sum formula
Ternary quadratic forms ax² + by² + cz² representing all positive integers 8k + 4
Under the assumption that the ternary form x² + 2y² + 5z² + xz represents all odd positive integers, we prove that a ternary quadratic form ax² + by² + cz² (a,b,c ∈ ℕ) represents all positive integers n ≡ 4(mod 8) if and only if it represents the eight integers 4,12,20,28,52,60,140 and 308.
On the least quadratic non-residue of a prime p ... 3 (mod 4).
Determination of modulo p
Some new evaluations of the Legendre symbol ((a+b√q)/p)
Normal integral bases for Emma Lehmer’s parametric family of cyclic quintics
Explicit normal integral bases are given for some cyclic quintic fields defined by Emma Lehmer’s parametric family of quintics.
An arithmetic formula of Liouville
An elementary proof is given of an arithmetic formula, which was stated but not proved by Liouville. An application of this formula yields a formula for the number of representations of a positive integer as the sum of twelve triangular numbers.
Ramanujan identities and Euler products for a type of Dirichlet series
A restricted Epstein zeta function and the evaluation of some definite integrals
On the number of representations of n by ax² + bxy + cy²
Sums of twelve squares
The genera representing a positive integer
On the number of representations of a positive integer by a binary quadratic form
A diophantine system of Dickson
Nei problemi di ciclotomia interessa conoscere il numero delle soluzioni della congruenza (), p dispari. Il caso e = 5 fu trattato completamente da L. E. Dickson; gli Autori trattano ora il caso e = 7.
The conductor of a cyclic quartic field using Gauss sums
Let denote the field of rational numbers. Let be a cyclic quartic extension of . It is known that there are unique integers , , , such that where The conductor of is , where A simple proof of this formula for is given, which uses the basic properties of quartic Gauss sums.
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