Determine at least three pairs of positive integers (x,y) with x< y such that xy(x+y) is not divisible by 7, but (x+y)⁷ - x⁷ - y⁷ is divisible by 7⁷

Does the given problem generate an infinite number of pairs as solutions?

Can you do this in a short time using pen and paper, and eventually a hand calculator, but no computer programs?

Remember that xy(x+y) is NOT divisible by 7, but:
(x+y)^7 - x^7 - y^7 IS divisible by 7^7.

(i) Start by noting the identity:
(x + y)^7 - x^7 - y^7 = 7xy(x + y)(x^2 + xy + y^2)^2.
(What does this yield?)

(ii) Apply Euler Fermat Theorem.

 

Edited on April 19, 2007, 1:24 am

Posted by K Sengupta on 2007-04-19 01:21:01