Solved with Enigma714 WLOG $(a,b,c)=1$. Choose $ n \equiv 1 \pmod{ 2\prod_{p \mid abc} (p-1) \prod_{p \mid (a+b)(b+c)(c+a)} p }$ with $n$ large enough. Define $t_n = a^n + b^n + c^n$ we want $t_n \mid t_{n+ \phi(t_n)} = t_{m}$ if this doesn't happen then exists some $q$ prime such that $v_q(a^n + b^n + c^n) > v_q(a^m + b^m + c^m)$ this is false unless $ q \mid abc $ by Euler-Fermat theorem. WLOG $q \mid a$ so $q\mid b^n + c^n \implies q \mid b+c$ by FLT since $(q,bc)=1$ by LTE $v_q(b^n+c^n)=v_q(b+c)$ so for $n$ large enough we get $v_q(a^n + b^n + c^n) = v_q(b+c) = v_q(a^m + b^m + c^m)$ contradiction. done