Every prime dividing the k=5 index divides the cyclotomic product

Let p be a prime congruent to 1 modulo 4, m an odd natural number not divisible by p, and q and r primes. Suppose the k=5 Dris equation 2 m squared equals sigma of p to the fifth times d1 squared q r in its already factorised form. Then every prime dividing the index d1 squared q r also divides the cyclotomic product (p+1)/2 times p squared plus p plus one times p squared minus p plus one. Equivalently, the squarefree kernel of that cyclotomic product is contained in the set of primes dividing the index. Since the index has squarefree kernel exactly the two primes q and r, this forces the squarefree kernel of the cyclotomic product to have at most two prime factors, which for all but two Euler primes up to fifteen hundred fails.