New type of elliptic curve point multiplication is proposed, where complex multiplication by 2 − or by 2 ) 7 1 ( − ± is used instead of point duplication. This allows speeding up multiplication about 1.34 times. Using higher radix makes it possible to use one point duplication instead of two and to speed-up computation about 1.6 times. The method takes prime group order factorization: ρ ρ = r a...

Sierpiński and Riesel numbers are not easy to find. To disqualify an odd positive integer as a Sierpiński number or a Riesel number, one need only locate a prime in the appropriate infinite list. With four exceptions, k = 47,103,143,197, all of the first 100 odd positive integers, 1 ≤ k ≤ 199, are disqualified as Sierpiński numbers by finding at least one prime in the first eight elements of th...

Fermat’s Little Theorem states that xp z(modp) for z E N and prime p, and so identifies an integer-valued polynomial (IVP) g,(z) (zr x)/p. Presented here are IVP’s gn for non-prime n that complete the sequence {gn n E N} in a natural way. Also presented are characterizations of the gn’S and an indication of the ideas from topological dynamics and algebra that brought these matters to our attent...

(1.2) f(x) = a1x k1 + · · ·+ arx with 0 < k1 < k2 < · · · < kr. We assume always that the content of f , (a1, a2, . . . , ar), is relatively prime to the modulus q. Let d = d(f) = kr denote the degree of f and for any prime p let dp(f) denote the degree of f read modulo p. A fundamental problem is to determine whether there exists an absolute constant C such that for an arbitrary positive integ...

Given a pair (Ut) and (Vt) of Lucas sequences, we establish various congruences involving sums of ratios Vt Ut . More precisely, let p be a prime divisor of the positive integer m. We establish congruences, modulo powers of p, for the sum ∑ Vt Ut , where t runs from 1 to r(m), the rank of m, and r(q) ∤ t for all prime factors q of m.

It is proved that for a given integer N and for all but (log N)B prime numbers k ≤ N5/48−ε the following is true: For any positive integers bi, i ∈ {1, 2, 3}, (bi, k) = 1 that satisfy N ≡ b1 + b2 + b3 (mod k), N can be written as N = p1+p2+p3, where the pi, i ∈ {1, 2, 3} are prime numbers that satisfy pi ≡ bi (mod k).

We examine n×nmatrices over Zm, with 0’s in the diagonal and nonzeros elsewhere. If m is a prime, then such matrices have large rank (i.e., n1/(p−1) − O(1) ). If m is a non-prime-power integer, then we show that their rank can be much smaller. For m = 6 we construct a matrix of rank exp(c √ log n log log n). We also show, that explicit constructions of such low rank matrices imply explicit cons...

This project is concerned with the set of primes modulo which some monic, irreducible polynomial f(x) ∈ Z[x] has a root, called the Prime Set of f . We completely characterise these sets for degree 2 polynomials, and develop sufficient machinery from algebraic number theory to show that if the Galois group of a monic, irreducible polynomial in Z[x] is abelian, then its Prime Set can be written ...

Let p a be prime number. Using algebraic methods from the factorization theory of abelian groups we will prove a result about the structure of the 1-error correcting t-shift integer codes over the alphabet Zp in the special case when t is a prime. The algorithms to construct such codes can take advantage of this extra structural information in a straightforward manner and the search for these c...

Let n > 1 be a positive integer and p be the smallest prime divisor of n. Let S be a sequence of elements from Zn = Z/nZ of length n + k where k ≥ np − 1. If every element of Zn appears in S at most k times, we prove that there must be a subsequence of S of length n whose sum is zero when n has only two distinct prime divisors.