Abstract
In this text, the exclusive exponential Diophantine equation px + (p + 1)y= z2such that the sum of integer powers and of two consecutive prime numbers engrosses a square is examined or estimating enormous integer solutions by exploiting the fundamental notion of Mathematics and the speculation of divisibility or all possibilities of x + y = 1, 2, 3, 4..
Highlights
The study of an exponential Diophantine equations has stimulated the curiosity of plentiful Mathematicians since ancient times as can be seen from [2,3,4,5,6, 9].BanyatSroysang [7] showed that 7x+ 8y= z2 has a unique non-negative integer solution (x, y, z) as (0,1,3) in 2013 and he proposed an open problem where x, y and z are non-negative integers and p is a positive odd prime number
The list of infinite numbers of integer solutions of the equationpx + (p + 1)y= z2 where p is a prime number by using the basic concept of Mathematics and the theory of divisibility
The approach of search out an integer solution to the equation under contemplation is proved by the following theorem
Summary
The study of an exponential Diophantine equations has stimulated the curiosity of plentiful Mathematicians since ancient times as can be seen from [2,3,4,5,6, 9].BanyatSroysang [7] showed that 7x+ 8y= z2 has a unique non-negative integer solution (x, y, z) as (0,1,3) in 2013 and he proposed an open problem where x, y and z are non-negative integers and p is a positive odd prime number. A [8] proved thatpx + (p + 1)y= z2has a unique solution (p, x, y, z) = (3, 1, 0, 2) and was disproved by Nechemia Burshtein [1] by few examples. In this text, the list of infinite numbers of integer solutions of the equationpx + (p + 1)y= z2 where p is a prime number by using the basic concept of Mathematics and the theory of divisibility
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