We generalize a formula of B. Litow [7] and propose several new formula linked with the parallel Integer Coprimality, Integer GCD and Modular Inverse problems as well. Particularly, we find a new trigonometrical definition of the GCD of two integers a, b ≥ 1 : gcd(a, b) = 1 π ∫ π 0 cos[ (b− a)x ] sin (abx) sin(ax) sin(bx) dx. We also suggest a generalization of the GCD function to real numbers.

We describe a new subquadratic left-to-right gcd algorithm, inspired by Schönhage’s algorithm for reduction of binary quadratic forms, and compare it to the first subquadratic gcd algorithm discovered by Knuth and Schönhage, and to the binary recursive gcd algorithm of Stehlé and Zimmermann. The new gcd algorithm runs slightly faster than earlier algorithms, and it is much simpler to implement....

On the lines of the binary gcd algorithm for rational integers, algorithms for computing the gcd are presented for the ring of integers in Q( √ d) where d ∈ {−2,−7,−11,−19}. Thus a binary gcd like algorithm is presented for a unique factorization domain which is not Euclidean (case d = −19). Together with the earlier known binary gcd like algorithms for the ring of integers in Q( √−1) and Q(√−3...

Let m, k be positive integers such that m gcd(m,k) ≥ 3, p be an odd prime and π be a primitive element of Fpm. Let h1(x) and h2(x) be the minimal polynomials of −π−1 and π pk+1 2 over Fp, respectively. In the case of odd m gcd(m,k) , when k is even, gcd(m,k) is odd or when k gcd(m,k) is odd, Zhou et al. in [25] obtained the weight distribution of a class of cyclic codes C over Fp with parity-ch...

K. Bibak et al. [arXiv:1503.01806v1 [math.NT], March 5 2015] proved that congruence ax ≡ b (mod n) has a solution x0 with t = gcd(x0, n) if and only if gcd ( a, n t ) = gcd ( b t , n t ) thereby generalizing the result for t = 1 proved by B. Alomair et al. [J. Math. Cryptol. 4 (2010), 121–148] and O. Grošek et al. [ibid. 7 (2013), 217–224]. We show that this generalized result for arbitrary t f...

Relativistic integral representation in terms of experimental neutron– proton scattering phase shifts alone is used to compute the charge form factor of the deuteron GCd(Q ). The results of numerical calculations of |GCd(Q)| are presented in the interval of the four–momentum transfers squared 0 ≤ Q ≤ 35 fm−2. Zero and the prominent secondary maximum in |GCd(Q)| are the direct consequences of th...

New Result EREW PRAM: Compute gcd(x, y) with probability 1 − o(1) in O(n log log n/ log n) time using n6+ processors. [16] Reduction Our inputs are integers x, y with x ≥ y > 0. • Choose a prime bound B > 0, and assume p | x or p | y implies p > B. • Choose a at random, 1 ≤ a ≤ y − 1. • Compute r := ax mod y. • Remove all prime divisors ≤ B from r producing s. Thus P (r/s) ≤ B. We use (x, y) → ...

One can prove the following three propositions: (1) For all natural numbers m, n holds gcd(m,n) = gcd(m, n + m). (2) For all natural numbers k, m, n such that gcd(k, m) = 1 holds gcd(k,m · n) = gcd(k, n). (3) For every real number s such that s > 0 there exists a natural number n such that n > 0 and 0 < 1 n and 1 n ¬ s. In this article we present several logical schemes. The scheme Fib Ind conc...

A number of social insect species have recently been shown to have genetically influenced caste determination (GCD), challenging the conventional view that caste determination should be strictly environmental. To date, GCD has been found in phylogenetically isolated species; examples of GCD being present in multiple species of a genus are lacking. Through crossing experiments of neotenic (juven...