Abstract: The article explores one of the important relations between numbers-the Pythagorean triples (triplets) which finds its application in distance measurement, construction of roads, towers, buildings and wherever Pythagoras theorem finds its application. The Pythagorean triples are numbers, that satisfy the condition “In a given set of three natural numbers, the sum of squares of two natural numbers is equal to the square of the other natural number”. There are numerous methods and equations to obtain the triplets, which have their own merits and demerits. Here, quadratic approach for generating triples uses the hypotenuse leg difference method. The advantage is that variables are few and finally only three independent variables are present.
Abstract: This paper presents two solutions to the Fermat’s Last Theorem (FLT). The first one using some algebraic basis related to the Pythagorean theorem, expression of equations, an analysis of their behavior, when compared with power and power and using " the “Well Ordering Principle” of natural numbers it is demonstrated that in Fermat equation . The second one solution is using the connection between and power through the Pascal’s triangle or Newton’s binomial coefficients, where de Fermat equation do not fulfill the first coefficient, then it is impossible that:
zn=xn+yn for n>2 and (x, y, z) E Z+ - {0}
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.