Consider the equation $$ x^3+y^3+z^3=(x+y+z)^3 $$ for triples of integers $(x, y, z) $.
I noticed that this has infinitely many solutions: $ x, y $ arbitrary and $ z=-y $.
Are there more solutions?
2026-04-23 21:56:11.1776981371
Integer solutions of $ x^3+y^3+z^3=(x+y+z)^3 $
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$$(x+y+z)^3-(x^3+y^3+z^3)=3(x+y)(y+z)(z+x)$$
so the only solutions are the ones the OP observed and their cyclically symmetric counterparts. There's no essential number theory going on here, just an algebraic identity.