E739 -- Regula facilis problemata Diophantea per numeros integros expedite resolvendi

(An easy rule for Diophantine problems which are to be resolved quickly by integral numbers)

Summary:

Euler returns to the problem of making formulas of the form axx + bx + g into squares. He generalizes to try to find a and b solving axx + bx + g = zyy + hy + t. Again, it seems to rely on an initial solution and a clever application of solutions to Pell’s equation. He does some nice examples:

To find all triagonal numbers that, at the same time, are squares.
To find all square numbers that, diminshed by one, are triangular numbers. Some such squares are 1, 4, 16, etc.
To find those triangular numbers that, tripled, are again triangular numbers, such as 1, whose triple is exactly a triangular number.

According to the records, it was presented to the St. Petersburg Academy on May 4, 1778.

Publication:

Originally published in Memoires de l'academie des sciences de St.-Petersbourg 4, 1813, pp. 3-17
Opera Omnia: Series 1, Volume 4, pp. 406 - 417
Reprinted in Commentat. arithm. 2, 1849, pp. 263-269 [E739a]

Documents Available:

Original publication: E739

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