give non trivial solutions?

Question

Give non trivial solutions to the equation
a^(2n) - b^(2n) = c^(n) - d^(n).
one example is
3^(6) - 1^(6) = 12^(3) - 10^(3).

Answer 1

there is no way to solve this equation but hit and trial .

Look at the power! n and 4 variables.
No one can ever solve it.

But a computer program can be created to solve it.

You should wait for an hour or so till I create it.

Ps- are all a, b, c, d and n integers or natural numbers?

visit http://www.geocities.com/himanshu_kapoor00/calc.html and get solutions!!!!!!!!!!!!!

Answer 2

It does seem a little trivial, but you used 1 in your example.

For any x

x^2n - 1^2n = (x^2)^n - 1^n

Another one: since every odd number 2n + 1 is the difference between (n + 1)^2 and n^2, choose any n even and m odd, and you can get

n^4 - m^4 = p^2 - q^2, where p and q are chosen to give the proper odd number.

I suspect this can be generalized to higher degrees, but since I just generated a whole class of nontrivial solutions I will rest now.