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A note on the diophantine equation x2=4pn-4pm + l2

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2021-11-11

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Indian Nat Sci Acad

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Abstract

Let l be a fixed odd positive integer. In this paper, using some classical results on the generalized Ramanujan-Nagell equation, we completely derive all solutions (p, x, m, n) of the equation x(2) = 4p(n)-4p(m)+l(2) with l(2) < 4p(m) for any l > 1, where p is a prime, x, m, n are positive integers satisfying gcd(x, l) = 1 and m < n. Meanwhile we give a method to solve the equation with l(2) > 4p(m). As an example of using this method, we find all solutions (p, x, m, n) of the equation for l is an element of {5, 7}.

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Polynomial-exponential diophantine equation, Generalized ramanujan-nagell equation, Baker's method, Science & technology, Physical sciences, Mathematics, Mathematics

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