Allan Wechsler <acwacw@gmail.com> wrote:

Michael Collins <mjcollins10@gmail.com> wrote:

...

I suspect that, more generally, a sum a_1 + ... + a_n with each a_k being either k^2 or -k^2 can never be zero?

Ah, no, I thought of that, and there are examples! 1 + 4 - 9 + 16 - 25 - 36 + 49 = 0

Here is another: 1 - 4 - 9 + 16 - 25 + 36 + 49 - 64 = 0

Apparently these have to end with the square of a number that's congruent to 0 or 3 mod 4. Can anyone find a counterexample? The number of solutions, starting with 0, with the 1 positive, are: 1,0,0,0,0,0,0,1,1,0,0,1,5,0,0,43,57,0,0,239,430,0,0,2904,5419 I see that it's already in OEIS. A083527. Then I tried it with cubes. Here's the first solution:

1 + 8 - 27 + 64 - 125 - 216 - 343 + 512 + 729 - 1000 - 1331 + 1728

Again, apparently these have to end with the cube of a number that's congruent to 0 or 3 mod 4. The numbers of solutions, starting with 12, with the 1 positive, are: 1,0,0,2,1,0,0,2,62,0,0,268,356. A113263. I also tried 4th powers. And 5th. And 1st. A058377. Again, congruent to 0 or 3 mod 4. Powers of 2? No solutions, unsurprisingly. Fibonacci series? It's past my bedtime so I'll check tomorrow.