The Diophantine equation $x^4 + y^4 = 1$ in algebraic number fields

We consider the diophantine equation (*) x^p - x = y^q - y in integers (x, p, y, q). We prove that for given p and q with 2 ≤ p < q, (*) has only finitely many solutions. Assuming the abc-conjecture we can prove that p and q are bounded. In the special case p = 2 and y a prime power we are able to solve (*) completely.

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Per Salberger (2005)

Annales scientifiques de l'École Normale Supérieure

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Masao Toyoizumi (1983)

Acta Arithmetica

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