Fermat's theorem implies that 2-1)/2 = +1 (mod p).] 11. apply theorem 5.5 to obtain two solutions to each of the quadratic congruences x -1 (mod 29) and x = -1 (mod 37). 12. show that ilp - 4k+ 3 is prime and a + b 0 (mod p), then amb0 (mod p). hint: if a 0 (mod p), then there exists an integer c such that ac i (mod p); use this fact to contradict theorem 5.5.)
theorem 5.5. the quadratic congruence x + 1 = 0 (mod p), where p is an odd prime, has a solution if and only if p = 1 (mod 4).