E753 -- Solution succincta et elegans problematis, quo quaeruntur tres numeri tales, ut tam summae quam differentiae binorum sint quadrata
(English Translation of Title)
"Let x, y and z be the three numbers being sought, of which the largest is x and the smallest z,
and let x = pp + qq and y = 2pq, so that x + y =(p + q)2 and
x - y = (p - q)2. In the same way, setting x = rr + ss and z = 2rs,
then x + z = (r + s)2 and x - z = (r - s)2. In addition to these four
conditions being satisfied, it must be that rr + ss = pp + qq. Then, two additional conditions msut be added,
that y + z = 2pq + 2rs and y - z = 2pq - 2rs must both be squares."
Euler gets x = 50, y = 50, z = 14, then x = 733025, y = 488000, z = 418304.
Then, characteristically, he proposes a slightly different problem (section 16) and solves it by the same means.
Originally published in Mémoires de l'académie des sciences de St.-Petersbourg 6, 1818, pp. 54-65
Opera Omnia: Series 1, Volume 5, pp. 20 - 27
- Reprinted in Commentat. arithm. 2, 1849, pp. 392-396 [E753a]
- Original publication: E753
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