English Title

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

Authors

Leonhard Euler

Enestrom Number

753

Fuss Index

43

Original Language

Latin

Published as

Journal article

Published Date

1818

Written Date

1780

Content Summary

The problem, according to Euler: "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 xy = (pq)2. In the same way, setting x=rr+ss and z=2rs, then x+z=(r+s)2 and xz=(rs)2. In addition to these four conditions being satisfied, it must be that rr+ss = pp+qq. Then, two additional conditions must be added, that y+z = 2pq+2rs and yz = 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.

Original Source Citation

Mémoires de l'académie des sciences de St.-Petersbourg, Volume 6, pp. 54-65.

Opera Omnia Citation

Series 1, Volume 5, pp.20-27.

Record Created

2018-09-25

Included in

Mathematics Commons

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