Table 1: Parameter sets that do not exist by Criterion 1. and parameters with asterisk indicate new results.
 ( 𝑣 , 𝑘 , 𝜆 ) 𝑚 𝑝 | 𝐺 / 𝑁 | Factoring of 𝑝 in ℤ [ 𝜁 | 𝐺 / 𝑁 | ] No. of groups of order 𝑣 Solutions in 𝐺 / 𝑁 1 (115, 19, 3) 4 2 23 Remark 2.3 1 − 4 + ⟨ 𝑥 ⟩ 2 (1333, 37, 1) 6 2, 3 43 𝑎 ≡ − 1 ( m o d 4 3 ) 𝑎 = 2 7 , 3 2 1 1 − 6 + ⟨ 𝑥 ⟩ 3 (221, 45, 9) 6 2, 3 17 𝑎 ≡ − 1 ( m o d 1 7 ) 𝑎 = 2 4 , 3 8 1 − 6 + 3 ⟨ 𝑥 ⟩ 4 (145, 64, 28) 6 2, 3 29 𝑎 1 4 ≡ − 1 ( m o d 2 9 ) , 𝑎 = 2 , 3 1 6 + 2 ⟨ 𝑥 ⟩ 5 (1463, 86, 5) 9 3 19 3 9 ≡ − 1 ( m o d 1 9 ) 1 − 9 + 5 ⟨ 𝑥 ⟩ 6 (583, 97, 16) 9 3 53 3 2 6 ≡ − 1 ( m o d 5 3 ) 1 − 9 + 2 ⟨ 𝑥 ⟩ 7 (345, 129, 48) 9 3 23 Remark 2.3 1 − 9 + 6 ⟨ 𝑥 ⟩ 8 (3503, 103, 3) 10 2, 5 113 𝑎 ≡ − 1 ( m o d 1 1 3 ) 𝑎 = 2 1 4 , 5 5 6 1 − 1 0 + ⟨ 𝑥 ⟩ 9 (2185, 105, 5) 10 2, 5 23 Remark 2.3 1 − 1 0 + 5 ⟨ 𝑥 ⟩ 10 (1309, 109, 9) 10 2, 5 17 𝑎 ≡ − 1 ( m o d 1 7 ) 𝑎 = 2 4 , 5 8 1 − 1 0 + 7 ⟨ 𝑥 ⟩ 11 (1037, 112, 12) 10 2, 5 61 𝑎 ≡ − 1 ( m o d 6 1 ) 𝑎 = 2 3 0 , 5 1 5 1 − 1 0 + 2 ⟨ 𝑥 ⟩ 1 2 ∗ (621, 125, 25) 10 2, 5 69 5 1 1 ≡ − 1 ( m o d 𝑏 ) 𝑏 = 2 3 , 6 9 2 factors trivially in ℤ [ 𝜁 𝑎 ] , 𝑎 = 2 3 , 6 9 5 1 0 + 5 ⟨ 𝑥 ⟩ in 𝐶 2 3 ; None in 𝐶 6 9 13 (469, 144, 44) 10 2, 5 67 𝑎 ≡ − 1 ( m o d 6 7 ) 𝑎 = 2 3 3 , 5 1 1 1 1 0 + 2 ⟨ 𝑥 ⟩ 14 (407, 175, 75) 10 2, 5 37 𝑎 1 8 ≡ − 1 ( m o d 3 7 ) 𝑎 = 2 , 5 1 − 1 0 + 5 ⟨ 𝑥 ⟩ 15 (3151, 126, 5) 11 11 137 1 1 3 4 ≡ − 1 ( m o d 1 3 7 ) 1 − 1 1 + ⟨ 𝑥 ⟩ 1 6 ∗ (483, 241, 120) 11 11 23 1 1 1 1 ≡ − 1 ( m o d 𝑏 ) 𝑏 = 2 3 , 6 9 2 1 1 + 1 0 ⟨ 𝑥 ⟩ in 𝐶 2 3 ; None in 𝐶 6 9 17 (561, 176, 55) 11 11 187 1 1 8 ≡ − 1 ( m o d 1 7 ) 11 factors trivially in ℤ [ 𝜁 𝑎 ] , 𝑎 = 1 7 , 1 8 7 1 − 1 1 + 1 1 ⟨ 𝑥 ⟩ in 𝐶 1 7 ; None in 𝐶 1 8 7 18 (20881, 145, 1) 12 2, 3 157 𝑎 ≡ − 1 ( m o d 1 5 7 ) 𝑎 = 2 2 6 , 3 3 9 1 − 1 2 + ⟨ 𝑥 ⟩ 19 (1591, 160, 16) 12 2, 3 43 𝑎 ≡ − 1 ( m o d 4 3 ) 𝑎 = 2 7 , 3 2 1 1 − 1 2 + 4 ⟨ 𝑥 ⟩ 20 (9805, 172, 3) 13 13 37 1 3 1 8 ≡ − 1 ( m o d 3 7 ) 1 − 1 3 + 5 ⟨ 𝑥 ⟩ 2 1 ∗ (3895, 177, 8) 13 13 19 1 3 9 ≡ − 1 ( m o d 1 9 ) 2 − 1 3 + 1 0 ⟨ 𝑥 ⟩ 2 2 ∗ (1711, 190, 21) 13 13 29 1 3 7 ≡ − 1 ( m o d 2 9 ) 2 − 1 3 + 7 ⟨ 𝑥 ⟩ 23 (2323, 216, 20) 14 14 23 1 4 1 1 ≡ − 1 ( m o d 2 3 ) 1 − 1 4 + 1 0 ⟨ 𝑥 ⟩ 2 4 ∗ (9951, 200, 4) 14 2, 7 107 𝑎 5 3 ≡ − 1 ( m o d 1 0 7 ) 𝑎 = 2 , 7 2 − 1 4 + 2 ⟨ 𝑥 ⟩ 25 (8041, 201, 5) 14 2, 7 43 𝑎 ≡ − 1 ( m o d 4 3 ) 𝑎 = 2 7 , 7 3 1 − 1 4 + 5 ⟨ 𝑥 ⟩ 26 (793, 352, 156) 14 2, 7 61 𝑎 3 0 ≡ − 1 ( m o d 6 1 ) 𝑎 = 2 , 7 1 − 1 4 + 6 ⟨ 𝑥 ⟩ 27 (50851, 226, 1) 15 3, 5 241 𝑎 ≡ − 1 ( m o d 2 4 1 ) 𝑎 = 3 6 0 , 5 2 0 1 − 1 5 + ⟨ 𝑥 ⟩ 2 8 ∗ (2871, 246, 21) 15 3, 5 29 𝑎 ≡ − 1 ( m o d 2 9 ) 𝑎 = 3 1 4 , 5 7 2 − 1 5 + 9 ⟨ 𝑥 ⟩ 29 (2491, 250, 25) 15 3, 5 53 𝑎 2 6 ≡ − 1 ( m o d 5 3 ) 𝑎 = 3 , 5 1 − 1 5 + 5 ⟨ 𝑥 ⟩ 3 0 ∗ (13573, 261, 5) 16 2 277 2 4 6 ≡ − 1 ( m o d 2 7 7 ) 2 − 1 6 + ⟨ 𝑥 ⟩ 31 (4879, 271, 15) 16 2 41 2 1 0 ≡ − 1 ( m o d 4 1 ) 1 − 1 6 + 7 ⟨ 𝑥 ⟩ 3 2 ∗ (26815, 328, 4) 18 2, 3 173 𝑎 8 6 ≡ − 1 ( m o d 1 7 3 ) 𝑎 = 2 , 3 2 − 1 8 + 2 ⟨ 𝑥 ⟩ 3 3 ∗ (4551, 351, 27) 18 2, 3 41 𝑎 ≡ − 1 ( m o d 4 1 ) 𝑎 = 2 1 0 , 3 4 2 − 1 8 + 9 ⟨ 𝑥 ⟩ 3 4 ∗ (16975, 369, 8) 19 19 97 1 9 1 6 ≡ − 1 ( m o d 9 7 ) 2 − 1 9 + 4 ⟨ 𝑥 ⟩ 3 5 ∗ (15171, 370, 9) 19 19 389 1 9 9 7 ≡ − 1 ( m o d 3 8 9 ) 2 − 1 9 + ⟨ 𝑥 ⟩ 36 (2599, 433, 72) 19 19 113 1 9 5 6 ≡ − 1 ( m o d 1 1 3 ) 1 − 1 9 + 4 ⟨ 𝑥 ⟩ 37 (11455, 415, 15) 20 2, 5 29 𝑎 ≡ − 1 ( m o d 2 9 ) 𝑎 = 2 1 4 , 5 7 1 − 2 0 + 1 5 ⟨ 𝑥 ⟩ 38 (3657, 457, 57) 20 2, 5 53 𝑎 2 6 ≡ − 1 ( m o d 5 3 ) 𝑎 = 2 , 5 1 − 2 0 + 9 ⟨ 𝑥 ⟩ 39 (194923, 442, 1) 21 3, 7 463 𝑎 ≡ − 1 ( m o d 4 6 3 ) 𝑎 = 3 2 3 1 , 7 7 7 1 − 2 1 + ⟨ 𝑥 ⟩ 40 (28609, 448, 7) 21 3, 7 67 𝑎 ≡ − 1 ( m o d 6 7 ) 𝑎 = 3 1 1 , 7 3 3 1 − 2 1 + 7 ⟨ 𝑥 ⟩ 4 1 ∗ (18533, 452, 11) 21 3, 7 43 𝑎 ≡ − 1 ( m o d 4 3 ) 𝑎 = 3 2 1 , 7 3 2 − 2 1 + 1 1 ⟨ 𝑥 ⟩ 4 2 ∗ (13833, 456, 15) 21 3, 7 53 𝑎 ≡ − 1 ( m o d 5 3 ) 𝑎 = 3 2 6 , 7 1 3 2 − 2 1 + 9 ⟨ 𝑥 ⟩ 43 (4891, 490, 49) 21 3, 7 73 𝑎 ≡ − 1 ( m o d 7 3 ) 𝑎 = 3 6 , 7 1 2 1 − 2 1 + 7 ⟨ 𝑥 ⟩ 44 (3649, 513, 72) 21 3, 7 89 𝑎 4 4 ≡ − 1 ( m o d 8 9 ) 𝑎 = 3 , 7 1 − 2 1 + 6 ⟨ 𝑥 ⟩ 45 (2941, 540, 99) 21 3, 7 173 𝑎 8 6 ≡ − 1 ( m o d 1 7 3 ) 𝑎 = 3 , 7 1 2 1 + 3 ⟨ 𝑥 ⟩ 46 (1919, 686, 245) 21 3, 7 101 𝑎 5 0 ≡ − 1 ( m o d 1 0 1 ) 𝑎 = 3 , 7 1 − 2 1 + 7 ⟨ 𝑥 ⟩ 47 (1769, 833, 392) 21 3, 7 61 𝑎 ≡ − 1 ( m o d 6 1 ) 𝑎 = 3 5 , 7 3 0 1 − 2 1 + 1 4 ⟨ 𝑥 ⟩ 4 8 ∗ (78895, 487, 3) 22 2, 11 509 𝑎 ≡ − 1 ( m o d 5 0 9 ) 𝑎 = 2 2 5 4 , 1 1 1 2 7 2 − 2 2 + ⟨ 𝑥 ⟩ 49 (20461, 496, 12) 22 2, 11 37 𝑎 ≡ − 1 ( m o d 3 7 ) 𝑎 = 2 1 8 , 1 1 3 1 − 2 2 + 1 4 ⟨ 𝑥 ⟩ 50 (4081, 561, 77) 22 2, 11 53 𝑎 ≡ − 1 ( m o d 5 3 ) 𝑎 = 2 2 6 , 1 1 1 3 1 − 2 2 + 1 1 ⟨ 𝑥 ⟩ 51 (3835, 568, 84) 22 2, 11 59 𝑎 2 9 ≡ − 1 ( m o d 5 9 ) 𝑎 = 2 , 1 1 1 − 2 2 + 1 0 ⟨ 𝑥 ⟩ 52 (3601, 576, 92) 22 2, 11 277 𝑎 ≡ − 1 ( m o d 2 7 7 ) 𝑎 = 2 4 6 , 1 1 1 3 8 1 2 2 + 2 ⟨ 𝑥 ⟩ 53 (94165, 532, 3) 23 23 37 2 3 6 ≡ − 1 ( m o d 3 7 ) 1 − 2 3 + 1 5 ⟨ 𝑥 ⟩ 5 4 ∗ (18531, 545, 16) 23 23 71 2 3 7 ≡ − 1 ( m o d 7 1 ) 2 − 2 3 + 8 ⟨ 𝑥 ⟩ 55 (8557, 621, 45) 24 2, 3 43 𝑎 ≡ − 1 ( m o d 4 3 ) 𝑎 = 2 7 , 3 2 1 1 − 2 4 + 1 5 ⟨ 𝑥 ⟩ 56 (2959, 783, 207) 24 2, 3 269 𝑎 1 3 4 ≡ − 1 ( m o d 2 6 9 ) 𝑎 = 2 , 3 1 − 2 4 + 3 ⟨ 𝑥 ⟩ 5 7 ∗ (131253, 628, 3) 25 5 653 5 3 2 6 ≡ − 1 ( m o d 6 5 3 ) 2 − 2 5 + ⟨ 𝑥 ⟩ 58 (11289, 664, 39) 25 5 53 5 2 6 ≡ − 1 ( m o d 5 3 ) 1 − 2 5 + 1 3 ⟨ 𝑥 ⟩ 59 (6205, 705, 80) 25 5 73 5 3 6 ≡ − 1 ( m o d 7 3 ) 1 − 2 5 + 1 0 ⟨ 𝑥 ⟩ 60 (3115, 865, 240) 25 5 89 5 2 2 ≡ − 1 ( m o d 8 9 ) 1 − 2 5 + 1 0 ⟨ 𝑥 ⟩