# Question: is there a pattern when (p-1)/2 ! = +1 or -1 mod p, for p = -1 mod 4? # Sept 17, 2007 > ; > for i from 2 to 1000 do p := ithprime(i); if `mod`(p, 4) = 3 then a := 1; for k to (p-1)*1/2 do a := `mod`(a*k, p) end do; print(p, a) end if end do; 3, 1 7, 6 11, 10 19, 18 23, 1 31, 1 43, 42 47, 46 59, 1 67, 66 71, 1 79, 78 83, 1 103, 102 107, 1 127, 126 131, 130 139, 1 151, 1 163, 162 167, 1 179, 178 191, 190 199, 198 211, 1 223, 1 227, 226 239, 1 251, 1 263, 262 271, 1 283, 1 307, 1 311, 1 331, 1 347, 346 359, 1 367, 366 379, 1 383, 382 419, 418 431, 430 439, 1 443, 442 463, 1 467, 1 479, 478 487, 1 491, 490 499, 1 503, 502 523, 522 547, 1 563, 562 571, 570 587, 1 599, 598 607, 606 619, 618 631, 630 643, 1 647, 1 659, 1 683, 682 691, 690 719, 1 727, 726 739, 738 743, 742 751, 1 787, 786 811, 1 823, 822 827, 1 839, 838 859, 1 863, 862 883, 1 887, 886 907, 1 911, 1 919, 1 947, 946 967, 1 971, 1 983, 1 991, 990 1019, 1018 1031, 1 1039, 1 1051, 1050 1063, 1 1087, 1086 1091, 1090 1103, 1 1123, 1122 1151, 1150 1163, 1 1171, 1 1187, 1186 1223, 1 1231, 1 1259, 1 1279, 1 1283, 1 1291, 1290 1303, 1 1307, 1 1319, 1318 1327, 1 1367, 1366 1399, 1 1423, 1422 1427, 1 1439, 1 1447, 1 1451, 1450 1459, 1 1471, 1 1483, 1 1487, 1486 1499, 1498 1511, 1510 1523, 1 1531, 1 1543, 1 1559, 1 1567, 1 1571, 1570 1579, 1578 1583, 1582 1607, 1 1619, 1 1627, 1 1663, 1662 1667, 1666 1699, 1 1723, 1722 1747, 1746 1759, 1 1783, 1782 1787, 1 1811, 1 1823, 1822 1831, 1 1847, 1 1867, 1866 1871, 1870 1879, 1 1907, 1906 1931, 1930 1951, 1950 1979, 1 1987, 1 1999, 1 2003, 2002 2011, 1 2027, 1 2039, 2038 2063, 2062 2083, 1 2087, 1 2099, 1 2111, 2110 2131, 2130 2143, 2142 2179, 1 2203, 2202 2207, 1 2239, 1 2243, 1 2251, 1 2267, 1 2287, 2286 2311, 2310 2339, 1 2347, 2346 2351, 1 2371, 2370 2383, 2382 2399, 1 2411, 1 2423, 2422 2447, 2446 2459, 1 2467, 1 2503, 2502 2531, 2530 2539, 1 2543, 1 2551, 2550 2579, 2578 2591, 2590 2647, 1 2659, 2658 2663, 1 2671, 1 2683, 2682 2687, 1 2699, 1 2707, 1 2711, 2710 2719, 2718 2731, 1 2767, 2766 2791, 1 2803, 2802 2819, 2818 2843, 1 2851, 1 2879, 2878 2887, 2886 2903, 1 2927, 1 2939, 2938 2963, 2962 2971, 1 2999, 2998 3011, 3010 3019, 1 3023, 1 3067, 1 3079, 3078 3083, 3082 3119, 3118 3163, 3162 3167, 3166 3187, 1 3191, 3190 3203, 1 3251, 1 3259, 3258 3271, 1 3299, 1 3307, 3306 3319, 3318 3323, 3322 3331, 1 3343, 1 3347, 1 3359, 3358 3371, 3370 3391, 3390 3407, 3406 3463, 1 3467, 1 3491, 1 3499, 1 3511, 3510 3527, 3526 3539, 1 3547, 3546 3559, 3558 3571, 1 3583, 3582 3607, 1 3623, 3622 3631, 1 3643, 3642 3659, 3658 3671, 3670 3691, 3690 3719, 1 3727, 1 3739, 1 3767, 1 3779, 1 3803, 1 3823, 3822 3847, 1 3851, 3850 3863, 3862 3907, 1 3911, 1 3919, 1 3923, 1 3931, 1 3943, 1 3947, 3946 3967, 3966 4003, 4002 4007, 4006 4019, 1 4027, 4026 4051, 1 4079, 4078 4091, 4090 4099, 1 4111, 1 4127, 4126 4139, 1 4159, 1 4211, 1 4219, 1 4231, 1 4243, 4242 4259, 1 4271, 4270 4283, 4282 4327, 1 4339, 4338 4363, 4362 4391, 1 4423, 4422 4447, 4446 4451, 4450 4463, 1 4483, 4482 4507, 4506 4519, 4518 4523, 4522 4547, 4546 4567, 4566 4583, 4582 4591, 4590 4603, 1 4639, 1 4643, 4642 4651, 4650 4663, 4662 4679, 1 4691, 4690 4703, 1 4723, 4722 4751, 1 4759, 1 4783, 1 4787, 4786 4799, 1 4831, 4830 4871, 1 4903, 1 4919, 1 4931, 1 4943, 1 4951, 1 4967, 1 4987, 4986 4999, 4998 5003, 1 5011, 5010 5023, 5022 5039, 1 5051, 5050 5059, 1 5087, 5086 5099, 1 5107, 1 5119, 1 5147, 1 5167, 5166 5171, 1 5179, 1 5227, 1 5231, 1 5279, 1 5303, 1 5323, 1 5347, 5346 5351, 5350 5387, 1 5399, 1 5407, 1 5419, 5418 5431, 5430 5443, 5442 5471, 1 5479, 1 5483, 5482 5503, 5502 5507, 1 5519, 5518 5527, 1 5531, 1 5563, 1 5591, 1 5623, 5622 5639, 1 5647, 5646 5651, 1 5659, 1 5683, 1 5711, 5710 5743, 5742 5779, 5778 5783, 5782 5791, 5790 5807, 5806 5827, 1 5839, 5838 5843, 5842 5851, 5850 5867, 5866 5879, 5878 5903, 5902 5923, 1 5927, 1 5939, 1 5987, 1 6007, 1 6011, 1 6043, 6042 6047, 1 6067, 1 6079, 6078 6091, 1 6131, 1 6143, 6142 6151, 1 6163, 1 6199, 1 6203, 6202 6211, 1 6247, 1 6263, 6262 6271, 1 6287, 1 6299, 1 6311, 6310 6323, 6322 6343, 6342 6359, 6358 6367, 6366 6379, 6378 6427, 6426 6451, 6450 6491, 1 6547, 1 6551, 6550 6563, 1 6571, 1 6599, 6598 6607, 6606 6619, 6618 6659, 1 6679, 1 6691, 6690 6703, 1 6719, 6718 6763, 6762 6779, 1 6791, 6790 6803, 1 6823, 6822 6827, 6826 6863, 6862 6871, 6870 6883, 6882 6899, 1 6907, 6906 6911, 1 6947, 6946 6959, 1 6967, 6966 6971, 6970 6983, 6982 6991, 1 7019, 1 7027, 1 7039, 1 7043, 1 7079, 7078 7103, 7102 7127, 1 7151, 7150 7159, 7158 7187, 7186 7207, 7206 7211, 1 7219, 1 7243, 7242 7247, 1 7283, 7282 7307, 7306 7331, 7330 7351, 7350 7411, 7410 7451, 1 7459, 1 7487, 7486 7499, 7498 7507, 1 7523, 1 7547, 1 7559, 1 7583, 1 7591, 7590 7603, 1 7607, 7606 7639, 1 7643, 7642 7687, 7686 7691, 1 7699, 1 7703, 7702 7723, 7722 7727, 7726 7759, 7758 7823, 1 7867, 1 7879, 7878 7883, 7882 7907, 7906 7919, 7918 > #