A rational triangle has rational edge-lengths and area; a rational tetrahedron has rational faces and volume; either is Heronian when its edge-lengths are integer, and proper when its content is nonzero. A variant proof is given, via complex number GCD, of the previously known result that any Heronian triangle may be embedded in the Cartesian lattice Z; it is then shown that, for a proper trian...

The enzyme quinoprotein glucose dehydrogenase (GDH) catalyses the oxidation of glucose to gluconic acid by direct oxidation in the periplasmic space of several Gram-negative bacteria. Acidification of the external environment with the release of gluconic acid contributes to the solubilization of the inorganic phosphate by biofertilizer strains of the phosphate-solubilizing bacteria. Glucose deh...

A nonempty subset A of {1, 2, . . . , n} is said to be relatively prime if gcd(A) = 1. Nathanson [4] defined f(n) to be the number of relatively prime subsets of {1, 2, . . . , n} and, for k ≥ 1, fk(n) to be the number of relatively prime subsets of {1, 2, . . . , n} of cardinality k. Nathanson [4] defined Φ(n) to be the number of nonempty subsets A of the set {1, 2, . . . , n} such that gcd(A)...

We present a parallel implementation of Schönhage’s integer GCD algorithm on distributed memory architectures. Results are generalized for the extended GCD algorithm. Experiments on sequential architectures show that Schönhage’s algorithm overcomes other GCD algorithms implemented in two well known multiple-precision packages for input sizes larger than about 50000 bytes. In the extended case t...

This paper proposed an efficient implementation of digital circuit based on the Euclidean Algorithm with modular arithmetic to find Greatest Common Divisor (GCD) of two Binary Numbers given as input to the circuit. Output of the circuit is the GCD of the given inputs. In this paper subtraction-based narrative defined by Euclid is described, the remainder calculation replaced by repeated subtrac...

In [3] and [5] the authors ask how many primes are of the form xY + yX, where gcd (x, y) = 1 and x, y 2: 2. Moreover, Jose Castillo (see [2]) asks how many primes are of the Smarandache form xil + X2 X3 + ... + Xn Xl , where n > 1, Xl, X2, ••• , Xn > 1 and gcd (Xl, X2, ••• , X n ) = 1 (see [9]). In this article we announce a lower bound for the size of the largest prime divisor of an expression...

The general fractional conformable derivative (GCD) and its attributes have been described by researchers in the recent times. Compared with other definitions, this presents a generalization of follows same derivation formulae. For electrical circuits, such as RLC, RC, LC, we obtain new class fractional-order differential equations using novel derivative, use GCD to depict circuits has shown be...

The bistatic backscatter architecture, with its extended range, enables flexible deployment opportunities for devices. In this paper, we study the placement of power beacons (PBs) in networks to maximize guaranteed coverage distance (GCD), defined as from reader within which devices are able satisfy a given quality-of-service constraint. This work departs conventional energy source problems by ...

This paper presents an algorithm and its implementation for computing the approximate GCD (greatest common divisor) of multivariate polynomials whose coefficients may be inexact. The method and the companion software appears to be the first practical package with such capabilities. The most significant features of the algorithm are its robustness and accuracy as demonstrated in the results of c...

We present a new GCD algorithm for two integers that combines both the Euclidean and the binary gcd approaches. We give its worst case time analysis and prove that its bit-time complexity is still O(n) for two n-bit integers. However, our preliminar experiments show that it is very fast for small integers. A parallel version of this algorithm matches the best presently known time complexity, na...