Sets of Integers and Quasi–integers with Pairwise Common Divisor

Sets of Integers with Pairwise Common Divisor and a Factor from a Specified Set of Primes

Maximal sets of integers not containing k + 1 pairwise coprimes and having divisors from a specified set of primes

We find the formula for the cardinality of maximal set of integers from [1,. .. , n] which does not contain k + 1 pairwise coprimes and has divisors from a specified set of primes. This formula is defined by the set of multiples of the generating set, which does not depend on n. 1 Formulation of the result Let P = {p 1 < p 2 ,. . .} be the set of primes and N be the set of natural numbers. Writ...

EEH: AGGH-like public key cryptosystem over the eisenstein integers using polynomial representations

GGH class of public-key cryptosystems relies on computational problems based on the closest vector problem (CVP) in lattices for their security. The subject of lattice based cryptography is very active and there have recently been new ideas that revolutionized the field. We present EEH, a GGH-Like public key cryptosystem based on the Eisenstein integers Z [ζ3] where ζ3 is a primitive...

Consecutive Integers with Equally Many Principal Divisors

Classifying the positive integers as primes, composites, and the unit, is so familiar that it seems inevitable. However, other classifications can bring interesting relationships to our attention. In that spirit, let us classify positive integers by the number of principal divisors they possess, where we define a principal divisor of a positive integer n to be any prime-power divisor pa|n which...

Linear Combinations of Sets of Consecutive Integers

Let k-l,ml,..~+k denote non-negative integers, and suppose the greatest common divisor of ml,...,mk is 1 . We show that if '1, "',sk are sufficiently long blocks of consecutive integers, then the set mlSl+ . ..+mkSk contains a sizable block of consecutive integers. For example; if m and n are relatively prime natural numbers, and U, U ? Vt V are integers with U-u 2 n-l , V-v 2 m-l 1 then the se...