Prime Power Modulus Research Articles

Implementation of number-theoretic transforms

We propose various methods for the implementation of number-theoretic transforms to a prime or prime-power modulus. Such transforms may be performed in a processor or by special hardware. We also give, without proof, the condition for the existence of transforms in an arbitrary ring.

Primes in progressions to prime-power modulus

Primes in progressions to prime-power modulus

The solubility of certain Diophantine inequalities

The method devised by Hardy and Littlewood for the solution of Waring's Problem, and further developed by Vinogradov, applies quite generally to Diophantine equations of an additive type. One particular result that can be proved by this method is that if a 1 , …, a s are integers not all of the same sign, and if s is greater than a certain number depending only on k , the Diophantine equation has infinitely many solutions in positive integers x 1 , …, x s , provided that the corresponding congruence has a non-zero solution to every prime power modulus. A result of a similar kind holds if on the right of (1) there stands an arbitrary integer in place of 0.