On consecutive quadratic non-residues: a conjecture of Issai Schur

Issai Schur once asked if it was possible to determine a bound, preferably using elementary methods, such that for all prime numbers p greater than the bound, the greatest number of consecutive quadratic non-residues modulo p is always less than p1/2. This paper uses elementary methods to prove that 13 is the only prime number for which the greatest number of consecutive quadratic non-residues modulo p exceeds p1/2.