On the Diophantine equation F(x)=G(y)
Sz. Tengely (2003)
Acta Arithmetica
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Sz. Tengely (2003)
Acta Arithmetica
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Manisha Kulkarni, B. Sury (2005)
Acta Arithmetica
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M. Filaseta, F. Luca, P. Stănică, R. G. Underwood (2007)
Acta Arithmetica
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Wolfgang M. Schmidt (2012)
Acta Arithmetica
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Susil Kumar Jena (2014)
Bulletin of the Polish Academy of Sciences. Mathematics
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The Diophantine equation A² + nB⁴ = C³ has infinitely many integral solutions A, B, C for any fixed integer n. The case n = 0 is trivial. By using a new polynomial identity we generate these solutions, and then give conditions when the solutions are pairwise co-prime.
S. Akhtari, A. Togbé, P. G. Walsh (2009)
Acta Arithmetica
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Makoto Nagata (2003)
Acta Arithmetica
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Utz, W.R. (1982)
International Journal of Mathematics and Mathematical Sciences
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Csaba Rakaczki (2003)
Acta Arithmetica
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L. Carlitz (1969)
Acta Arithmetica
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Pingzhi Yuan, Jiagui Luo (2010)
Acta Arithmetica
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Ernst, Bruno (1996)
General Mathematics
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H. Kleiman (1976)
Journal für die reine und angewandte Mathematik
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W. J. Ellison (1970-1971)
Séminaire de théorie des nombres de Bordeaux
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Jianhua Chen (2001)
Acta Arithmetica
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Yang Hai, P. G. Walsh (2010)
Acta Arithmetica
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Shin-ichi Katayama, Claude Levesque (2003)
Acta Arithmetica
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Maciej Ulas (2007)
Colloquium Mathematicae
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Let f ∈ ℚ [X] and deg f ≤ 3. We prove that if deg f = 2, then the diophantine equation f(x)f(y) = f(z)² has infinitely many nontrivial solutions in ℚ (t). In the case when deg f = 3 and f(X) = X(X²+aX+b) we show that for all but finitely many a,b ∈ ℤ satisfying ab ≠ 0 and additionally, if p|a, then p²∤b, the equation f(x)f(y) = f(z)² has infinitely many nontrivial solutions in rationals.
Alan Filipin (2009)
Acta Arithmetica
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A. Togbe, P. M. Voutier, P. G. Walsh (2005)
Acta Arithmetica
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