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} - x^{7} - y^{7} is divisible by 7^{7}

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**