## E753 -- Solution succincta et elegans problematis, quo quaeruntur tres numeri tales, ut tam summae quam differentiae binorum sint quadrata

(English Translation of Title)

Summary:

"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.

Publication:
• 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]
Documents Available:
• Original publication: E753