円周率の式(≒√３)

ﾘｭｶ数列(+β)(mod5)

時間部分f(t)⋍e^i√ℓ･t⋍e^i√2k+1･t
微分して　i√ℓ･f(t)⋍i√2k+1･f(t)
∴⓪⋍√ℓ･f(t)⋍√2k+1･f(t)
k	文献値	文/11	+(s/11)	√Q⋍√2k+1	s	ﾘｭｶ数列(+β)
1	14.13472	1.284975	0.454545	1.73952	5	⋆	(mod5)
2	21.02203	1.911094	0.363636	2.27473	4	⋆	5+4⋍9-5⋍4
3	25.01085	2.273714	0.363636	2.63735	4	⋆	4+4⋍8
4				3		⋆
5	30.42487	2.765897	0.727273	3.49317	8	⋆	4+8⋍12-5⋍7
6	32.93506	2.994096	0.636364	3.63046	7	⋆	8+7⋍15-10⋍5
7	37.58617	3.416925	0.454545	3.87147	5	~	7+5⋍12-9⋍3
8	40.91871	3.719883	0.272727	3.99261	3	~	5+3⋍8ｰ3⋍5
9	43.32707	3.938825	0.454545	4.39337	5	~	3+5⋍8ｰ3⋍5
10				4.58257569		⋆
11	48.00515	4.364105	0.454545	4.81865	5	~	5+5⋍10-6⋍4
12	49.77383	4.524894	0.363636	4.88853	4	~	5+4⋍9-4⋍5
13	52.97	4.815455	0.454545	5.27	5	~	4+5⋍9-4⋍5
14	56.446	5.131455	0.454545	5.586	5	~	5+5⋍10-8⋍2
15	59.347	5.395182	0.181818	5.577	2	~	5+2⋍7-6⋍1
16	60.831	5.530091	0.090909	5.621	1	⋆	2+1⋍3
17				5.91607978		⋆
18	65.112	5.919273	0.272727	6.192	3	⋆	1+4⋍5⋍10
29				7.61577311		⋆
30	75.8	6.890909	0.909091	7.8	10	⋆	…

∴2s+1三p (mod 4)

⓪Hψ⋍-iaψ/at (ｼｭﾚﾃﾞｲﾝｶﾞ-)				∴2s+1三p (mod 4)
時間部分f(t)⋍e^i√ℓ･t⋍e^i√2k+1･t				ｶﾞｳｽ素数p→4n+1,4n+3
微分して　i√ℓ･f(t)⋍i√2k+1･f(t)					∴2s+1三p (mod 4)
∴⓪⋍√ℓ･f(t)⋍√2k+1･f(t)							Σ⋍	ｶﾞｳｽp
k	文献値	文/11	+(s/11)	√Q⋍√2k+1	s	s+1	2s+1	mod 4
1	14.13472	1.284975	0.454545	1.73952	5	6	11	3
2	21.02203	1.911094	0.363636	2.27473	4	5	9	1
3	25.01085	2.273714	0.363636	2.63735	4	5	9	1
4				3	0	1	1	1
5	30.42487	2.765897	0.727273	3.49317	8	9	17	1
6	32.93506	2.994096	0.636364	3.63046	7	8	15	3
7	37.58617	3.416925	0.454545	3.87147	5	6	11	3
8	40.91871	3.719883	0.272727	3.99261	3	4	7	3
9	43.32707	3.938825	0.454545	4.39337	5	6	11	3
10				4.58257569	0	1	1	1
11	48.00515	4.364105	0.454545	4.81865	5	6	11	3
12	49.77383	4.524894	0.363636	4.88853	4	5	9	1
13	52.97	4.815455	0.454545	5.27	5	6	11	3
14	56.446	5.131455	0.454545	5.586	5	6	11	3
15	59.347	5.395182	0.181818	5.577	2	3	5	1
16	60.831	5.530091	0.090909	5.621	1	2	3	3
17				5.91607978	0	1	1	1
18	65.112	5.919273	0.272727	6.192	3	4	7	3
29				7.61577311	0	1	1	1
30	75.8	6.890909	0.909091	7.8	10	11	21	1

⓪⋍√ℓ･f(t)⋍√2k+1･f(t)

⓪Hψ⋍-iaψ/at (ｼｭﾚﾃﾞｲﾝｶﾞ-)
時間部分f(t)⋍e^i√ℓ･t⋍e^i√2k+1･t
微分して　i√ℓ･f(t)⋍i√2k+1･f(t)
∴⓪⋍√ℓ･f(t)⋍√2k+1･f(t)
k	文献値	文/11	+(s/11)	√Q⋍√2k+1
1	14.13472	1.284975	0.454545	1.73952
2	21.02203	1.911094	0.363636	2.27473
3	25.01085	2.273714	0.363636	2.63735
4				3
5	30.42487	2.765897	0.727273	3.49317
6	32.93506	2.994096	0.636364	3.63046
7	37.58617	3.416925	0.454545	3.87147
8	40.91871	3.719883	0.272727	3.99261
9	43.32707	3.938825	0.454545	4.39337
10				4.58257569
11	48.00515	4.364105	0.454545	4.81865
12	49.77383	4.524894	0.363636	4.88853
13	52.97	4.815455	0.454545	5.27
14	56.446	5.131455	0.454545	5.586
15	59.347	5.395182	0.181818	5.577
16	60.831	5.530091	0.090909	5.621
17				5.91607978
18	65.112	5.919273	0.272727	6.192
29				7.61577311
30	75.8	6.890909	0.909091	7.8

(ﾙｼﾞｬﾝﾄﾞﾙ) ℓ^2+ℓ⋍α^2+α

ﾙｼﾞｬﾝﾄﾞ		ℓ^2+ℓ⋍α^2+α
∵Δ(M)⋍ℓ^2⋍(2k+1)^2　(同形)							△(m)	⋍	2k+1
∴△	⋍	4k＾2+4k+1			⋍	ℓ＾2	質量項				⋍	ℓ
(M行列)		1	2		Δ⋍	9	(m)		0	√3	Δ⋍	3
M1⋍		-2	5				m1⋍		-√3	0
(M行列)		1	4		Δ⋍	25	(m)		0	√5	Δ⋍	5
M2⋍		-4	9				m2⋍		-√5	0
(M行列)		1	6		Δ⋍	49	(m)		0	√7	Δ⋍	7
M3⋍		-6	13				m3⋍		-√7	0
(M行列ツ)		1	8		Δ⋍	81	(m)		0	√9	Δ⋍	9
M4⋍		-8	17				m4⋍		-√9	0
(M行列)		1	10		Δ⋍	121	(m)		0	√11	Δ⋍	11
M5⋍		-10	21				m5⋍		-√11	0
(M行列)		1	2k		Δ⋍	4k＾2+4k+	(m)		0	√2k+1	Δ⋍	2k+1
Mk⋍	-	2k	4k+(4Q+1)				mｋ⋍	-	√2k+1	0