Question Number 226569 by hardmath last updated on 05/Dec/25
$$\mathrm{a}^{\mathrm{4}} \:+\:\mathrm{b}^{\mathrm{4}} \:+\:\mathrm{c}^{\mathrm{4}} \:=\:\mathrm{2d}^{\mathrm{2}} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{has}\:\mathrm{an}\:\mathrm{infinite} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{natural}\:\mathrm{solutions} \\ $$
Commented by mr W last updated on 06/Dec/25
$${and}\:{a}\neq{b}\neq{c}\:? \\ $$
Commented by hardmath last updated on 06/Dec/25
$$\mathrm{no}\:\mathrm{dear}\:\mathrm{professor} \\ $$
Answered by peace2 last updated on 06/Dec/25
$${d}={x}^{\mathrm{2}} ;{x}\in\mathbb{N} \\ $$$${let}\:{S}\:{set}\:{of}\:{solution} \\ $$$${we}\:{have}\:{A}=\:\left\{\left(\mathrm{0},{x},{x},{x}^{\mathrm{2}} \right),{x}\in\mathbb{N}\right\}\subseteq{S} \\ $$