To say that R is a root set modulo n means that R is a subset of Z n , the ring of integers modulo n , and there is a polynomial the roots of which modulo n are exactly the elements of R . Note that [ and Z n are always root sets modulo n . It seems that only two papers have appeared which mention the nature of root sets modulo n , and then only at a very basic level : Sierpin ́ ski [3] and Choj...

In this paper, we prove that every sufficiently large positive integer satisfying some necessary congruence conditions can be represented by the sum of a fourth power of integer and twelve fourth powers of prime numbers. MSC: 11P05, 11P32, 11P55.

Let N denote a sufficiently large even integer. In this paper it is proved that for 0.941 ≤ θ ≤ 1, the equation N = p + P 2 , p≤ N θ is solvable, where p is a prime and P 2 is an almost prime with at most two prime factors. The range 0.941 ≤ θ ≤ 1 extended the previous one 0.945 ≤ θ ≤ 1.

For every prime integer p, M. Hochster conjectured the existence of certain p-torsion elements in a local cohomology module over a regular ring of mixed characteristic. We show that Hochster’s conjecture is false. We next construct an example where a local cohomology module over a hypersurface has p-torsion elements for every prime integer p, and consequently has infinitely many associated prim...

These notes contain as little theory as possible, and most results are stated without proof. Any introductory book on algebra will contain proofs and put the results in a more general, and more beautiful framework. For example, a book by Childs [C95] covers all the required material without getting too abstract. It also points out the cryptographic applications. By integer, we mean a positive o...

Integers without large prime factors, dubbed smooth numbers, are by now firmly established as a useful and versatile tool in number theory. More than being simply a property of numbers that is conceptually dual to primality, smoothness has played a major role in the proofs of many results, from multiplicative questions to Waring’s problem to complexity analyses of factorization and primality-te...

Prime factoring is important for multiple reasons. One is its central role in several cryptographic schemes. The central element of factoring is to find two prime numbers p, q where n = p ∗ q, given the value of n. A mathematical expression is developed that is amenable to iterative resolution of prime factors for a composite. The expression is then shown to admit a factoring process of P-compl...

Prime Factorization based on Quantum approach in two phases has been performed. The first phase has been achieved at Quantum computer and the second phase has been achieved at the classic computer (Post Processing). At the second phase the goal is to estimate the period r of equation 1 N r x ≡ and to find the prime factors of the composite integer N in classic computer. In this paper we present...

Let $R$ be a prime ring with extended centroid $C$, $H$ a generalized derivation of $R$ and $ngeq 1$ a fixed integer. In this paper we study the situations: (1) If $(H(xy))^n =(H(x))^n(H(y))^n$ for all $x,yin R$; (2) obtain some related result in case $R$ is a noncommutative Banach algebra and $H$ is continuous or spectrally bounded.

We introduce a two person game played with a pair of nonnegative integers; a move consists of subtracting from the larger integer, a positive integer no greater than the smaller integer. The player who reduces one of the integers to zero wins. The game is curious in several respects: in particular, its Sprague-Grundy values have an interesting connection with prime numbers.