We describe a method to perform scalar multiplication on two classes of ordinary elliptic curves, namely [Formula: see text] in prime characteristic [Formula: see text], and [Formula: see text] in prime characteristic [Formula: see text]. On these curves, the 4-th and 6-th roots of unity act as (computationally efficient) endomorphisms. In order to optimise the scalar multiplication, we conside...

Primitive Pythagorean triples (PPT) may be put into different equivalence classes using residues with respect to primes. We show that the probability that the smaller odd number associated with the PPT triple is divisible by prime p is 2/(p+1). We have determined the autocorrelation function of the Baudhāyana sequences obtained from the residue classes and we show these sequences have excellent...

We prove that any set of integers A ⊂ [1, x] with |A| (log x)r lies in at least νA(p) p r r+1 many residue classes modulo most primes p (log x)r+1. (Here r is a positive constant.) This generalizes a result of Erdős and Ram Murty, who proved in connection with Artin’s conjecture on primitive roots that the integers below x which are multiplicatively generated by the coprime integers a1, . . . ,...

Residue Number Systems require the selection of ring moduli whose product is greater than the predicted dynamic range of the computation being performed. The restriction that the moduli be relatively prime usually limits the set of available moduli and hence the maximum dynamic range. This is particularly the case when small moduli are to be considered for efficient hardware implementation. Sev...

We estimate character sums with Catalan numbers and middle binomial coefficients modulo a prime p. We use this bound to show that the first at most p13/2(log p)6 elements of each sequence already fall in all residue classes modulo every sufficiently large p, which improves the previously known result requiring pO(p) elements. We also study, using a different technique, similar questions for seq...

For p∈N, a coloring λ of the vertices graph G is p-centered if for every connected subgraph H G, either receives more than p colors under or there color that appears exactly once in H. Centered colorings play an important role theory sparse classes introduced by Nešetřil and Ossona de Mendez [31], [32], as they structurally characterize bounded expansion — one key sparsity notions this theory. ...

Let E/K be an abelian extension of number fields, with E 6= Q. Let ∆ and n denote the absolute discriminant and degree of E. Let σ denote an element of the Galois group of E/K. We prove the following theorems, assuming the Extended Riemann Hypothesis: (1) There is a degree-1 prime p of K such that ( p E/K ) = σ, satisfying Np ≤ (1 + o(1))(log ∆ + 2n)2. (2) There is a degree-1 prime p of K such ...