Five peculiar theorems on simultaneous representation of primes by quadratic forms

Five peculiar theorems on simultaneous representation of primes by quadratic forms

It is a theorem of Kaplansky that a prime p ≡ 1 (mod 16) is representable by both or none of x2 + 32y2 and x2 + 64y2, whereas a prime p ≡ 9 (mod 16) is representable by exactly one of these binary quadratic forms. In this paper five similar theorems are proved. As an example, one theorem states that a prime p ≡ 1 (mod 20) is representable by both or none of x2 + 20y2 and x2 + 100y2, whereas a p...

On Representation of Integers by Quadratic Forms.

The operation of finding the limit of an infinite series has been one of the most fruitful operations of all mathematics. While this is not a group operation the theory of continuous transformation groups inaugurated by S. Lie has thrown much new light on this operation. The assumption that only two elements are combined at a time applies to continuous groups as well as to those which are disco...

Representation of Quadratic Forms by Integral Quadratic Forms

Representation by Ternary Quadratic Forms

The problem of determining when an integral quadratic form represents every positive integer has received much attention in recent years, culminating in the 15 and 290 Theorems of Bhargava-Conway-Schneeberger and Bhargava-Hanke. For ternary quadratic forms, there are always local obstructions, but one may ask whether there are ternary quadratic forms which represent every locally represented in...

Representation of Quadratic forms