E774 -- Investigatio binorum numerorum formae xy(x4-y4), quorum productum sive quotus sit quadratum
(An investigation of two numbers of the form xy(x4-y4), of which the product and the quotient will be a square)
Summary:
Euler previously solved the following problem in Chapter XV of the Novi Commentarii: to find two rational numbers \(\frac{x}{z}\), \(\frac{y}{z}\) such that their product increased or decreased by their sum or difference will produce a square number. Or equivalently, to find integers \(x\), \(y\), \(z\) such that \(xy \pm z(x+y)\) and \(xy \pm z(x-y)\) are all squares. In this paper, he describes a more elegant solution, which first involves finding when \(xy(x^4-y^4)\) is a square. He then uses a double-quadratic technique to find a chain of solutions, given one.
Publication:
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Originally published in Memoires de l'academie des sciences de St.-Petersbourg 11, 1830, pp. 31-45
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Opera Omnia: Series 1, Volume 5, pp. 116 - 130
- Reprinted in Commentat. arithm. 2, 1849, pp. 438-446 [E774a]
Documents Available:
- Original publication: E774
- English translation (Chris Goff): E774
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