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Only For Disciples of Diophantus

To find out three numbers, ye sages refin'd,
Your assistance be pleas'd to contribute:
The sum of each two less the number remaining
Square numbers must always exhibit.
Ye famous disciples of fam'd Diophantus
Of this you must also take care,---
That the sum of all three universally be
In all cases a true perfect square.

Solution:

Source: The Gentlemanís Mathematical Companion, 1803