Same as to prove that exists $a$ such that $2^k|a^2+7$ Let $a_1=1$ and if $2^m|a_i^2+7$ and $2^{m+1}\not |a_i^2+7$ then $a_{i+1}=a_i+2^{m-1}$ Then $2^3|a_1^2+7$ And we have $a_i^2+7=2^{m} (2u+1)$ and $a_{i+1}^2+7=a_i^2+2^{2m-2}+2^{m}a_i+7=2^m(2u+1+a_i+2^{m-2})$and $2|2u+1+a_i+2^{m-2}$ so $2^{m+1}|a_{i+1}^2+7$

It's ISL 1990 N1: https://artofproblemsolving.com/community/c6h3293p9808.