We investigate number theoretic aspects of the integer sequence seq (n) with identification number A000522 in Sloane’s On-Line Encyclopedia of Integer Sequences: seq (n) counts the number of sequences without repetition one can build with n distinct objects. By introducing the the notion of the “shadow” of an integer function, we examine divisibility properties of the combinatorial function seq...

We propose a new class of pseudorandom number generators based on Pohlig-Hellman exponentiation ciphers. The method generates uniform pseudorandom streams by encrypting simple sequences of short integer messages into ciphertexts by exponentiation modulo prime numbers. The advantages of the method are: the method is trivially parallelizable by parameterization with each pseudorandom number gener...

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...

In 1974, Vegh proved that if k is a prime and m a positive integer, there is an m term permutation chain of kth power residue for infinitely many primes [E.Vegh, kth power residue chains, J.Number Theory, 9(1977), 179-181]. In fact, his proof showed that 1, 2, 22, ..., 2m−1 is an m term permutation chain of kth power residue for infinitely many primes. In this paper, we prove that for k being a...

In Shamir’s scheme that the security based on the numbers of the field of a prime number P which the coefficients' polynomial reduced to modulo P (takes a value from some field p Z , where P is a large prime number). Thus, the adversary must know only the free coefficient of the polynomial in order to break the scheme. Our scheme which based on representation integer using the so-called g a d i...

Every integer n > 0 є N defines an increasing monotonic series of integers: n1, n2, ...nk, such that nk = nk +k(k-1)/2. We define as s(m) the number of such series that an integer m belongs to. We prove that there are infinite number of integers with s=1, all of the form 2 (they belong only to the series that they generate, not to any series generated by a smaller integer). We designate them as...

Abstract: Let N be a (large) positive integer, let b be an integer satisfying 1 < b < N that is relatively prime to N , and let r be the order of b modulo N . Finally, let QC be a quantum computer whose input register has the size specified in Shor’s original description of his order-finding algorithm. We prove that when Shor’s algorithm is implemented on QC, then the probability P of obtaining...

Abstract: Let N be a positive integer, let b < N be a positive integer relatively prime to N , and let r be the order of b modulo N . Finally, let QC be a quantum computer whose input register has the size specified in Shor’s original description of his order-finding algorithm. We prove that when Shor’s algorithm is implemented on QC, then the probability P of obtaining a (nontrivial) divisor o...

1. First we introduce some notations. elements of the finite set S is denoted by ISI. We write ex = exp x. We denote the i th prime number by p i ,. v(n) denotes the number of distinct prime factors of n. mk is the smallest integer m for which v(1k» > k. P(n) is the greatest, p(n) the smallest prime factor of n. nk is the smallest integer n for which P(n+i) > k for all 1 < i <_ k. Mk > c I k 2 ...

The multicolor Ramsey number rk(C4) is the smallest integer n for which any k-coloring of the edges of the complete graph Kn must produce a monochromatic 4-cycle. In [6] and [3] it was proved that rk(C4) ≥ k2−k+2 for k − 1 being a prime power. In this note we establish rk(C4) ≥ k2 + 2 for k being an odd prime power. ( Journal of Combinatorial Theory, Series B 79, 172–176 (2000))