Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.7 AIME, harder Prove it

5. Prove: The indeterminate equation
$(7 a+1) x^{3}+(7 b+2) y^{3}+(7 c+4) z^{3}+(7 d+1) x y z=0$

has only the trivial solution $x=y=z=0$ .

Solution

5. Let's assume $(x, y, z)=1$ , and note that $u^{3} \equiv 0, \pm 1(\bmod 7)$ . If there are non-trivial solutions, consider the congruence equation modulo 7, and discuss the cases $(z, 7)=1,(z, 7)=7$ separately.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.