Abstract: The balancing numbers are natural numbers n satisfying
the Diophantine equation 1 + 2 + 3 + · · · + (n - 1) = (n + 1) +
(n + 2) + · · · + (n + r); r is the balancer corresponding to the
balancing number n.The nth balancing number is denoted by Bn
and the sequence {Bn}1
n=1 satisfies the recurrence relation Bn+1 =
6Bn-Bn-1. The balancing numbers posses some curious properties,
some like Fibonacci numbers and some others are more interesting.
This paper is a study of recurrent sequence {xn}1
n=1 satisfying the
recurrence relation xn+1 = Axn - Bxn-1 and possessing some
curious properties like the balancing numbers.
Abstract: In this work, we consider the number of integer solutions
of Diophantine equation D : y2 - 2yx - 3 = 0 over Z and
also over finite fields Fp for primes p ≥ 5. Later we determine the
number of rational points on curves Ep : y2 = Pp(x) = yp
1 + yp
2
over Fp, where y1 and y2 are the roots of D. Also we give a formula
for the sum of x- and y-coordinates of all rational points (x, y) on
Ep over Fp.
Abstract: A lot of Scientific and Engineering problems require the solution of large systems of linear equations of the form bAx in an effective manner. LU-Decomposition offers good choices for solving this problem. Our approach is to find the lower bound of processing elements needed for this purpose. Here is used the so called Omega calculus, as a computational method for solving problems via their corresponding Diophantine relation. From the corresponding algorithm is formed a system of linear diophantine equalities using the domain of computation which is given by the set of lattice points inside the polyhedron. Then is run the Mathematica program DiophantineGF.m. This program calculates the generating function from which is possible to find the number of solutions to the system of Diophantine equalities, which in fact gives the lower bound for the number of processors needed for the corresponding algorithm. There is given a mathematical explanation of the problem as well. Keywordsgenerating function, lattice points in polyhedron, lower bound of processor elements, system of Diophantine equationsand : calculus.
Abstract: Let k ≥ 1 and t ≥ 0 be two integers and let d = k2 + k be a positive non-square integer. In this paper, we consider the integer solutions of Pell equation x2 - dy2 = 2t. Further we derive a recurrence relation on the solutions of this equation.