Can 422215686281216 be expressed as the sum or difference of two fifth powers? I have a large integer, 422215686281216.  I am looking for two fifth powers  which when added together, or subtracted from one another, equal this number.  How can I tell whether an integer  is the sum , or difference,  of two fifth powers ?
 A: You can show that sums and differences of two fifth powers cannot be congruent to $3$ modulo $11$.
Then note that your number is congruent to $3$ modulo $11$.
Conclude that you cannot write your number as a sum or difference of two fifth powers.
Added: Oh, so many opinions and helpful comments.  To clarify, I merely calculated sums and differences of fifth powers modulo $2, 3, 4,$ etc. until I found a modulus for which not all congruence classes were possible.  This happened to be $11$, and it just happened that one class left out, $3$, was the class of the given number.  This is a standard method for treating these sums-of-powers questions.  For further reading, take a look at most any introductory number theory text (let me know if you would like a more specific reference).
A: Here's something you could try if Matthew Conroy's trick didn't work.
More generally, suppose you have a positive integer $N$ and you want to write $N = x^5 - y^5$ where $x$ and $y$ are integers (this includes a sum of $5$'th powers if you take $y$ to be negative).  Note that 
$$ N = (x - y)(x^4 + x^3 y + x^2 y^2 + x y^3 + y^4)$$ 
so $x - y$ must be one of the (positive or negative) divisors of $N$.  Enumerate all divisors of $N$ (which you can do from the prime factorization); for each such divisor $d$, check whether the polynomial $(Y+d)^5 - Y^5 - N$ has an integer root.  If it does (say $r$), then $y = r$, $x = r+d$ is a solution.  If none of these has an integer root, it is not possible to write
$N$ as a sum or difference of $5$'th powers.
