On the Existence of (𝑣,𝑘,𝜆) Difference Sets with 𝑘<1250 and 𝑘−𝜆 Is a Square

Table 1

Parameter sets that do not exist by Criterion 1. 𝐶_|𝐺/𝑁|=⟨𝑥⟩ 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(mod43); 𝑎=2^7,3^21	1	−6+⟨𝑥⟩
3	(221, 45, 9)	6	2, 3	17	𝑎≡−1(mod17); 𝑎=2^4,3^8	1	−6+3⟨𝑥⟩
4	(145, 64, 28)	6	2, 3	29	𝑎^14≡−1(mod29),; 𝑎=2,3	1	6+2⟨𝑥⟩
5	(1463, 86, 5)	9	3	19	3^9≡−1(mod19)	1	−9+5⟨𝑥⟩
6	(583, 97, 16)	9	3	53	3^26≡−1(mod53)	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(mod113); 𝑎=2^14,5^56	1	−10+⟨𝑥⟩
9	(2185, 105, 5)	10	2, 5	23	Remark 2.3	1	−10+5⟨𝑥⟩
10	(1309, 109, 9)	10	2, 5	17	𝑎≡−1(mod17); 𝑎=2^4,5^8	1	−10+7⟨𝑥⟩
11	(1037, 112, 12)	10	2, 5	61	𝑎≡−1(mod61); 𝑎=2^30,5^15	1	−10+2⟨𝑥⟩
12^∗	(621, 125, 25)	10	2, 5	69	5^11≡−1(mod𝑏); 𝑏=23,69; 2 factors trivially in ℤ[𝜁𝑎], 𝑎=23,69	5	10+5⟨𝑥⟩ in 𝐶23; None in 𝐶69
13	(469, 144, 44)	10	2, 5	67	𝑎≡−1(mod67); 𝑎=2^33,5^11	1	10+2⟨𝑥⟩
14	(407, 175, 75)	10	2, 5	37	𝑎^18≡−1(mod37); 𝑎=2,5	1	−10+5⟨𝑥⟩
15	(3151, 126, 5)	11	11	137	11^34≡−1(mod137)	1	−11+⟨𝑥⟩
16^∗	(483, 241, 120)	11	11	23	11^11≡−1(mod𝑏); 𝑏=23,69	2	11+10⟨𝑥⟩ in 𝐶23; None in 𝐶69
17	(561, 176, 55)	11	11	187	11^8≡−1(mod17); 11 factors trivially in ℤ[𝜁𝑎], 𝑎=17,187	1	−11+11⟨𝑥⟩ in 𝐶17; None in 𝐶187
18	(20881, 145, 1)	12	2, 3	157	𝑎≡−1(mod157); 𝑎=2^26,3^39	1	−12+⟨𝑥⟩
19	(1591, 160, 16)	12	2, 3	43	𝑎≡−1(mod43); 𝑎=2^7,3^21	1	−12+4⟨𝑥⟩
20	(9805, 172, 3)	13	13	37	13^18≡−1(mod37)	1	−13+5⟨𝑥⟩
21^∗	(3895, 177, 8)	13	13	19	13^9≡−1(mod19)	2	−13+10⟨𝑥⟩
22^∗	(1711, 190, 21)	13	13	29	13^7≡−1(mod29)	2	−13+7⟨𝑥⟩
23	(2323, 216, 20)	14	14	23	14^11≡−1(mod23)	1	−14+10⟨𝑥⟩
24^∗	(9951, 200, 4)	14	2, 7	107	𝑎^53≡−1(mod107); 𝑎=2,7	2	−14+2⟨𝑥⟩
25	(8041, 201, 5)	14	2, 7	43	𝑎≡−1(mod43); 𝑎=2^7,7^3	1	−14+5⟨𝑥⟩
26	(793, 352, 156)	14	2, 7	61	𝑎^30≡−1(mod61); 𝑎=2,7	1	−14+6⟨𝑥⟩
27	(50851, 226, 1)	15	3, 5	241	𝑎≡−1(mod241); 𝑎=3^60,5^20	1	−15+⟨𝑥⟩
28^∗	(2871, 246, 21)	15	3, 5	29	𝑎≡−1(mod29); 𝑎=3^14,5^7	2	−15+9⟨𝑥⟩
29	(2491, 250, 25)	15	3, 5	53	𝑎^26≡−1(mod53); 𝑎=3,5	1	−15+5⟨𝑥⟩
30^∗	(13573, 261, 5)	16	2	277	2^46≡−1(mod277)	2	−16+⟨𝑥⟩
31	(4879, 271, 15)	16	2	41	2^10≡−1(mod41)	1	−16+7⟨𝑥⟩
32^∗	(26815, 328, 4)	18	2, 3	173	𝑎^86≡−1(mod173); 𝑎=2,3	2	−18+2⟨𝑥⟩
33^∗	(4551, 351, 27)	18	2, 3	41	𝑎≡−1(mod41); 𝑎=2^10,3^4	2	−18+9⟨𝑥⟩
34^∗	(16975, 369, 8)	19	19	97	19^16≡−1(mod97)	2	−19+4⟨𝑥⟩
35^∗	(15171, 370, 9)	19	19	389	19^97≡−1(mod389)	2	−19+⟨𝑥⟩
36	(2599, 433, 72)	19	19	113	19^56≡−1(mod113)	1	−19+4⟨𝑥⟩
37	(11455, 415, 15)	20	2, 5	29	𝑎≡−1(mod29); 𝑎=2^14,5^7	1	−20+15⟨𝑥⟩
38	(3657, 457, 57)	20	2, 5	53	𝑎^26≡−1(mod53); 𝑎=2,5	1	−20+9⟨𝑥⟩
39	(194923, 442, 1)	21	3, 7	463	𝑎≡−1(mod463); 𝑎=3^231,7^77	1	−21+⟨𝑥⟩
40	(28609, 448, 7)	21	3, 7	67	𝑎≡−1(mod67); 𝑎=3^11,7^33	1	−21+7⟨𝑥⟩
41^∗	(18533, 452, 11)	21	3, 7	43	𝑎≡−1(mod43); 𝑎=3^21,7^3	2	−21+11⟨𝑥⟩
42^∗	(13833, 456, 15)	21	3, 7	53	𝑎≡−1(mod53); 𝑎=3^26,7^13	2	−21+9⟨𝑥⟩
43	(4891, 490, 49)	21	3, 7	73	𝑎≡−1(mod73); 𝑎=3^6,7^12	1	−21+7⟨𝑥⟩
44	(3649, 513, 72)	21	3, 7	89	𝑎^44≡−1(mod89); 𝑎=3,7	1	−21+6⟨𝑥⟩
45	(2941, 540, 99)	21	3, 7	173	𝑎^86≡−1(mod173); 𝑎=3,7	1	21+3⟨𝑥⟩
46	(1919, 686, 245)	21	3, 7	101	𝑎^50≡−1(mod101); 𝑎=3,7	1	−21+7⟨𝑥⟩
47	(1769, 833, 392)	21	3, 7	61	𝑎≡−1(mod61); 𝑎=3^5,7^30	1	−21+14⟨𝑥⟩
48^∗	(78895, 487, 3)	22	2, 11	509	𝑎≡−1(mod509); 𝑎=2^254,11^127	2	−22+⟨𝑥⟩
49	(20461, 496, 12)	22	2, 11	37	𝑎≡−1(mod37); 𝑎=2^18,11^3	1	−22+14⟨𝑥⟩
50	(4081, 561, 77)	22	2, 11	53	𝑎≡−1(mod53); 𝑎=2^26,11^13	1	−22+11⟨𝑥⟩
51	(3835, 568, 84)	22	2, 11	59	𝑎^29≡−1(mod59); 𝑎=2,11	1	−22+10⟨𝑥⟩
52	(3601, 576, 92)	22	2, 11	277	𝑎≡−1(mod277); 𝑎=2^46,11^138	1	22+2⟨𝑥⟩
53	(94165, 532, 3)	23	23	37	23^6≡−1(mod37)	1	−23+15⟨𝑥⟩
54^∗	(18531, 545, 16)	23	23	71	23^7≡−1(mod71)	2	−23+8⟨𝑥⟩
55	(8557, 621, 45)	24	2, 3	43	𝑎≡−1(mod43); 𝑎=2^7,3^21	1	−24+15⟨𝑥⟩
56	(2959, 783, 207)	24	2, 3	269	𝑎^134≡−1(mod269); 𝑎=2,3	1	−24+3⟨𝑥⟩
57^∗	(131253, 628, 3)	25	5	653	5^326≡−1(mod653)	2	−25+⟨𝑥⟩
58	(11289, 664, 39)	25	5	53	5^26≡−1(mod53)	1	−25+13⟨𝑥⟩
59	(6205, 705, 80)	25	5	73	5^36≡−1(mod73)	1	−25+10⟨𝑥⟩
60	(3115, 865, 240)	25	5	89	5^22≡−1(mod89)	1	−25+10⟨𝑥⟩