2: a^2-2d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 3: If n is of the form x^2+y^2, then 3*n is not of the form x^2+y^2 and vice versa. Numbers of form x^2+y^2: No prime factors congruent to 3 (mod 4) to an odd power. 5: a^2-5d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 7: a^2-3d^2 == 0 (mod 7) implies a == 0 (mod 7) and d == 0 (mod 7) 8: a^2-8d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 10: a^2-3d^2 == 0 (mod 5) implies a == 0 (mod 5) and d == 0 (mod 5) 11: a^2-11d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 12: If n is of the form x^2+4y^2, then 3*n is not of the form x^2+4y^2 and vice versa. Numbers of form x^2+4y^2: No prime factors congruent to 3 (mod 4) to an odd power. 14: a^2-14d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 15: If n is of the form x^2+5y^2, then 3*n is not of the form x^2+5y^2 and vice versa. Numbers of form x^2+5y^2: Sum of exponents of 2, 3, 7 (mod 20) prime factors is even. 17: a^2-17d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 18: Substituting a = 3*p, c = 3*q reduces the case to n = 2. 19: a^2-3d^2 == 0 (mod 19) implies a == 0 (mod 19) and d == 0 (mod 19) 20: a^2-20d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 21: If n is of the form x^2+7y^2, then 3*n is not of the form x^2+7y^2 and vice versa. Numbers of form x^2+7y^2: No prime factors congruent to 3, 5, 13 (mod 14) to an odd power. 23: a^2-23d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 26: a^2-26d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 27: Substituting a = 3*p, c = 3*q reduces the case to n = 3. 28: a^2-3d^2 == 0 (mod 7) implies a == 0 (mod 7) and d == 0 (mod 7) 29: a^2-29d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 30: If n is of the form x^2+10y^2, then 3*n is not of the form x^2+10y^2 and vice versa. Numbers of form x^2+10y^2: No prime factors congruent to 3, 17, 21, 27, 29, 31, 33, 39 (mod 40) to an odd power. 31: a^2-3d^2 == 0 (mod 31) implies a == 0 (mod 31) and d == 0 (mod 31) 32: a^2-32d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 34: a^2-3d^2 == 0 (mod 17) implies a == 0 (mod 17) and d == 0 (mod 17) 35: a^2-35d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 38: a^2-38d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 39: If n is of the form x^2+13y^2, then 3*n is not of the form x^2+13y^2 and vice versa. Numbers of form x^2+13y^2: No prime factors congruent to 3, 5, 21, 23, 27, 33, 35, 37, 41, 43, 45, 51 (mod 52) to an odd power. 40: a^2-3d^2 == 0 (mod 5) implies a == 0 (mod 5) and d == 0 (mod 5) 41: a^2-41d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 42: If n is of the form x^2+14y^2, then 3*n is not of the form x^2+14y^2 and vice versa. Numbers of form x^2+14y^2: Sum of exponents of 3, 5, 13, 19, 27, 45 (mod 56) prime factors is even. 43: a^2-3d^2 == 0 (mod 43) implies a == 0 (mod 43) and d == 0 (mod 43) 44: a^2-44d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 45: Substituting a = 3*p, c = 3*q reduces the case to n = 5. 47: a^2-47d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 48: If n is of the form x^2+16y^2, then 3*n is not of the form x^2+16y^2 and vice versa. Numbers of form x^2+16y^2: No prime factors congruent to 3 (mod 4) to an odd power. 50: a^2-50d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 51: If n is of the form x^2+17y^2, then 3*n is not of the form x^2+17y^2 and vice versa. Numbers of form x^2+17y^2: Sum of exponents of 3, 7, 11, 23, 27, 31, 39, 63 (mod 68) prime factors is even. 53: a^2-53d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 55: a^2-3d^2 == 0 (mod 5) implies a == 0 (mod 5) and d == 0 (mod 5) 56: a^2-56d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 57: If n is of the form x^2+19y^2, then 3*n is not of the form x^2+19y^2 and vice versa. Numbers of form x^2+19y^2: No prime factors congruent to 2, 3, 8, 10, 12, 13, 14, 15, 18 (mod 19) to an odd power. 58: a^2-3d^2 == 0 (mod 29) implies a == 0 (mod 29) and d == 0 (mod 29) 59: a^2-59d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 60: If n is of the form x^2+20y^2, then 3*n is not of the form x^2+20y^2 and vice versa. Numbers of form x^2+20y^2: Sum of exponents of 2, 3, 7 (mod 20) prime factors is even. 62: a^2-62d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 63: Substituting a = 3*p, c = 3*q reduces the case to n = 7. 65: a^2-65d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 66: If n is of the form x^2+22y^2, then 3*n is not of the form x^2+22y^2 and vice versa. Numbers of form x^2+22y^2: No prime factors congruent to 3, 5, 7, 17, 27, 37, 39, 41, 45, 53, 57, 59, 63, 65, 67, 69, 73, 75, 79, 87 (mod 88) to an odd power. 67: a^2-3d^2 == 0 (mod 67) implies a == 0 (mod 67) and d == 0 (mod 67) 68: a^2-68d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 70: a^2-3d^2 == 0 (mod 7) implies a == 0 (mod 7) and d == 0 (mod 7) 71: a^2-71d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 72: Substituting a = 3*p, c = 3*q reduces the case to n = 8. 74: a^2-74d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 75: If n is of the form x^2+25y^2, then 3*n is not of the form x^2+25y^2 and vice versa. Numbers of form x^2+25y^2: No prime factors congruent to 3, 7, 11, 19 (mod 20) to an odd power. 76: a^2-3d^2 == 0 (mod 19) implies a == 0 (mod 19) and d == 0 (mod 19) 77: a^2-77d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 79: a^2-3d^2 == 0 (mod 79) implies a == 0 (mod 79) and d == 0 (mod 79) 80: a^2-80d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 82: a^2-3d^2 == 0 (mod 41) implies a == 0 (mod 41) and d == 0 (mod 41) 83: a^2-83d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 84: If n is of the form x^2+28y^2, then 3*n is not of the form x^2+28y^2 and vice versa. Numbers of form x^2+28y^2: No prime factors congruent to 3, 5, 13, 17, 19, 27 (mod 28) to an odd power. 85: a^2-3d^2 == 0 (mod 5) implies a == 0 (mod 5) and d == 0 (mod 5) 86: a^2-86d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 87: If n is of the form x^2+29y^2, then 3*n is not of the form x^2+29y^2 and vice versa. Numbers of form x^2+29y^2: Sum of exponents of prime factors generated by 3x^2+2xy+10y^2 and Sum of exponents of prime factors generated by 2x^2+2xy+15y^2 are of same parity. Sum of exponents of prime factors generated by 3x^2+2xy+10y^2 + Sum of exponents of prime factors generated by 5x^2+2xy+6y^2 != 1. 89: a^2-89d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 90: Substituting a = 3*p, c = 3*q reduces the case to n = 10. 91: a^2-3d^2 == 0 (mod 7) implies a == 0 (mod 7) and d == 0 (mod 7) 92: a^2-92d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 93: If n is of the form x^2+31y^2, then 3*n is not of the form x^2+31y^2 and vice versa. Numbers of form x^2+31y^2: No prime factors congruent to 3, 11, 13, 15, 17, 21, 23, 27, 29, 37, 43, 53, 55, 57, 61 (mod 62) to an odd power. 95: a^2-95d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 98: a^2-98d^2 == 0 (mod 3) implies a == 0 (mod 3) and d == 0 (mod 3) 99: Substituting a = 3*p, c = 3*q reduces the case to n = 11.