The Dris index at exponent five is divisible by three

Let p and s be natural numbers with s nonzero, and suppose the first Dris equation at exponent five holds: twice the square of m equals the sum of the divisors of p to the fifth times s, where m is a natural number. Assume p is a prime different from two. Then three divides s. Indeed the sum of the six terms 1 plus p plus p squared plus p cubed plus p to the fourth plus p to the fifth is always divisible by three when p is a prime other than three: if p is one modulo three all six terms are one modulo three, and if p is minus one modulo three the six terms alternate one and minus one and sum to zero. Hence three divides twice the square of m, so three divides m squared and therefore three divides m as well. This is an unconditional consequence of the first Dris equation alone, with no hypothesis on the square-free structure of the index.