Pell’s Equation

Marcin Acewicz; Karol Pąk

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These are expository notes that accompany my talk at the 25th Journées Arithmétiques, July 2–6, 2007, Edinburgh, Scotland. I aim to shed light on the following two questions: (i) Given a Diophantine equation, what information can be obtained by following the strategy of Wiles’ proof of Fermat’s Last Theorem? ...

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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.

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We consider Diophantine quintuples a, b, c, d, e. These are sets of positive integers, the product of any two elements of which is one less than a perfect square. It is conjectured that there are no Diophantine quintuples; we improve on current estimates to show that there are at most $5.441 \cdot 10^{26}$ Diophantine quintuples.

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This lecture is a survey of recent results in the theory of diophantine equations, especially for dimension 1. The unit equation and its generalizations are examined in detail, as well as Baker's theory and the consequences of the abc-conjecture.