Displaying similar documents to “On the Diophantine equation q n - 1 q - 1 = y ”

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 ) . ...

A ternary Diophantine inequality over primes

Roger Baker, Andreas Weingartner (2014)

Acta Arithmetica

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Let 1 < c < 10/9. For large real numbers R > 0, and a small constant η > 0, the inequality | p ₁ c + p ₂ c + p ₃ c - R | < R - η holds for many prime triples. This improves work of Kumchev [Acta Arith. 89 (1999)].

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 a ternary Diophantine problem with mixed powers of primes

Alessandro Languasco, Alessandro Zaccagnini (2013)

Acta Arithmetica

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Let 1 < k < 33/29. We prove that if λ₁, λ₂ and λ₃ are non-zero real numbers, not all of the same sign and such that λ₁/λ₂ is irrational, and ϖ is any real number, then for any ε > 0 the inequality | λ ₁ p ₁ + λ ₂ p ² ₂ + λ ₃ p ₃ k + ϖ | ≤ ( m a x j p j ) - ( 33 - 29 k ) / ( 72 k ) + ε has infinitely many solutions in prime variables p₁, p₂, p₃.

A note on the number of S -Diophantine quadruples

Florian Luca, Volker Ziegler (2014)

Communications in Mathematics

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Let ( a 1 , ⋯ , a m ) be an m -tuple of positive, pairwise distinct integers. If for all 1 ≤ i < j ≤ m the prime divisors of a i a j + 1 come from the same fixed set S , then we call the m -tuple S -Diophantine. In this note we estimate the number of S -Diophantine quadruples in terms of | S | = r .