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

Table 2

Partial results in groups of order 𝑣 by Criterion 1. 𝐶_|𝐺/𝑁|=⟨𝑥⟩ and parameters with asterisk indicate new results. ? means the number of groups of order 𝑣 is unknown.

	(𝑣, 𝑘, 𝜆)	𝑚	𝑝	|𝐺/𝑁|	Factoring of 𝑝 in ℤ[𝜁|𝐺/𝑁|]	No. of groups of order 𝑣	No. of groups ruled out	Solutions in 𝐺/𝑁
1	(171, 51, 15)	6	2,3	19	𝑎^9≡−1(mod19); 𝑎=2,3	5	2	−6+3⟨𝑥⟩
2	(155, 56, 20)	6	2,3	31	3^15≡−1(mod31) 2 factors trivially [20]	2	1	−6+2⟨𝑥⟩
3	(231, 70, 21)	7	7	77	7^5≡−1(mod11) 7 factors trivially in ℤ[𝜁𝑏], 𝑏=11,77	2	1	−7+7⟨𝑥⟩ in 𝐶7; None in 𝐶77
4^∗	(2325, 84, 3)	9	3	31	3^15≡−1(mod31)	10	3	−9+3⟨𝑥⟩,
5^∗	(10101, 101, 1)	10	2,5	37	𝑎^18≡−1(mod37); 𝑎=2,3	14	5	−10+3⟨𝑥⟩
6	(715, 154, 33)	11	11	143	11^6≡−1(mod13) 11 factors trivially in ℤ[𝜁𝑏], 𝑏=13,143	2	1	11+11⟨𝑥⟩ in 𝐶13; None in 𝐶143
7^∗	(7155, 147, 3)	12	2,3	53	𝑎^26≡−1(mod53); 𝑎=2,3	?	?	−12+3⟨𝑥⟩
8^∗	(38613, 197, 1)	14	2,7	211	𝑎^105≡−1(mod211); 𝑎=2,7	5	2	−14+⟨𝑥⟩
9^∗	(5859, 203, 7)	14	2,7	31	7^15≡−1(mod31); 2 factors trivially [20]	?	?	−14+7⟨𝑥⟩
10^∗	(903, 287, 91)	14	2,7	43	𝑎≡−1(mod43); 𝑎=2^7,7^3	7	2	−14+7⟨𝑥⟩
11^∗	(2255, 392, 68)	18	2,3	41	𝑎≡−1(mod41); 𝑎=2^10,3^4	7	2	−18+10⟨𝑥⟩
12^∗	(160401, 401, 1)	20	2,5	421	𝑎≡−1(mod421); 𝑎=2^210,5^105	5	2	−20+⟨𝑥⟩
13^∗	(23607, 407, 7)	20	2,5	61	𝑎≡−1(mod61); 𝑎=2^30,5^15	11	5	−20+7⟨𝑥⟩
14^∗	(22451, 450, 9)	21	3,7	157	𝑎≡−1(mod157); 𝑎=3^39,7^26	2	1	−21+3⟨𝑥⟩
15^∗	(2619, 561, 120)	21	3,7	97	𝑎≡−1(mod97); 𝑎=3^24,7^48	13	5	−21+6⟨𝑥⟩
16	(2211, 715, 231)	22	2,11	67	𝑎^33≡−1(mod67); 𝑎=2,11	4	1	−22+11⟨𝑥⟩
17	(7450, 573, 44)	23	23	149	23^74≡−1(mod149)	10	5	−23+4⟨𝑥⟩
18	(111555, 579, 3)	24	2,3	67	𝑎≡−1(mod67); 𝑎=2^33,3^11	?	?	−24+9⟨𝑥⟩
19	(37961, 585, 9)	24	2,3	29	𝑎^14≡−1(mod29); 𝑎=2,3	2	1	−24+21⟨𝑥⟩
20	(23247, 591, 15)	24	2,3	41	𝑎≡−1(mod41); 𝑎=2^10,3^4	?	?	−24+15⟨𝑥⟩
21	(25641, 641, 16)	25	5	37	5^18≡−1(mod37)	14	4	−25+18⟨𝑥⟩