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)(x�+y�) 
where (x-y) is the smallest factor, (x+y) the middle-size factor,
and x�+y� the largest factor. Thus the largest prime factor 12239 
must be a factor of x�+y�, and only 1,2,and 41 can be factors of
x-y and x+y 

(x-y)(x+y)(x�+y�) = 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, x�+y� = 21�+20� = 841
Then (x-y)(x+y)(x�+y�) = 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, 
x�+y� = 42�+40� = 3364
Then (x-y)(x+y)(x�+y�) = 2�82�3364 which = 551696, not 2007196.

So there are no solutions, so the number of solutions is zero.

Edwin