Supplementary data for the paper "On class numbers inside the real <i>p</i>th cyclotomic field, II; computational aspect"

Supplementary data for the paper "On class numbers inside the real pth cyclotomic field, II; computational aspect"

ℓ=3	n=1	2	4	5	7	8	10	11	13	14	16	17	19	20	22	23	25	26	28	29	31
Proposition 5.3. The following assertions hold for every prime number r which is a primitive root modulo ℓ². (I) The case ℓ=3. Let n be an integer with ¬3\|n. (I-i) When 1 ≤ n ≤ 29, ¬r\|h_f/h_f-1 for every f ≥ 2. (I-ii) When 1 ≤ n ≤ 10, ¬r\|h_f/h_f-2 for every f ≥ 2. (I-iii) When n=1 or 2, ¬r\|h_f/h_f-3 for every f ≥ 3. (II) The case ℓ=5. Let n be an integer with ¬5\|n. (II-i) When 1 ≤ n ≤ 29, ¬r\|h_f/h_f-1 for every f ≥ 2. (II-ii) When 1 ≤ n ≤ 6, ¬r\|h_f/h_f-2 for every f ≥ 2. (II-iii) When n=1, ¬r\|h_f/h_f-3 for every f ≥ 3. (III) The case ℓ=7. Let n be an integer with ¬7\|n. (III-i) When 2 ≤ n ≤ 30, ¬r\|h_f/h_f-1 for every f ≥ 2. (III-ii) When n=2 or 3, ¬r\|h_f/h_f-2 for every f ≥ 2.
(I) ℓ=3
s=0	O	O	O	O	T	O	O	O	O	O	O	O	O	O	T	O	T	O	O	O	R
s=1	O	O	A	O	T	O	O	R
s=2	O	T	R
(II) ℓ=5
ℓ=5	n=1	2	3	4	6	7	8	9	11	12	13	14	16	17	18	19	21	22	23	24	26	27	28	29	31
s=0	O	O	O	O	O	O	O	O	O	O	O	O	O	O	O	O	O	O	O	O	O	O	O	O	R
s=1	O	O	T	O	O	R
s=2	T	R
(III) ℓ=7
ℓ=7	2	3	5	6	8	9	11	12	15	17	18	20	23	24	26	27	29	30	32
s=0	O	O	O	O	O	O	O	O	O	O	O	O	O	O	O	O	O	O	R
s=1	O	O	R
s=2	R
Notation for the table above: (See also Proof of Proposition 5.3) • Cells with symbol 'O' : cases (a), (b), (c) appear and deg D_ts,r ≡ 0 in case (c) • Cells with symbol 'A' : cases (a), (b), (c) appear and deg D_ts,r = 0 in case (c) except one (f,r) • Cells with symbol 'T' : all cases (a)-(e) appear and deg D_ts,r ≡ 0 in cases (c) and (e). • Cells with symbol 'R' or cells to the right of it : nℓ^s > 30 so that not considered in the propositions Notation for the table in the link destination pages: (See also Example 5.1 / Figure 1) • Cells with symbol 'P2' : case (a). (2^{2nℓ^s}≠1 mod p is verified computationally.) • Cells with symbol 'P2*' : also in case (a). (2^{2nℓ^s}≠1 mod p is satisfied since f >θ, see Remark 2.1) • Cells with symbol 'P1' : f > f (ℓ,n,s,r) holds. case (b) (or case (a) if r > 𝔯₀)) • Cells of case (d) are not shown in the table. See footnote ※ below. • Other cells in the table : cases (c) or (e). For each of them, we checked whether or not deg D_{t_s,r} = 0 with the help of computer. ••• Cells with symbol 'O' : deg D_{t_s,r} = 0 ••• Cells with symbol 'A' : deg D_{t_s,r} ≥ 1 ※ Case (d), given by Proposition 2.4, appears in 7 tables. Parameters of those tables are shown in the next table.
ℓ	s	n	f	𝔯₁ (end of case (e))		RHS of (2.2)
3	0	7	2	239		241.5
3	0	22	2	947		967.5
3	0	25	4	1571		1582.1
3	1	7	2	677		724.6
3	2	2	3	587		598.2
5	1	3	2	523		539.1
5	2	1	3	997		1004.2
Proposition 5.1. (I) The case ℓ=3. Let n be an integer with 1 ≤ n ≤ 100 and ¬3\|n. (I-i) When f=2, h_f/h_f-2 is odd except for the case n=22, 52 or 56. In the exceptional case, h₂/h₁ is odd but h₁ is even. (I-ii) h_f/h_f-3 is always odd for every f ≥ 3. (II) The case ℓ=5. Let n be an integer with 1 ≤ n ≤ 99 and ¬5\|n. (II-i) When f = 2, h_f/h_f-2 is odd except for the case n = 66. In the exceptional case, h₂/h₁ is odd and h₁ is even. (II-ii) h_f/h_f-3 is always odd for every f ≥ 3.
r=2	s=0	s=1	s=2	exceptional cases (for Table 1 in the paper)
ℓ=3	O	A	O	[DB]
ℓ=5	O	A	O	[DB]
Proposition 5.2. (I) The case ℓ=3. Let 1 ≤ n ≤ 100 and ¬3\|n, and let r∈{5,11,23,29,41,47}. (I-i) ¬r\|h_f/h_f-2 for f=2. (I-ii) ¬r\|h_f/h_f-3 for every f ≥ 3. (II) The case ℓ=5. Let n be an integer with ¬5\|n, and let r∈{3,13,17,23,37,47}. (II-i) When 1 ≤ n ≤ 99, ¬r\|h_f/h_f-2 for every f ≥ 2. (II-ii) When 1 ≤ n ≤ 69, ¬r\|h_f/h_f-3 for every f ≥ 3. (III) The case ℓ=7. Let n be an integer with ¬7\|n, and let r∈{3,5,17,47}. (III-i) When 2 ≤ n ≤ 99, ¬r\|h_f/h_f-2 for every f ≥ 2. (III-ii) When 2 ≤ n ≤ 30, ¬r\|h_f/h_f-3 for every f ≥ 3.
ℓ=3	r=5	r=11	r=23	r=29	r=41	r=47	exceptional cases (for Table 1 in the paper)
s=0	O	O	O	O	O	O	[DB]
s=1	A	A	O	A	O	O
s=2	A	A	O	O	O	O
ℓ=5	r=3	r=13	r=17	r=23	r=37	r=47	check of no exceptional cases
s=0	O	O	O	O	O	O	[DB]
s=1	O	O	O	O	O	O
s=2	O(n≤69)	O(n≤69)	O(n≤69)	O(n≤69)	O(n≤69)	O(n≤69)
ℓ=7	r=3	r=5	r=17	r=47
s=0	O	O	O	O
s=1	O	O	O	O
s=2	O(n≤30)	O(n≤30)	O(n≤30)	O(n≤30)
Notation for the tables to Propositions 5.1 and 5.2: • Cells with symbol 'O' : All computational results is deg D_{t_s,r} = 0 • Cells with symbol 'A' : Some computational results is deg D_{t_s,r} ≥ 1 Notation for the tables in the link destination pages: • At f 's with subscript 'P1' : f > f (ℓ,n,s,r) holds. • At f 's with subscript 'O' : deg D_{t_s,r} = 0 is obtained by computation. • At f 's with subscript 'A' : deg D_{t_s,r} ≥ 1 is obtained by computation. Columns in the DB: • primroot = 'O' means that r is a primitive root modulo ℓ² • koyo = '1' means that it appears in the list of Koyama-Yoshino[15] • compresult = 'A' means that deg D_{t_s,r} ≥ 1 is obtained by computation.
§ 5.2 The case where r is not a primitive root
as r≤47 is quite small, we see that the assumption r≠p=2nℓ^f+1 in Lemmas 3.1, 3.4, 4.1 and 4.2 is satisfied except for the case where (ℓ,n,f,p)=(3,1,2,19) with r=19 or (ℓ,n,f,p)=(3,2,2,37) with r=37. For these exceptional cases, we have h_p⁺=1 (see [20,page 421]) and hence ¬r\|h_f for any prime r.
Verification by searching all (ℓ,n,f,r,s) records where p ≤ r and r is not a primroot [in database] Observation on the n-r plane: [s=0] [s=1]. In these link destination pages, ·superscripts 'T' mean that indivisibility is assured by Washington[20] ·subscripts 'O' mean computation results were deg D_{t_s,r}=0 ·subscripts 'A' mean computation results were deg D_{t_s,r} ≥ 1
Table 2: The number of prime numbers of the form p=2nℓ^f+1 with n and f in the range (5.1)
Verification by searching all (ℓ,n,f,r=19,s) records where n and f in the range (5.1) [in database], where r=19 is adopted since it is always not a primroot modulo ℓ² for ℓ=3,5,7.
Table 3: The number of experimental ones for each (ℓ,s,r) [DB:the number] [DB:among them p<10⁴]
l=3	r=7	r=13	r=17	r=19	r=31	r=37	r=43
s=0	O	O	O	A(14)	O	A(5)	O
s=1	A(8)	A(6)	O	A(3·10)	A(1)	A(1·4)	O
s=2	A(·7)	A(·5)	O	A(·3·7)	O	A(·1·4)	A(·2)
Observation of exceptional results on the n-s plane				[f=2] [f=3]		[f=2] [f=3]
l=5	r=7	r=11	r=19	r=29	r=31	r=41	r=43
s=0	O	O	O	O	O	O	O
s=1	O	A(8)	O	O	A(4)	A(3)	O
s=2	O	A(·7)	A(·1)	O	A(·5)	A(·3)	O
l=7	r=2	r=11	r=13	r=19	r=23	r=29	r=31	r=37	r=41	r=43
s=0	O	O	O	O	O	O	O	O	O	O
s=1	A(10)	O	O	O	O	A(6)	O	O	A(1)	A(4)
s=2	A(·1)	O	A(·2)	O	O	A(·2)	O	O	O	A(·3)
Notation in the above tables: The symbol "A(3·10)" in the cell (ℓ,s,r)=(3,1,19) (for instance) means that some exceptional results occurs there and among them 3 ones are at f=2 and 10 ones are at f=3. Similarly, the symbol "A(·3·7)" means that 3 ones at f=3 and 7 ones at f=4.
Table 4: List of (p; n, f, s) with p = 2nℓ^f+1 < 10⁴ for which deg D_{t_s,r} ≥ 1 and the pair (r, p) is contained in the list of [15]
ℓ	r	p	n	f	s=0	s=1	s=2
3	7	577	32	2	O	A
	7	1567	29	3	O	O	A
	13	1063	59	2	O	A
	19	1153	64	2	A	O
	19	4591	85	3	O	A	O
5	11	2351	47	2	O	A
		3001	12	3	O	O	A
		4201	84	2	O	A
		5501	22	3	O	O	A
7	2	491	5	2	O	A
7	2	7841	80	2	O	A
Example 5.3: We found examples of p = 2nℓ^f+1 or (ℓ,n,f) for which, with several r or s, deg D_{t_s,r} ≥ 1 (or equivalently H(t_s,r) ≥ 1). Let us give some of them.
(I) (p; ℓ, n, f)=(883; 3, 49, 2), (p; ℓ, n, f)=(883; 7, 9, 2) (p; ℓ, n, f)=(4591; 3, 85, 3), (II) (p; ℓ, n, f)=(1531; 3, 85, 2) (p; ℓ, n, f)=(3673; 3, 68, 3)
§5.3 computation code
The largest p we dealt with in this paper is p = 2nℓ^f+1 ≈ 10³⁴⁷⁵⁰ with (ℓ, n, f)=(7, 29, 41117), which appeared in the proof of Proposition 5.2(III-ii) with s=2 and r ∈ {17, 47}. Here, 41117=f (7,29,2,r) for those r. [Search five largest p in DB] [Observation on r-s plane] [Observation on n-f plane, r=17] [Observation on n-f plane, r=47]