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?*

Start by expanding (x+y)^7-x^7-y^7 and simplifying:

7x^6*y + 21x^5*y^2 + 35x^4*y^3 + 35x^3*y^4 + 21x^2*y^5 + 7x*y^6

Factoring that expression yields 7xy(x+y)*(x^2+xy+y^2)^2

For that expression to be a multiple of 7^7 with xy(x+y) coprime to 7, then x^2+xy+y^2 must be a multiple of 343.

A brute force search of (y,x) pairs with 342 >= y > x >= 1 finds 294 different solutions, such as (18,1), (20,17), (36,2), (52,41), etc.

Each of those solutions is a generator for more solutions since if (x,y) is a solution then (x+343,y) and (x,y+343) are also solutions.