Find the minimum odd integer(positive) such that:

(1) for some positive integers .

(2) for some positive integers .

(3) .

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- September 12th 2010, 07:34 AMShanksFind the minimum integer satisying certain condition
Find the minimum odd integer(positive) such that:

(1) for some positive integers .

(2) for some positive integers .

(3) . - September 12th 2010, 09:47 AMwonderboy1953
- September 13th 2010, 05:39 AMShanks
I think， you misread the problem, we are supposed to find the minimum number satisfying all the three condition. maybe that is my fault, but thanks anyway.

- September 13th 2010, 12:32 PMOpalg
From (2), is a Pythagorean triple. If this triple is primitive then there are integers s, t such that , and . So we can try taking and . In that case, (3) says that . Therefore . This says that is a square triangular number. The smallest such number apart from 1 is . That suggests taking t = 6 and s – t = 9, so that s = 15.

That gives the solution , with . This certainly satisfies all the conditions (1), (2), (3), but I do not know whether it is the smallest solution apart from a = 5.