There is no Diophantine quintuple. (English) Zbl 1430.11044
Summary: A set of \(m\) positive integers \(\{a_1, a_2, \ldots , a_m\}\) is called a Diophantine \(m\)-tuple if \( a_i a_j + 1\) is a perfect square for all \(1 \le i < j \le m\). Dujella proved that there is no Diophantine sextuple and that there are at most finitely many Diophantine quintuples. In particular, a folklore conjecture concerning Diophantine \(m\)-tuples states that no Diophantine quintuple exists at all. We prove this conjecture.
MSC:
| 11D09 | Quadratic and bilinear Diophantine equations |
| 11J86 | Linear forms in logarithms; Baker’s method |
| 11Y50 | Computer solution of Diophantine equations |
| 11J68 | Approximation to algebraic numbers |
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