05 Jul 05
Originally posted by elopawnI find two pairs: 1 and 3, and 2 and 3
Find all pairs of positive integers a and b such that...
a^2 + b^2 - 7 = ab
A hint: a^2 - ab - (7-b^2) = 0
from a^2 + b^2 -7 =ab also follows that
a^2 -2ab + b^2 = 7 - ab , or
(a-b)^2= 7 - ab
that means 7 >=ab , leaving only (1,1), (1,2), (1,3), (2,2) and (2,3) to check. No heavy calculations necessary for that.