E.H. Dudeney's recreational math classic "Amusements in Mathematics" (1917) can get quite arcane quite quickly sometimes. I could really use some help picking apart his train of logic in answering one of his questions.
The challenge is to find a pair of integers x,y, such that 0<x,y<10, and x^2 + y^2 + 1xy = a perfect square.
The answer Dudeney gives is comparatively generous by his usual standards: There are two solutions with numbers less than ten: 3 and 5, and 7 and 8. The general solution to this problem is as follows: Calling the numbers a and b, we have: X2 + y2 + xy = sq = (x-iy)2 = x2 - 2xiy + y2i2 Y+x = -2xi + yi2 Y=(x(2i+1)/i2-1) X=i2-1 and y=2i+1
Now, I can see how with a bit of shimmying, you can say that x^2 + xy + y^2 means that (x - iy)^2 must also be square -- for at least i = -1/2 -- by adding (y/2)^2 to both sides, and taking the y^2 to the RHS.
I can also see how you'd unpack x+y = yi^2 - 2xi to give you x = i^2-1 and y = 2i-1, by adding (-y+2ax) to both sides and simplifying aggressively.
What I just can't wrap my head around is the leap from x^2 - 2axy + a^2y^2 to x+y = ya^2 2ax
Can anyone help my poor, old addled brain see where Dudeney is coming from?