Equality of the sum of powers Hi everyone, I got a problem when proving lemmas for some combinatorial problems,
and it is a question about integers.
Let 
$\sum_{k=1}^m a_k^t = \sum_{k=1}^n b_k^t$ 
be an equation,
where $m, n, t, a_i, b_i$ are positive integers, and
$a_i \neq a_j$ for all $i, j$,
$b_i \neq b_j$ for all $i, j$,
$a_i \neq b_j$ for all $i, j$.

Does the equality have no solutions?

For $n \neq m$, it is easy to find solutions for $t=2$ by Pythagorean theorem,
and even for $n = m$, we have solutions like 
$1^2 + 4^2 + 6^2 + 7^2 = 2^2 + 3^2 + 5^2 + 8^2$.
For $t > 2$, similar equalities hold:
$1^2 + 4^2 + 6^2 + 7^2 + 10^2 + 11^2 + 13^2 + 16^2 = 2^2 + 3^2 + 5^2 + 8^2 + 9^2 + 12^2 + 14^2 + 15^2$
and 
$1^3 + 4^3 + 6^3 + 7^3 + 10^3 + 11^3 + 13^3 + 16^3 = 2^3 + 3^3 + 5^3 + 8^3 + 9^3 + 12^3 + 14^3 + 15^3$,
and we can extend this trick to all $t > 2$.

The question is, if we introduce one more restriction, that is,
  $|a_i - a_j| \geq 2$ and $|b_i - b_j| \geq 2$ for all $i, j$,
  is it still possible to find solutions for the equation?

For $t = 2$ we can combine two Pythagorean triples, say,
$5^2 + 12^2 + 25^2 = 7^2 + 13^2 + 24^2$,
but how about the cases for $t > 2$ and $n = m$?

One set of solutions for t = 3 is the class of numbers known as Taxicab Numbers, named after the number of a taxicab G. H. Hardy took, 1729, that Ramanujan mentioned was equal to 13 + 123 = 93 + 103.  This particular example fails, as |10 - 9| = 1 < 2, but there are other Taxicab numbers, such as:
1673 + 4363 = 2283 + 4233 = 2553 + 4143.
This might be a helpful site for your question.
-Gabriel Benamy
