x^4 - y^4 = 2007196
To subtract to give an even number x^4 and y^4 must be either both
odd or both even.
Therefore x and y must be both odd or both even.
2007196 breaks into prime factors as 2·2·41·12239
x^4 - y^4 breaks into prime polynomial factors as (x-y)(x+y)(x²+y²)
where (x-y) is the smallest factor, (x+y) the middle-size factor,
and x²+y² the largest factor. Thus the largest prime factor 12239
must be a factor of x²+y², and only 1,2,and 41 can be factors of
x-y and x+y
(x-y)(x+y)(x²+y²) = 2·2·41·12239
The only way x-y and x+y can both be odd is for x-y = 1 and x+y = 41.
Solving that system gives x=21, y=20, x²+y² = 21²+20² = 841
Then (x-y)(x+y)(x²+y²) = 1·41·841 which = 34481, not 2007196.
The only way x-y and x+y can both be even is for x-y = 2 and
x+y = 2·41 = 82. Solving that system gives x=42, y=40,
x²+y² = 42²+40² = 3364
Then (x-y)(x+y)(x²+y²) = 2·82·3364 which = 551696, not 2007196.
So there are no solutions, so the number of solutions is zero.
Edwin
