On the septimic character of 2 and 3

Extract

Let e be an integer greater than 1 and let pbe a prime such that p ≡ 1 (mod e). Criteria for 2 to be a residue of degree e modulo p have been obtained in various forms for e = 2, 3, 4 and 5. Thus, Euler and Lagrange proved that 2 is a quadratic residue mod p if and only if p ≡ ± 1 (mod 8). Gauss showed that 2 is a cubic residue mod p if and only if p is representable as p = l² + 27m² with integers l and m, and that 2 is a quartic residue mod p if and only if p is representable as p = 1² + 64m². Lehmer (5) and Alderson (1) have found similar but more complicated conditions for 2 to be a quintic residue mod p. There are analogous results about 3. For example, it follows from quadratic reciprocity that 3 is a quadratic residue mod p if and only if p ≡ ± 1 (mod 12), and Jacobi(4) showed that 3 is a cubic residue mod p if and only if 4p is representable as l² + 3⁵m².