Displaying similar documents to “On Bumby’s equation ( x 2 - 2 ) 2 - 2 = 2 y 2 : a solution via pythagorean triples”

Diophantine equation q n - 1 q - 1 = y for four prime divisors of y - 1

Zdeněk Polický (2005)

Commentationes Mathematicae Universitatis Carolinae

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In this paper the special diophantine equation q n - 1 q - 1 = y with integer coefficients is discussed and integer solutions are sought. This equation is solved completely just for four prime divisors of y - 1 .

On the Diophantine equation q n - 1 q - 1 = y

Amir Khosravi, Behrooz Khosravi (2003)

Commentationes Mathematicae Universitatis Carolinae

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There exist many results about the Diophantine equation ( q n - 1 ) / ( q - 1 ) = y m , where m ≥ 2 and n ≥ 3 . In this paper, we suppose that m = 1 , n is an odd integer and q a power of a prime number. Also let y be an integer such that the number of prime divisors of y - 1 is less than or equal to 3 . Then we solve completely the Diophantine equation ( q n - 1 ) / ( q - 1 ) = y for infinitely many values of y . This result finds frequent applications in the theory of finite groups.

On the Diophantine equation x² - dy⁴ = 1 with prime discriminant II

D. Poulakis, P. G. Walsh (2006)

Colloquium Mathematicae

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Let p denote a prime number. P. Samuel recently solved the problem of determining all squares in the linear recurrence sequence {Tₙ}, where Tₙ and Uₙ satisfy Tₙ² - pUₙ² = 1. Samuel left open the problem of determining all squares in the sequence {Uₙ}. This problem was recently solved by the authors. In the present paper, we extend our previous joint work by completely solving the equation Uₙ = bx², where b is a fixed positive squarefree integer. This result also extends previous work...

The method of infinite ascent applied on A 4 ± n B 3 = C 2

Susil Kumar Jena (2013)

Czechoslovak Mathematical Journal

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Each of the Diophantine equations A 4 ± n B 3 = C 2 has an infinite number of integral solutions ( A , B , C ) for any positive integer n . In this paper, we will show how the method of infinite ascent could be applied to generate these solutions. We will investigate the conditions when A , B and C are pair-wise co-prime. As a side result of this investigation, we will show a method of generating an infinite number of co-prime integral solutions ( A , B , C ) of the Diophantine equation a A 3 + c B 3 = C 2 for any co-prime integer pair ( a , c ) . ...