Is 406558142992290754819586 the sum of two fourth powers? I suspect that this number is the sum of two fourth powers. Can anyone use Wolfram Mathematica , SAGE , or some computer program, to check whether this number is the sum of two fourth powers ?
The complete factorization of this number is given by : 406558 142992 290754 819586 = 2 × 17 x 17 × 21529 × 10 052297 × 3250 166849 Notice that all the factors are of the form 16n+1 or 16n+9 ( they are all congruent to 1 mod 8 ). Hence this number may be the sum of two fourth powers.
No; your number is congruent to $11$ modulo $13$, whereas fourth powers are congruent to either $0$, $1$, $3$ or $9$ modulo $13$.