That is, what is the smallest nontrivial 'd' (in absolute terms) in the equation a^n + b^n = c^n + d?

After excluding the all-1 cases, and the cases where a = c, here are the best solutions for a,b,c < 10000:

8^3 + 6^3 = 9^3 + -1

5^4 + 5^4 = 6^4 + -46

16^5 + 13^5 = 17^5 + 12

2^6 + 2^6 = 3^6 + -601

2^7 + 2^7 = 3^7 + -1931

Here are some more computational number theory results on x^3 + y^2 = z^5: http://qr.ae/RO864s

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