The question of which terms of a recurrence sequence fail to have primitive prime divisors has been significantly studied for several classes of linear recurrence sequences and for elliptic divisibility sequences. In this paper, we consider the question for sequences generated by the iteration of a polynomial. For two classes of polynomials f(x) ∈ Z[x] and initial values a1 ∈ Z, we show that th...

The ideal class groups of hyperelliptic curves(HECs) can be used in cryptosystems based on the discrete loga-rithm problem. Recent developments of computational technolo-gies for scalar multiplications of divisor classes have shown thatthe performance of hyperelliptic curve cryptosystems (HECC) iscompatible to that of elliptic curve cryptosystems (ECC). Espe-cially, genu...

Asymptotic formulae for Titchmarsh-type divisor sums are obtained with strong error terms that uniform in the shift parameter. This applies to more general arithmetic functions such as of two squares, improving term representation number a sum prime and Fourier coefficients cusp forms, generalizing result Pitt.

On primitive roots of 1 mod p k , divisors of p ± 1, Wieferich primes, and quadratic analysis mod p Abstract Primitive roots of 1 mod p k (k > 2 and odd prime p) are sought, in cyclic units group G k ≡ A k B k mod p k , coprime to p, of order (p − 1)p k−1. 'Core' subgroup A k has order p − 1 independent of precision k, and 'extension' subgroup B k of all p k−1 residues 1 mod p is generated by p...

A classical result of Sylvester [21] (see also [16], [17]), generalizing Bertrand’s Postulate, states that the greatest prime divisor of a product of k consecutive integers greater than k exceeds k. More recent work in this vein, well surveyed in [18], has focussed on sharpening Sylvester’s theorem, or upon providing lower bounds for the number of prime divisors of such a product. As noted in [...

• An arithmetic function takes positive integers as inputs and produces real or complex numbers as outputs. • If f is an arithmetic function, the divisor sum Df(n) is the sum of the values of f at the positive divisors of n. • τ (n) is the number of positive divisors of n; σ(n) is the sum of the positive divisors of n. • The Möbius function μ(n) is 1 if n = 1 and 0 if n has a repeated prime fac...

Proof. Suppose first that (m− 1)! ≡ −1 (mod m) for some positive integer m. If m is not prime then there exists a divisor d of m with 1 < d < m, so d|(m− 1)!. But d|m, so d| − 1, a contradiction. Thus, m must be prime. Suppose now that m is prime. If some residue class x modulo m has got a multiplicative inverse x with x 6≡ x (mod m) then they both drop out of (m−1)!. Hence, (m−1)! is congruent...

The question of which terms of a recurrence sequence fail to have primitive prime divisors has been significantly studied for several classes of linear recurrence sequences and for elliptic divisibility sequences. In this paper, we consider the question for sequences generated by the iteration of a polynomial. For two classes of polynomials f(x) ∈ Z[x] and initial values a1 ∈ Z, we show that th...

One of the basic facts of group theory is that each finite group contains a Sylow p-subgroup for each prime p which divides the order of the group. In this note we show that each vertex-transitive selfcomplementary graph has an analogous property. As a consequence of this fact, we obtain that each prime divisor p of the order of a vertex-transitive self-complementary graph satisfies the congrue...

Take a rational elliptic curve defined by the equation $y^2=x^3+ax$ in minimal form and consider sequence $B_n$ of denominators abscissas iterate non-torsion point; we show that $B_{5m}$ has primitive divisor for every $m$. Then, how to generalize this method terms $B_{mp}$ with $p$ prime congruent $1$ modulo $4$.