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BBP系公式とは、David Bailey, Peter Borwein, Simon Plouffeの三人によって発見されたBBP公式を発展させた公式群である。[1][2][3]

高校数学程度の数学の知識でも理解可能であるにも関わらず、1990年代まで発見されなかったこと、BBP公式は円周率をコンピュータで求めることに特化しているにも関わらず、円周率以外の様々な値を求めることに応用できること、2進数において任意の桁の値を、たとえ十兆桁目でも、そこだけを特定することが可能であることなど、様々な興味深い性質を持つ。

一般形 編集

BBP公式の一般系は

P(s,b,n,A)=\sum_{j=0}^\infty\frac{1}{b^j}\sum_{k=1}^n\frac{a_k}{(nj+k)^s}

と書ける。ただし、s,nは自然数、bは0以外の整数、A=(a_1,a_2,\cdots,a_n)a_1,a_2,\cdots,a_nは整数。

この記事でもこれに従って記述する。

公式集 編集

BBP系公式 編集

基底が1 編集

P(1,-1,4,(1,0,1,0))=\pi\sqrt{2}

P(2,-1,1,(1))=\frac{\pi^2}{12}

P(2,-1,2,(1,0))=K

P(2,-1,3,(1,1,0))=\text{Cl}\left(\frac{1}{3}\pi\right)

P(2,1,6,(64,-160,-56,-40,4,-1))=32(\log 2)^2

P(2,1,7,(1,1,-1,1,-1,-1,0))=\text{L}_{-2}(7)

基底が2 編集

P(1,2,1,(1))=2\log 2

P(1,2,2,(1,0))=\sqrt{2}\log (1+\sqrt{2})

P(1,-2,2,(1,0))=\sqrt{2}\arctan\left(\frac{1}{\sqrt{2}}\right)

P(1,4,2,(1,0))=\log 3

P(1,-4,4,(2,1,1,0))=2\arctan\phi^3

P(1,-4,4,(2,2,1,0))=\pi

P(1,-4,4,(2,4,1,0))=4\arctan\phi

P(1,-4,4,(2,8,1,0))=4\arctan\phi^5

P(1,-4,4,(6,0,-3,-1))=2\log 10

P(1,-8,3,(2,1,0))=\frac{4}{9}\pi\sqrt{3}

P(1,-8,6,(4,0,1,0,1,0))=\pi\sqrt{2}

P(1,16,4,(24,20,6,1))=16\log 10

P(1,16,8,(0,8,4,4,0,0,-1,0))=2\pi

P(1,16,8,(4,0,0,-2,-1,-1,0,0))=\pi

P(1,16,8,(8,0,4,0,-2,0,-1,0))=\arctan 2

P(1,16,8,(16,8,-8,4,-4,2,2,1))=8\log 10

P(1,64,6,(2,0,6,-1,-3,-1))=4\pi\sqrt{3}

P(1,-2^6,12,(0,32,24,0,0,4,0,0,3,2,0,0))=8\pi

P(1,-2^6,12,(2^5,0,2^4,0,-2^3,-2^4,-2^2,0,2,0,1,0))=\frac{2^6}{3}\arctan\phi^7

P(1,-64,12,(64,16,0,8,-16,8,12,-2,4,1,-3,-1))=32\log 10

64P(1,-2^{10},4,(32,8,1,0))+4P(1,-2^6,4,(8,4,1,0))=\pi

P(1,-2^{10},20,(0,512,0,0,-160,-128,0,0,0,-8,0,0,0,-8,-5,0,0,2,0,0))=2^6\pi

P(1,-2^{10},20,(2^9,2^9,-2^8,0,0,2^7,2^6,0,2^5,0,2^4,0,2^3,2^3,0,0,-2,2,1,0))=\frac{2^9}{5}\pi\sqrt{5}

\begin{align}
 &P(1,-2^{10},10,(2^8,0,-2^6,0,-4,0,-4,0,1,0))-P(1,-2^{10},4,(2^5,0,1,0)) \\
&=64\pi\end{align}

\begin{align}
&P(1,2^{12},24,(0,0,768,512,0,128,0,0,0,0,0,-16,0,-16,-12,0,0,0,0,2,0,-1,0)) \\
&=2^7\pi
\end{align}

\begin{align}
&P(1,2^{20},40,(0,2^{19},0,-3\cdot2^{17},-15\cdot2^{15},0,0,5\cdot2^{15},0,2^{15},0,-3\cdot2^{13},0,0,5\cdot2^{10}, \\
&5\cdot2^{11},0,2^{11},0,2^10,0,0,0,5\cdot2^7,15\cdot2^5,128,0,-96,0,0,0,40,0,8,-5,-6,0,0,0,0)) \\
&=2^{17}\arctan \frac{4}{5}
\end{align}

-4P(2,2,1(1))+P(2,-8,3,(8,-4,-1))=\frac{2\pi^2}{9}

-2P(2,2,2(0,2))+P(2,-4,4,(2,-1,-1))=\frac{\pi^2}{8}

P(2,-4,4,(2,2,1,0))=16\text{G}-2\pi\log2

24P(2,-4,4,(2,-2,1,0))-P(2,-2^6,4,(8,4,1,0))=2^5\text{G}

P(2,16,8,(16,-16,-8,-16,-4,-4,2,0))=\pi^2

P(2,16,8,(16,-40,-8,-28,-4,-10,2,-3))=6\log^22

P(2,-2^6,12,(32,-64,-48,-8,0,4,12,8,0,-1,0))=\frac{20}{9}\pi^2

\frac{9}{8}P(2,2^6,6,(16,0,-8,0,1,0))+\frac{3}{64}P(2,2^6,1,(1))-\frac{27}{4}P(2,4,1,(1))=2^5\log^22

P(2,64,6,(16,-24,-8,-6,1,0))=\frac{8}{9}\pi^2

P(2,64,6,(64,-160,-56,-40,4,-1))=32\log^22

P(2,-2^6,12,(2^5,-2^5,-2^5,0,-2^3,-2^4,-2^2,0,-2^2,-2,1,0))=\frac{2^6}{3}\text{G}

P(2,-2^6,12,(32,-56,0,-8,-36,-4,-7,0,10,0))-16P(2,-4,2,(1,0))=4\pi\log2

P(2,2^{12},12,(2^{10},-3\cdot2^9,-2^9,-3\cdot2^7,2^6,0,2^4,-3\cdot2^3,-8,-6,1,0))=\frac{2^9}{9}\pi^2

\begin{align}
&P(2,2^{12},24,(0,2^{10},0,-3\cdot2^9,0,-2^9,0,-3\cdot2^7,0,2^6,0,0,0,2^4,0,-3\cdot2^3,0,-8,0, \\
&-6,0,1,0,0))=\frac{2^7}{9}\pi^2
\end{align}

\begin{align}
&P(2,2^{12},24,(0,5\cdot2^{11},-23\cdot2^9,-5\cdot2^{11},-3^2\cdot2^8,-2^9,0,-3^2\cdot2^6,5\cdot2^7,2^7,2^9,0, \\
&3^2\cdot2^5,23\cdot2^3,0,0,4,8,-40,-9,18,-2,0))=256\pi\log2
\end{align}

\begin{align}
&P(2,2^{12},24,(2^{10},2^{10},-2^9,-3\cdot2^10,-256,-2^11,-256,-9\cdot2^7, \\
&-5\cdot2^6,64,64,0,-16,64,8,-72,4,-8,4,-12,5,4,-1,0))=2^{10}\text{G}
\end{align}

\begin{align}
&P(2,2^{12},24,(2^{11},-3\cdot2^{11},0,-2^{11},2^9,0,2^8,2^10,0,3\cdot2^7,2^6, \\
&0,-2^5,-3\cdot2^5,0,-2^6,-2^3,0,-2^2,2^3,0,6,-1,0))=\frac{2^{10}}{9}\pi\sqrt{3}\log2
\end{align}

\begin{align}
&P(2,2^{12},24,(2^{11},-2^{12},-3^2\cdot2^9,0,-2^9,-5\cdot2^8,-2^8,-7\cdot2^6,-2^8,2^6,0,-2^5,2^6, \\
&7\cdot2^3,0,8,20,4,0,7,4,-1,0))=256\pi\log2
\end{align}

\begin{align}
&P(2,2^{12},24,(2^{11},-2^{12},0,-2^{10},2^9,0,2^8,3\cdot2^8,0,2^8,2^6,0,-2^5,-2^6,0, \\
&-3\cdot2^4,-2^3,0,-2^2,2^2,0,2^2,-1,0))=\frac{2^{10}\cdot 5\sqrt{3}}{9}\text{Cl}\left(\frac{\pi}{3}\right) \\
\end{align}

\begin{align}
&P(2,2^{12},12,(2^{13},0,-3^2\cdot2^9,0,2^9,3^2\cdot2^6,0,0,-40,0,0,9)) \\
&+2^{10}P(2,16,4,(-4,0,1,0))-5\cdot2^9P(2,4,1,(1))=2^{11}
\end{align}

\begin{align}
&P(2,2^{12},24,(2^{12},-2^{13},-51\cdot2^9,15\cdot2^{10},-2^{10},39\cdot2^8,0,45\cdot2^8,37\cdot2^6,-2^9,0 \\
&,3\cdot2^8,-64,0,51\cdot2^3,45\cdot2^4,16,196,0,60,-37,0,0,0))=256\pi\log2
\end{align}

\begin{align}
&P(2,2^{12},24,(9\cdot2^9,-27\cdot2^9,-9\cdot2^11,27\cdot2^9,0,81\cdot2^7,9\cdot2^6,45\cdot2^8,9\cdot2^8,0,0, \\
&9\cdot2^6,-72,-216,9\cdot2^5,9\cdot2^6,0,162,-9,72,-36,0,0,0))=128\pi\sqrt{3}\log2
\end{align}

\begin{align}
&P(2,2^{60},120,(0,0,0,0,5\cdot2^{55},-3\cdot2^{55},0,-2^{57},0,0,0,3\cdot2^{52},0,0,-5\cdot2^{50},-2^{53}, \\
&0,-3\cdot2^{49},0,-5\cdot2^{48},0,0,0,-5\cdot2^{46},-5\cdot2^{45},0,0,0,0,-3\cdot2^{43},0,-2^{45},0,0,5\cdot2^{40}, \\
&3\cdot2^{40},0,0,0,-3\cdot2^{38},0,3\cdot2^{37},0,0,5\cdot2^{35},0,0,-5\cdot2^{34},0,0,0,0,0,-3\cdot2^{31}, \\
&-5\cdot2^{30},-2^{33},0,0,0,-2^{29},0,0,0,-2^{29},-5\cdot2^{25},-3\cdot2^{25},0,0,0,0,0,-5\cdot2^{22},0,0, \\
&5\cdot2^{20},0,0,-3\cdot2^{19},0,-3\cdot2^{18},0,0,0,3\cdot2^{16},5\cdot2^{15},0,0,-2^{17},0,-3\cdot2^{13}, \\
&0,0,0,0,-5\cdot2^{10},-5\cdot2^{10},0,0,0,-5\cdot2^8,0,-3\cdot2^7,0,-2^9,-5\cdot2^5,0,0,3\cdot2^4,0,0,0, \\
&-2^5,0,-6,5,0,0,0,0,0))=\frac{2^{43}}{45}\pi^2
\end{align}

4P(3,2^4,8,(16,-88,-8,92,-4,-22,2,27))-P(3,-8,1,(1))=21\zeta(3)

\begin{align}
&P(3,2^{12},24,(3\cdot2^{11},-21\cdot2^{11},3\cdot2^{13},15\cdot2^{11},-3\cdot2^9,3\cdot2^{10},3\cdot2^8,0,-3\cdot2^{10}, \\
&-21\cdot2^7,-192,-3\cdot2^9,-96,-21\cdot2^5,-3\cdot2^7,0,24,48,-12,120,48,-42,3,0)) \\
&=7\cdot2^8\zeta(3)
\end{align}

\begin{align}
&P(3,2^{12},24,(0,3\cdot2^{13},-27\cdot2^{12},3\cdot2^{14},0,93\cdot2^9,0,3\cdot2^{14},27\cdot2^9,3\cdot2^9,0,75\cdot2^6, \\
&0,3\cdot2^7,27\cdot2^6,3\cdot2^{10},0,93\cdot2^3,0,192,-216,24,0,3))=2^8\log^32
\end{align}

\begin{align}
&P(3,2^{12},24,(0,9\cdot2^{11},-135\cdot2^9,9\cdot2^{11},0,99\cdot2^8,0,27\cdot2^{10},135\cdot2^6,9\cdot2^7,0,45\cdot2^6, \\
&0,9\cdot2^5,135\cdot2^3,27\cdot2^6,0,396,0,72,-135,18,0,0))=2^5\pi^2\log2
\end{align}

\begin{align}&P(3,64,6,(16,-24,-8,-6,1,0))=\frac{280}{9}\zeta(3)-\frac{8}{9}\pi^2\log2\end{align}

\begin{align}
&5\cdot2^3P(3,-2^6,12,(32,-192,88,-8,84,-4,11,-2,1,0))+P(3,-2^{10},4,(32,8,1,0)) \\
&=2^4\pi^3
\end{align}

\begin{align}
&8P(3,-2^6,12,(7\cdot2^5,-37\cdot2^6,13\cdot17\cdot2^3,-7\cdot2^3,5\cdot71\cdot2^2,-7\cdot2^2,0, \\
&13\cdot17,-37\cdot2^2,7,0))+3P(3,-2^{10},4,(32,8,1,0))=64\pi\log^22
\end{align}

\begin{align}
&9P(3,2^{12},24,(0,2^{11},-15\cdot2^9,2^{11},0,11\cdot2^8,3\cdot2^{10},15\cdot2^6,2^7,0,5\cdot2^6, \\
&0,2^5,15\cdot2^8,3\cdot2^6,11\cdot2^2,0,8,-15,2,0,0))
\end{align}

\begin{align}
&P(3,2^{12},24,(2^{13},-5\cdot2^{14},-2^{12},17\cdot2^{13},-2^{11},5\cdot19\cdot2^9,2^{10},9\cdot2^{12},2^9,-5\cdot2^{10}, \\
&-2^8,2^6,-2^7,-5\cdot2^8,2^6,9\cdot2^8,2^5,5\cdot19\cdot2^3,-2^4,17\cdot2^5,-2^3,-5\cdot2^4,2^2,9)) \\
&=3\cdot2^8\log^32
\end{align}

\begin{align}
&P(3,2^{12},24,(5\cdot2^{11},-41\cdot2^{11},-5\cdot2^{10},49\cdot2^{11},-5\cdot2^9,67\cdot2^9,5\cdot2^8,27\cdot2^{10}, \\
&5\cdot2^7,-41\cdot2^7,-5\cdot2^6,-5\cdot2^7,-5\cdot2^5,-41\cdot2^5,5\cdot2^4,27\cdot2^6,5\cdot2^3, \\
&67\cdot2^3,-5\cdot2^2,49\cdot2^3,-5\cdot2^1,-41\cdot2^1,5,0))=3\cdot2^6\pi^2\log2
\end{align}

\begin{align}
&P(3,2^{12},24,(2^{11},-11\cdot2^{10},-2^{10},23\cdot2^9,-2^9,2^{12},2^8,27\cdot2^7,2^7,-11\cdot2^6,-2^6, \\
&-2^7,-2^5,-11\cdot2^4,2^4,27\cdot2^3,2^3,2^6,-2^2,23\cdot2^1,-2^1,-11,1,0)) \\
&=21\cdot2^5\zeta(3)
\end{align}

\begin{align}
&P(3,2^{60},120,(7\cdot2^{59},-37\cdot2^{60},-63\cdot2^{58},85\cdot2^{59},3861\cdot2^{56},-3357\cdot2^{55},0, \\
&-655\cdot2^{58},347\cdot2^{54},79\cdot2^{53},0,4703\cdot2^{52},-7\cdot2^{53},0,-1687\cdot2^{52},-655\cdot2^{54}, \\
&7\cdot2^{51},-4067\cdot2^{49},0,-6695\cdot2^{48},-347\cdot2^{48},0,0,-7375\cdot2^{46},-3861\cdot2^{46}, \\
&-37\cdot2^{48},-63\cdot2^{46},85\cdot2^{47},-7\cdot2^{45},-933\cdot2^{45},0,-655\cdot2^{46},347\cdot2^{42}, \\
&-37\cdot2^{44},875\cdot2^{43},4703\cdot2^{40},-7\cdot2^{41},0,63\cdot2^{40},-3105\cdot2^{38},7\cdot2^{39}, \\
&-4067\cdot2^{37},0,85\cdot2^{39},441\cdot2^{39},0,0,-7375\cdot2^{34},7\cdot2^{35},79\cdot2^{33},-63\cdot2^{34}, \\
&85\cdot2^{35},-7\cdot2^{33},-3357\cdot2^{31},-875\cdot2^{33},-655\cdot2^{34},347\cdot2^{30},-37\cdot2^{32},0, \\
&-167\cdot2^{32},-7\cdot2^{29},0,63\cdot2^{28},-655\cdot2^{30},-3861\cdot2^{26},-4067\cdot2^{25},0,85\cdot2^{27}, \\
&-347\cdot2^{24},-375\cdot2^{23},0,-7375\cdot2^{22},7\cdot2^{23},-37\cdot2^{24},1687\cdot2^{22},85\cdot2^{23},-7\cdot2^{21}, \\
&-3357\cdot2^{19},0,-3105\cdot2^{18},347\cdot2^{18},-37\cdot2^{20},0,4703\cdot2^{16},3861\cdot2^{16},0,63\cdot2^{16}, \\
&-655\cdot2^{18},7\cdot2^{15},-923\cdot2^{15},0,85\cdot2^{15},-347\cdot2^{12},0,-875\cdot2^{13},-7375\cdot2^{10} \\
&,7\cdot2^{11},-37\cdot2^{12},-63\cdot2^{10},-6695\cdot2^8,-7\cdot2^9,-3357\cdot2^7,0,-655\cdot2^{10}, \\
&-441\cdot2^9,-37\cdot2^8,0,4703\cdot2^4,-224,-375\cdot2^3,63\cdot2^4,-655\cdot2^6,56,-8134, \\
&875\cdot2^3,85\cdot2^3,-347,0,0,0))=2^{56}\pi\log^22
\end{align}

\begin{align}
&P(3,2^{60},120,(5\cdot2^{59},-15\cdot2^{60},-225\cdot2^{58},95\cdot2^{59},4115\cdot2^{56},-3735\cdot2^{55},0, \\
&-685\cdot2^{58},505\cdot2^{54},5\cdot2^{53},0,5485\cdot2^{52},-5\cdot2^{53},0,-1775\cdot2^{52},-685\cdot2^{54}, \\
&5\cdot2^{51},-3945\cdot2^{49},0,-7365\cdot2^{48},-505\cdot2^{48},0,0,-8125\cdot2^{46},-4115\cdot2^{46}, \\
&-15\cdot2^{48},-225\cdot2^{46},95\cdot2^{47},-5\cdot2^{45},-965\cdot2^{45},0,-685\cdot2^{46},505\cdot2^{42}, \\
&-15\cdot2^{44},125\cdot2^{46},5485\cdot2^{40},-5\cdot2^{41},0,225\cdot2^{40},-2835\cdot2^{38},5\cdot2^{39}, \\
&-3945\cdot2^{37},0,95\cdot2^{39},905\cdot2^{38},0,0,-8125\cdot2^{34},5\cdot2^{35},5\cdot2^{33},-225\cdot2^{34}, \\
&95\cdot2^{35},-5\cdot2^{33},-3735\cdot2^{31},-125\cdot2^{36},-685\cdot2^{34},505\cdot2^{30},-15\cdot2^{32},0, \\
&-165\cdot2^{32},-5\cdot2^{29},0,225\cdot2^{28},-685\cdot2^{30},-4115\cdot2^{26},-3945\cdot2^{25},0,95\cdot2^{27}, \\
&-505\cdot2^{24},-125\cdot2^{23},0,-8125\cdot2^{22},5\cdot2^{23},-15\cdot2^{24},1775\cdot2^{22},95\cdot2^{23},-5\cdot2^{21}, \\
&-3735\cdot2^{19},0,-2835\cdot2^{18},505\cdot2^{18},-15\cdot2^{20},0,5485\cdot2^{16},4115\cdot2^{16},0,225\cdot2^{16}, \\
&-685\cdot2^{18},5\cdot2^{15},-955\cdot2^{15},0,95\cdot2^{15},-505\cdot2^{12},0,-125\cdot2^{16},-8125\cdot2^{10}, \\
&5\cdot2^{11},-15\cdot2^{12},-225\cdot2^{10},-7365\cdot2^8,-5\cdot2^9,-3735\cdot2^7,0,-685\cdot2^{10},-905\cdot2^8, \\
&-15\cdot2^8,0,5485\cdot2^4,-160,-125\cdot2^3,225\cdot2^4,-685\cdot2^6,40,-7890,125\cdot2^6, \\
&95\cdot2^3,-505,0,0,0))=2^{54}\pi^3
\end{align}

\begin{align}
&P(3,2^{60},120,(2^{59},-3\cdot2^{60},11\cdot2^{57},0,23\cdot2^{56},3\cdot7\cdot2^{56},-2^{56},0,11\cdot2^{54},13\cdot2^{54}, \\
&2^{54},0,-2^{53},3\cdot2^{54},7\cdot2^{52},0,2^{51},-3\cdot7\cdot2^{50},2^{50},0,-11\cdot2^{48},3\cdot2^{50},-2^{48},0, \\
&-23\cdot2^{46},-3\cdot2^{48},11\cdot2^{45},0,-2^{45},-2^{46},-2^{44},0,11\cdot2^{42},-3\cdot2^{44},-23\cdot2^{41},0, \\
&-2^{41},3\cdot2^{42},-11\cdot2^{39},0,2^{39},-3\cdot7\cdot2^{38},2^{38},0,7\cdot2^{37},3\cdot2^{38},-2^{36},0,2^{35}, \\
&13\cdot2^{34},11\cdot2^{33},0,-2^{33},3\cdot7\cdot2^{32},23\cdot2^{31},0,11\cdot2^{30},-3\cdot2^{32},2^{30},0,-2^{29}, \\
&3\cdot2^{30},-11\cdot2^{27},0,-23\cdot2^{26},-3\cdot7\cdot2^{26},2^{26},0,-11\cdot2^{24},-13\cdot2^{24},-2^{24},0,2^{23}, \\
&-3\cdot2^{24},-7\cdot2^{22},0,-2^{21},3\cdot7\cdot2^{20},-2^{20},0,11\cdot2^{18},-3\cdot2^{20},2^{18},0,23\cdot2^{16}, \\
&3\cdot2^{18},-11\cdot2^{15},0,2^{15},2^{16},2^{14},0,-11\cdot2^{12},3\cdot2^{14},23\cdot2^{11},0,2^{11},-3\cdot2^{12}, \\
&11\cdot2^9,0,-2^9,3\cdot7\cdot2^8,-2^8,0,-7\cdot2^7,-3\cdot2^8,2^6,0,-2^5,-13\cdot2^4,-11\cdot2^3,0,2^3, \\
&-3\cdot7\cdot2^2,-23\cdot2,0,-11,3\cdot2^2,-1,0))=\frac{2^{55}}{5}\pi^3
\end{align}

\begin{align}
&P(3,2^{60},120,(7\cdot2^{59},-37\cdot2^{60},13\cdot17\cdot2^{57},0,192^\cdot2^{56},5\cdot71\cdot2^{56},-7\cdot2^{56},0, \\
&13\cdot17\cdot2^{54},227\cdot2^{54},7\cdot2^{54},0,-7\cdot2^{53},37\cdot2^{54},7\cdot11\cdot2^{52},0,7\cdot2^{51}, \\
&-5\cdot71\cdot2^{50},7\cdot2^{50},0,-13\cdot17\cdot2^{48},37\cdot2^{50},-7\cdot2^{48},0,-192^\cdot2^{46},-37\cdot2^{48}, \\
&13\cdot17\cdot2^{45},0,-7\cdot2^{45},-5\cdot2^{46},-7\cdot2^{44},0,13\cdot17\cdot2^{42},-37\cdot2^{44},-192^\cdot2^{41}, \\
&0,-7\cdot2^{41},37\cdot2^{42},-13\cdot17\cdot2^{39},0,7\cdot2^{39},-5\cdot71\cdot2^{38},7\cdot2^{38},0, \\
&7\cdot11\cdot2^{37},37\cdot2^{38},-7\cdot2^{36},0,7\cdot2^{35},227\cdot2^{34},13\cdot17\cdot2^{33},0,-7\cdot2^{33}, \\
&5\cdot71\cdot2^{32},192^\cdot2^{31},0,13\cdot17\cdot2^{30},-37\cdot2^{32},7\cdot2^{30},0,-7\cdot2^{29},37\cdot2^{30}, \\
&-13\cdot17\cdot2^{27},0,-192^\cdot2^{26},-5\cdot71\cdot2^{26},7\cdot2^{26},0,-13\cdot17\cdot2^{24},-227\cdot2^{24}, \\
&-7\cdot2^{24},0,7\cdot2^{23},-37\cdot2^{24},-7\cdot11\cdot2^{22},0,-7\cdot2^{21},5\cdot71\cdot2^{20},-7\cdot2^{20},0, \\
&13\cdot17\cdot2^{18},-37\cdot2^{20},7\cdot2^{18},0,192^\cdot2^{16},37\cdot2^{18},-13\cdot17\cdot2^{15},0,7\cdot2^{15}, \\
&5\cdot2^{16},7\cdot2^{14},0,-13\cdot17\cdot2^{12},37\cdot2^{14},192^\cdot2^{11},0,7\cdot2^{11},-37\cdot2^{12}, \\
&13\cdot17\cdot2^9,0,-7\cdot2^9,5\cdot71\cdot2^8,-7\cdot2^8,0,-7\cdot11\cdot2^7,-37\cdot2^8,7\cdot2^6,0,-7\cdot2^5, \\
&-227\cdot2^4,-13\cdot17\cdot2^3,0,7\cdot2^3,-5\cdot71\cdot2^2,-192^\cdot2,0,-17\cdot13,37\cdot2^2,-7,0)) \\
&=2^{57}\pi\log^22
\end{align}

\begin{align}
&P(3,2^{60},120,(2^{59},0,-83\cdot2^{57},11\cdot2^{59},3\cdot41\cdot2^{56},0,2^{56},-11\cdot2^{57},83\cdot2^{54},0, \\
&-2^{54},53\cdot2^{53},-2^{53},0,-3\cdot7\cdot2^{52},-11\cdot2^{53},2^{51},0,-2^{50},-34\cdot2^{49},-83\cdot2^{48},0, \\
&2^{48},-53\cdot2^{47},-3\cdot41\cdot2^{46},0,-83\cdot2^{45},11\cdot2^{47},-2^{45},0,2^{44},-11\cdot2^{45},83\cdot2^{42},0, \\
&3\cdot41\cdot2^{41},53\cdot2^{41},-2^{41},0,83\cdot2^{39},34\cdot2^{39},2^{39},0,-2^{38},11\cdot2^{39},3\cdot7\cdot2^{37},0, \\
&2^{36},-53\cdot2^{35},2^{35},0,-83\cdot2^{33},11\cdot2^{35},-2^{33},0,-3\cdot41\cdot2^{31},-11\cdot2^{33},83\cdot2^{30},0, \\
&-2^{30},0,-2^{29},0,83\cdot2^{27},-11\cdot2^{29},-3\cdot41\cdot2^{26},0,-2^{26},11\cdot2^{27},-83\cdot2^{24},0,2^{24}, \\
&-53\cdot2^{23},2^{23},0,3\cdot7\cdot2^{22},11\cdot2^{23},-2^{21},0,2^{20},34\cdot2^{19},83\cdot2^{18},0,-2^{18}, \\
&53\cdot2^{17},3\cdot41\cdot2^{16},0,83\cdot2^{15},-11\cdot2^{17},2^{15},0,-2^{14},11\cdot2^{15},-83\cdot2^{12},0, \\
&-3\cdot41\cdot2^{11},-53\cdot2^{11},2^{11},0,-83\cdot2^9,-34\cdot2^9,-2^9,0,2^8,-11\cdot2^9, \\
&-3\cdot7\cdot2^7,0,-2^6,53\cdot2^5,-2^5,0,83\cdot2^3,-11\cdot2^5,2^3,0,3\cdot41\cdot2,11\cdot2^3,-83,0,1,0)) \\
&=7\cdot2^{55}\zeta(3)
\end{align}

\begin{align}
&P(3,2^{60},120,(7\cdot2^{59},0,-1031\cdot2^{57},19\cdot2^{62},32^\cdot179\cdot2^{56},0,7\cdot2^{56},-19\cdot2^{60}, \\
&1031\cdot2^{54},0,-7\cdot2^{54},53\cdot13\cdot2^{53},-7\cdot2^{53},0,-33\cdot11\cdot2^{52},-19\cdot2^{56}, \\
&7\cdot2^{51},0,-7\cdot2^{50},-32^\cdot113\cdot2^{49},-1031\cdot2^{48},0,7\cdot2^{48},-53\cdot13\cdot2^{47}, \\
&-32^\cdot179\cdot2^{46},0,-1031\cdot2^{45},19\cdot2^{50},-7\cdot2^{45},0,7\cdot2^{44},-19\cdot2^{48},1031\cdot2^{42}, \\
&0,32^\cdot179\cdot2^{41},53\cdot13\cdot2^{41},-7\cdot2^{41},0,1031\cdot2^{39},32^\cdot113\cdot2^{39},7\cdot2^{39},0, \\
&-7\cdot2^{38},19\cdot2^{42},33\cdot11\cdot2^{37},0,7\cdot2^{36},-53\cdot13\cdot2^{35},7\cdot2^{35},0,-1031\cdot2^{33}, \\
&19\cdot2^{38},-7\cdot2^{33},0,-32^\cdot179\cdot2^{31},-19\cdot2^{36},1031\cdot2^{30},0,-7\cdot2^{30},0,-7\cdot2^{29},0, \\
&1031\cdot2^{27},-19\cdot2^{32},-32^\cdot179\cdot2^{26},0,-7\cdot2^{26},19\cdot2^{30},-1031\cdot2^{24},0,7\cdot2^{24}, \\
&-53\cdot13\cdot2^{23},7\cdot2^{23},0,33\cdot11\cdot2^{22},19\cdot2^{26},-7\cdot2^{21},0,7\cdot2^{20}, \\
&32^\cdot113\cdot2^{19},1031\cdot2^{18},0,-7\cdot2^{18},53\cdot13\cdot2^{17},32^\cdot179\cdot2^{16},0,1031\cdot2^{15}, \\
&-19\cdot2^{20},7\cdot2^{15},0,-7\cdot2^{14},19\cdot2^{18},-1031\cdot2^{12},0,-32^\cdot179\cdot2^{11},-53\cdot13\cdot2^{11}, \\
&7\cdot2^{11},0,-1031\cdot2^9,-32^\cdot113\cdot2^9,-7\cdot2^9,0,7\cdot2^8,-19\cdot2^{12},-33\cdot11\cdot2^7,0,-7\cdot2^6, \\
&53\cdot13\cdot2^5,-7\cdot2^5,0,1031\cdot2^3,-19\cdot2^8,7\cdot2^3,0,32^\cdot179\cdot2,19\cdot2^6,-1031,0,7,0)) \\
&=\frac{5\cdot2^{56}}{3}\pi^2\log2
\end{align}

\begin{align}
&P(3,2^{60},120,(2^{59},0,-11\cdot19\cdot2^{57},5\cdot2^{62},373\cdot2^{56},0,2^{56},-5\cdot2^{60}, \\
&11\cdot19\cdot2^{54},0,-2^{54},367\cdot2^{53},-2^{53},0,-83\cdot2^{52},-5\cdot2^{56},2^{51},0,-2^{50}, \\
&-5\cdot43\cdot2^{49},-11\cdot19\cdot2^{48},0,2^{48},-367\cdot2^{47},-373\cdot2^{46},0,-11\cdot19\cdot2^{45}, \\
&5\cdot2^{50},-2^{45},0,2^{44},-5\cdot2^{48},11\cdot19\cdot2^{42},0,373\cdot2^{41},367\cdot2^{41},-2^{41},0, \\
&11\cdot19\cdot2^{39},5\cdot43\cdot2^{39},2^{39},0,-2^{38},5\cdot2^{42},83\cdot2^{37},0,2^{36},-367\cdot2^{35}, \\
&2^{35},0,-11\cdot19\cdot2^{33},5\cdot2^{38},-2^{33},0,-373\cdot2^{31},-5\cdot2^{36},11\cdot19\cdot2^{30},0, \\
&-2^{30},-2^{32},-2^{29},0,11\cdot19\cdot2^{27},-5\cdot2^{32},-373\cdot2^{26},0,-2^{26},5\cdot2^{30}, \\
&-11\cdot19\cdot2^{24},0,2^{24},-367\cdot2^{23},2^{23},0,83\cdot2^{22},5\cdot2^{26},-2^{21},0,2^{20}, \\
&5\cdot43\cdot2^{19},11\cdot19\cdot2^{18},0,-2^{18},367\cdot2^{17},373\cdot2^{16},0,11\cdot19\cdot2^{15}, \\
&-5\cdot2^{20},2^{15},0,-2^{14},5\cdot2^{18},-11\cdot19\cdot2^{12},0,-373\cdot2^{11},-367\cdot2^{11},2^{11},0, \\
&-11\cdot19\cdot2^9,-5\cdot43\cdot2^9,-2^9,0,2^8,-5\cdot2^{12},-83\cdot2^7,0,-2^6,367\cdot2^5,-2^5,0, \\
&11\cdot19\cdot2^3,-5\cdot2^8,2^3,0,373\cdot2,5\cdot2^6,-19\cdot11,0,1,2^2))=\frac{2^{58}}{3}\log^33
\end{align}

\begin{align}
&P(3,16,8,(8,0,-4,-4,-2,0,1,1))=\frac{1}{3}\log^32-\frac{5}{12}\pi^2\log2+\frac{35}{4}\zeta(3)
\end{align}

\begin{align}
&P(4,2^{12},24,(27\cdot2^{11},-513\cdot2^{11},135\cdot2^{14},-27\cdot2^{11},-27\cdot2^9,-621\cdot2^{10},27\cdot2^8, \\
&-729\cdot2^{10},-135\cdot2^{11},-513\cdot2^7,-27\cdot2^6,-189\cdot2^9,-27\cdot2^5,-513\cdot2^5,-135\cdot2^8, \\
&-729\cdot2^6,216,-621\cdot2^4,-108,-216,135\cdot2^5,-1026,27,0))=164\pi^4
\end{align}

\begin{align}
&P(4,2^{12},24,(73\cdot2^{12},-2617\cdot2^{12},8455\cdot2^{12},-2533\cdot2^{12},-73\cdot2^{10},-25781\cdot2^9, \\
&73\cdot2^9,-6891\cdot2^{11},-8455\cdot2^9,-2617\cdot2^8,-73\cdot2^7,-23551\cdot2^6,-73\cdot2^6,-2617\cdot2^6, \\
&-8455\cdot2^6,-6891\cdot2^7,73\cdot2^4,-25781\cdot2^3,-73\cdot2^3, \\
&-2533\cdot2^4,8455\cdot2^3,-10468,146,-615))=205\cdot2^5\log^42
\end{align}

\begin{align}
&P(4,2^{12},24,(121\cdot2^{11},-3775\cdot2^{11},10375\cdot2^{11},-1597\cdot2^{11},-121\cdot2^9,-3421\cdot2^{11}, \\
&121\cdot2^8,-7695\cdot2^{10},-10375\cdot2^8,-3775\cdot2^7,-121\cdot2^6,-3539\cdot2^8,-121\cdot2^5, \\
&-3775\cdot2^5,-10375\cdot2^5,-7695\cdot2^6,121\cdot2^3,-3421\cdot2^5,-484,-1597\cdot2^3, \\
&41500,-7550,121,0))=41\cdot2^5\pi^2\log^22
\end{align}

\begin{align}
&P(4,2^{60},120,(259,-5\cdot2^{61},11\cdot17\cdot2^{57},-2^{61},-127\cdot2^{56},-5\cdot2^{59},2^{56},-2^{61}, \\
&-11\cdot17\cdot2^{54},-5\cdot2^{57},-2^{54},-5\cdot41\cdot2^{53},-2^{53},-5\cdot2^{55},-31\cdot2^{52},-2^{57}, \\
&2^{51},-5\cdot2^{53},-2^{50},109\cdot2^{49},11\cdot17\cdot2^{48},-5\cdot2^{51},2^{48},53\cdot2^{47},127\cdot2^{46}, \\
&-5\cdot2^{49},11\cdot17\cdot2^{45},-2^{49},-2^{45},-5\cdot2^{47},2^{44},-2^{49},-11\cdot17\cdot2^{42},-5\cdot2^{45}, \\
&-127\cdot2^{41},-5\cdot41\cdot2^{41},-2^{41},-5\cdot2^{43},-11\cdot17\cdot2^{39},-33\cdot7\cdot2^{39},2^{39}, \\
&-5\cdot2^{41},-2^{38},-2^{41},31\cdot2^{37},-5\cdot2^{39},2^{36},53\cdot2^{35},2^{35},-5\cdot2^{37},11\cdot17\cdot2^{33}, \\
&-2^{37},-2^{33},-5\cdot2^{35},127\cdot2^{31},-2^{37},-11\cdot17\cdot2^{30},-5\cdot2^{33},-2^{30},-5\cdot2^{33},-2^{29}, \\
&-5\cdot2^{31},-11\cdot17\cdot2^{27},-2^{33},127\cdot2^{26},-5\cdot2^{29},-2^{26},-2^{29},11\cdot17\cdot2^{24},-5\cdot2^{27}, \\
&2^{24},53\cdot2^{23},2^{23},-5\cdot2^{25},31\cdot2^{22},-2^{25},-2^{21},-5\cdot2^{23},2^{20},-33\cdot7\cdot2^{19}, \\
&-11\cdot17\cdot2^{18},-5\cdot2^{21},-2^{18},-5\cdot41\cdot2^{17},-127\cdot2^{16},-5\cdot2^{19},-11\cdot17\cdot2^{15}, \\
&-2^{21},2^{15},-5\cdot2^{17},-2^{14},-2^{17},11\cdot17\cdot2^{12},-5\cdot2^{15},127\cdot2^{11},53\cdot2^{11},2^{11}, \\
&-5\cdot2^{13},11\cdot17\cdot2^9,109\cdot2^9,-2^9,-5\cdot2^{11},2^8,-2^{13},-31\cdot2^7,-5\cdot2^9,-2^6, \\
&-5\cdot41\cdot2^5,-2^5,-5\cdot2^7,-11\cdot17\cdot2^3,-2^9,2^3,-5\cdot2^5,-127\cdot2,-2^5,17\cdot11, \\
&-5\cdot2^3,1,0))=\frac{7\cdot71\cdot2^{51}}{675}\pi^4
\end{align}

\begin{align}
&P(4,2^{60},120,(823\cdot2^{59},-5\cdot3137\cdot2^{60},11\cdot40829\cdot2^{57},-18047\cdot2^{60}, \\
&-277\cdot1723\cdot2^{56},-5\cdot3137\cdot2^{58},823\cdot2^{56},1181\cdot2^{59},-11\cdot40829\cdot2^{54}, \\
&-5\cdot3137\cdot2^{56},-823\cdot2^{54},-595141\cdot2^{53},-823\cdot2^{53},-5\cdot3137\cdot2^{54}, \\
&29\cdot457\cdot2^{52},1181\cdot2^{55},823\cdot2^{51},-5\cdot3137\cdot2^{52},-823\cdot2^{50},331249\cdot2^{49}, \\
&11\cdot40829\cdot2^{48},-5\cdot3137\cdot2^{50},823\cdot2^{48},192^\cdot1301\cdot2^{47},277\cdot1723\cdot2^{46}, \\
&-5\cdot3137\cdot2^{48},11\cdot40829\cdot2^{45},-18047\cdot2^{48},-823\cdot2^{45},-5\cdot3137\cdot2^{46}, \\
&823\cdot2^{44},1181\cdot2^{47},-11\cdot40829\cdot2^{42},-5\cdot3137\cdot2^{44},-277\cdot1723\cdot2^{41}, \\
&-595141\cdot2^{41},-823\cdot2^{41},-5\cdot3137\cdot2^{42},-11\cdot40829\cdot2^{39},-3\cdot72^\cdot13\cdot239\cdot2^{39}, \\
&823\cdot2^{39},-5\cdot3137\cdot2^{40},-823\cdot2^{38},-18047\cdot2^{40},-29\cdot457\cdot2^{37},-5\cdot3137\cdot2^{38}, \\
&823\cdot2^{36},192^\cdot1301\cdot2^{35},823\cdot2^{35},-5\cdot3137\cdot2^{36},11\cdot40829\cdot2^{33},-18047\cdot2^{36}, \\
&-823\cdot2^{33},-5\cdot3137\cdot2^{34},277\cdot1723\cdot2^{31},1181\cdot2^{35},-11\cdot40829\cdot2^{30}, \\
&-5\cdot3137\cdot2^{32},-823\cdot2^{30},-29879\cdot2^{31},-823\cdot2^{29},-5\cdot3137\cdot2^{30},-11\cdot40829\cdot2^{27}, \\
&1181\cdot2^{31},277\cdot1723\cdot2^{26},-5\cdot3137\cdot2^{28},-823\cdot2^{26},-18047\cdot2^{28},11\cdot40829\cdot2^{24}, \\
&-5\cdot3137\cdot2^{26},823\cdot2^{24},192^\cdot1301\cdot2^{23},823\cdot2^{23},-5\cdot3137\cdot2^{24}, \\
&-29\cdot457\cdot2^{22},-18047\cdot2^{24},-823\cdot2^{21},-5\cdot3137\cdot2^{22},823\cdot2^{20}, \\
&-3\cdot72^\cdot13\cdot239\cdot2^{19},-11\cdot40829\cdot2^{18},-5\cdot3137\cdot2^{20},-823\cdot2^{18}, \\
&-595141\cdot2^{17},-277\cdot1723\cdot2^{16},-5\cdot3137\cdot2^{18},-11\cdot40829\cdot2^{15},1181\cdot2^{19}, \\
&823\cdot2^{15},-5\cdot3137\cdot2^{16},-823\cdot2^{14},-18047\cdot2^{16},11\cdot40829\cdot2^{12},-5\cdot3137\cdot2^{14} \\
&,277\cdot1723\cdot2^{11},192^\cdot1301\cdot2^{11},823\cdot2^{11},-5\cdot3137\cdot2^{12},11\cdot40829\cdot2^9, \\
&331249\cdot2^9,-823\cdot2^9,-5\cdot3137\cdot2^{10},823\cdot2^8,1181\cdot2^{11},29\cdot457\cdot2^7,-5\cdot3137\cdot2^8, \\
&-823\cdot2^6,-595141\cdot2^5,-823\cdot2^5,-5\cdot3137\cdot2^6,-11\cdot40829\cdot2^3,1181\cdot2^7,823\cdot2^3, \\
&-5\cdot3137\cdot2^4,-277\cdot1723\cdot2,-18047\cdot2^4,40829\cdot11,-5\cdot3137\cdot2^2,823, \\
&-3\cdot7\cdot71\cdot2))=7\cdot71\cdot2^{51}\log^42
\end{align}

\begin{align}
&P(4,2^{60},120,(13^2\cdot19\cdot2^{59},-37\cdot179\cdot2^{63},5\cdot13\cdot19973\cdot2^{57},-5\cdot1663\cdot2^{62}, \\
&-17\cdot179\cdot379\cdot2^{56},-37\cdot179\cdot2^{61},13^2\cdot19\cdot2^{56},-4931\cdot2^{60}, \\
&-5\cdot13\cdot19973\cdot2^{54},-37\cdot179\cdot2^{59},-13^2\cdot19\cdot2^{54},-43\cdot36529\cdot2^{53}, \\
&-13^2\cdot19\cdot2^{53},-37\cdot179\cdot2^{57},-5\cdot15137\cdot2^{52},-4931\cdot2^{56},13^2\cdot19\cdot2^{51}, \\
&-37\cdot179\cdot2^{55},-13^2\cdot19\cdot2^{50},5\cdot176159\cdot2^{49},5\cdot13\cdot19973\cdot2^{48}, \\
&-37\cdot179\cdot2^{53},13^2\cdot19\cdot2^{48},55\cdot367\cdot2^{47},17\cdot179\cdot379\cdot2^{46}, \\
&-37\cdot179\cdot2^{51},5\cdot13\cdot19973\cdot2^{45},-5\cdot1663\cdot2^{50},-13^2\cdot19\cdot2^{45}, \\
&-37\cdot179\cdot2^{49},13^2\cdot19\cdot2^{44},-4931\cdot2^{48},-5\cdot13\cdot19973\cdot2^{42},-37\cdot179\cdot2^{47}, \\
&-17\cdot179\cdot379\cdot2^{41},-43\cdot36529\cdot2^{41},-13^2\cdot19\cdot2^{41},-37\cdot179\cdot2^{45}, \\
&-5\cdot13\cdot19973\cdot2^{39},-35\cdot7\cdot13\cdot59\cdot2^{39},13^2\cdot19\cdot2^{39},-37\cdot179\cdot2^{43}, \\
&-13^2\cdot19\cdot2^{38},-5\cdot1663\cdot2^{42},5\cdot15137\cdot2^{37},-37\cdot179\cdot2^{41},13^2\cdot19\cdot2^{36}, \\
&55\cdot367\cdot2^{35},13^2\cdot19\cdot2^{35},-37\cdot179\cdot2^{39},5\cdot13\cdot19973\cdot2^{33},-5\cdot1663\cdot2^{38}, \\
&-13^2\cdot19\cdot2^{33},-37\cdot179\cdot2^{37},17\cdot179\cdot379\cdot2^{31},-4931\cdot2^{36},-5\cdot13\cdot19973\cdot2^{30}, \\
&-37\cdot179\cdot2^{35},-13^2\cdot19\cdot2^{30},-37\cdot179\cdot2^{35},-13^2\cdot19\cdot2^{29},-37\cdot179\cdot2^{33}, \\
&-5\cdot13\cdot19973\cdot2^{27},-4931\cdot2^{32},17\cdot179\cdot379\cdot2^{26},-37\cdot179\cdot2^{31},-13^2\cdot19\cdot2^{26}, \\
&-5\cdot1663\cdot2^{30},5\cdot13\cdot19973\cdot2^{24},-37\cdot179\cdot2^{29},13^2\cdot19\cdot2^{24},55\cdot367\cdot2^{23}, \\
&13^2\cdot19\cdot2^{23},-37\cdot179\cdot2^{27},5\cdot15137\cdot2^{22},-5\cdot1663\cdot2^{26},-13^2\cdot19\cdot2^{21}, \\
&-37\cdot179\cdot2^{25},13^2\cdot19\cdot2^{20},-35\cdot7\cdot13\cdot59\cdot2^{19},-5\cdot13\cdot19973\cdot2^{18}, \\
&-37\cdot179\cdot2^{23},-13^2\cdot19\cdot2^{18},-43\cdot36529\cdot2^{17},-17\cdot179\cdot379\cdot2^{16},-37\cdot179\cdot2^{21}, \\
&-5\cdot13\cdot19973\cdot2^{15},-4931\cdot2^{20},13^2\cdot19\cdot2^{15},-37\cdot179\cdot2^{19},-13^2\cdot19\cdot2^{14}, \\
&-5\cdot1663\cdot2^{18},5\cdot13\cdot19973\cdot2^{12},-37\cdot179\cdot2^{17},17\cdot179\cdot379\cdot2^{11},55\cdot367\cdot2^{11}, \\
&13^2\cdot19\cdot2^{11},-37\cdot179\cdot2^{15},5\cdot13\cdot19973\cdot2^9,5\cdot176159\cdot2^9,-13^2\cdot19\cdot2^9, \\
&-37\cdot179\cdot2^{13},13^2\cdot19\cdot2^8,-4931\cdot2^{12},-5\cdot15137\cdot2^7,-37\cdot179\cdot2^{11}, \\
&-13^2\cdot19\cdot2^6,-43\cdot36529\cdot2^5,-13^2\cdot19\cdot2^5,-37\cdot179\cdot2^9,-5\cdot13\cdot19973\cdot2^3, \\
&-4931\cdot2^8,13^2\cdot19\cdot2^3,-37\cdot179\cdot2^7,-17\cdot179\cdot379\cdot2,-5\cdot1663\cdot2^6, \\
&13\cdot19973\cdot5,-37\cdot179\cdot2^5,19\cdot13^2,0))=7\cdot71\cdot2^{54}\pi^2\log^22
\end{align}

\begin{align}
&P(4,2^{12},24,(2^11,-2^{12},-7\cdot2^9,0,-2^9,-5\cdot2^8,-2^8,0,-7\cdot2^6,-2^8,2^6,0,-2^5,2^6, \\
&7\cdot2^3,0,2^3,5\cdot2^2,2^2,0,7,2^2,-1,0))=\frac{5\cdot2^12}{9}\text{Cl}_4\left(\frac{\pi}{2}\right)+\frac{32}{3}\pi\log^32-24\pi^3\log2
\end{align}

\begin{align}
&P(4,2^{60},120,(260,23\cdot2^{60},-239\cdot2^{57},0,-3\cdot211\cdot2^{55},-5\cdot67\cdot2^{56},-2^{57},0, \\
&-239\cdot2^{54},-32^\cdot72^\cdot2^{53},2^{55},0,-2^{54},-23\cdot2^{54},-3\cdot72^\cdot2^{50},0,2^{52}, \\
&5\cdot67\cdot2^{50},2^{51},0,239\cdot2^{48},-23\cdot2^{50},-2^{49},0,3\cdot211\cdot2^{45},23\cdot2^{48}, \\
&-239\cdot2^{45},0,-2^{46},-32^\cdot5\cdot2^{43},-2^{45},0,-239\cdot2^{42},23\cdot2^{44},3\cdot211\cdot2^{40},0, \\
&-2^{42},-23\cdot2^{42},239\cdot2^{39},0,2^{40},5\cdot67\cdot2^{38},2^{39},0,-3\cdot72^\cdot2^{35},-23\cdot2^{38}, \\
&-2^{37},0,2^{36},-32^\cdot72^\cdot2^{33},-239\cdot2^{33},0,-2^{34},-5\cdot67\cdot2^{32},-3\cdot211\cdot2^{30},0, \\
&-239\cdot2^{30},23\cdot2^{32},2^{31},0,-2^{30},-23\cdot2^{30},239\cdot2^{27},0,3\cdot211\cdot2^{25},5\cdot67\cdot2^{26}, \\
&2^{27},0,239\cdot2^{24},32^\cdot72^\cdot2^{23},-2^{25},0,2^{24},23\cdot2^{24},3\cdot72^\cdot2^{20},0,-2^{22}, \\
&-5\cdot67\cdot2^{20},-2^{21},0,-239\cdot2^{18},23\cdot2^{20},2^{19},0,-3\cdot211\cdot2^{15},-23\cdot2^{18},239\cdot2^{15}, \\
&0,2^{16},32^\cdot5\cdot2^{13},2^{15},0,239\cdot2^{12},-23\cdot2^{14},-3\cdot211\cdot2^{10},0,2^{12},23\cdot2^{12}, \\
&-239\cdot2^9,0,-2^{10},-5\cdot67\cdot2^8,-2^9,0,3\cdot72^\cdot2^5,23\cdot2^8,2^7,0,-2^6,32^\cdot72^\cdot2^3,239\cdot2^3, \\
&0,2^4,5\cdot67\cdot2^2,211\cdot3,0,239,-23\cdot2^2,-2,0))=3\cdot2^{60}\text{Cl}_4\left(\frac{\pi}{2}\right)+\frac{29\cdot2^{52}}{3}\pi\log^32-47\cdot2^{50}\pi^3\log2
\end{align}

\begin{align}
&27P(5,2^{12},24,(782336,-54083584,296023040,-118439936,-195584,245927936, \\
&97792,36202086,-37002880,-3380224,-24448,-58359488,-12224,-845056, \\
&-4625360,226263043056,3843634,-1528,-462656,578170,-52816,382, \\
&1196875))+41024P(5,-2^{10},4,(-128,0,4,1))=16168\pi^4\log2
\end{align}

\begin{align}
&3P(5,2^{12},24,(29143040,-2536849408,16339911680,-7170572288,-7285760, \\
&17248845824,3642880,25002508288,-2042488960,-158553088,-910720, \\
&-3705698240,-455360,-39638272,-255311120,1562656768,113840, \\
&269513216,-56920,-28010048,31913890,-2477392,14230,78709375)) \\
&+302244P(5,-2^{10},4,(-128,0,4,1))=129344\pi^2\log^32
\end{align}

\begin{align}
&P(5,2^{12},24,(126976,-6610944,33418240,-12722176,-31744,25829376 \\
&,15782,38170624,-4177280,-413184,-3968,-6323008,-1984,-103296, \\
&-522160,2385564,496,403584,-248,-49696,65270,-6456,62,128125)) \\
&+1476P(5,-2^{10},4,(-128,0,4,1))=250604\zeta(5)
\end{align}

\begin{align}
&P(5,2^{12},24,(11399168,-1071480832,7344051200,-3072770048,-2849792, \\
&9616483328,1424896,13679460352,-918006400,-66067552,-356224, \\
&-1886951744,-178112,-16741888,-114750800,854966272,44528,150257552, \\
&-22264,-12003008,14343850,-1046368,5566,41307505)) \\
&+52796P(5,-2^{10},4,(-128,0,4,1))=64672\log^52
\end{align}

\begin{align}
&P(5,2^{60},120,(279\cdot2^{59},-7263\cdot2^{60},293715\cdot2^{57},-13977\cdot2^{60}, \\
&-1153683\cdot2^{56},28377\cdot2^{60},279\cdot2^{56},83871\cdot2^{59},-293715\cdot2^{54}, \\
&-7263\cdot2^{56},-279\cdot2^{54},-889173\cdot2^{53},-279\cdot2^{53},-7263\cdot2^{54}, \\
&429705\cdot2^{52},83871\cdot2^{55},279\cdot2^{51},28377\cdot2^{54},-279\cdot2^{50}, \\
&1041309\cdot2^{49},293715\cdot2^{48},-7263\cdot2^{50},279\cdot2^{48},1153125\cdot2^{47}, \\
&1153683\cdot2^{46},-7263\cdot2^{48},293715\cdot2^{45},-13977\cdot2^{48},-279\cdot2^{45}, \\
&28377\cdot2^{48},279\cdot2^{44},83871\cdot2^{47},-293715\cdot2^{42},-7263\cdot2^{44},-1153683\cdot2^{41}, \\
&-889173\cdot2^{41},-279\cdot2^{41},-7263\cdot2^{42},-293715\cdot2^{39},188811\cdot2^{39},279\cdot2^{39}, \\
&28377\cdot2^{42},-279\cdot2^{38},-13977\cdot2^{40},-429705\cdot2^{37},-7263\cdot2^{38},279\cdot2^{36}, \\
&1153125\cdot2^{35},279\cdot2^{35},-7263\cdot2^{36},293715\cdot2^{33},-13977\cdot2^{36},-279\cdot2^{33}, \\
&28377\cdot2^{36},1153683\cdot2^{31},83871\cdot2^{35},-293715\cdot2^{30},-7263\cdot2^{32},-279\cdot2^{30}, \\
&16497\cdot2^{33},-279\cdot2^{29},-7263\cdot2^{30},-293715\cdot2^{27},83871\cdot2^{31},1153683\cdot2^{26}, \\
&28377\cdot2^{30},-279\cdot2^{26},-13977\cdot2^{28},293715\cdot2^{24},-7263\cdot2^{26},279\cdot2^{24}, \\
&1153125\cdot2^{23},279\cdot2^{23},-7263\cdot2^{24},-429705\cdot2^{22},-13977\cdot2^{24},-279\cdot2^{21}, \\
&28377\cdot2^{24},279\cdot2^{20},188811\cdot2^{19},-293715\cdot2^{18},-7263\cdot2^{20},-279\cdot2^{18}, \\
&-889173\cdot2^{17},-1153683\cdot2^{16},-7263\cdot2^{18},-293715\cdot2^{15},83871\cdot2^{19},279\cdot2^{15}, \\
&28377\cdot2^{18},-279\cdot2^{14},-13977\cdot2^{16},293715\cdot2^{12},-7263\cdot2^{14},1153683\cdot2^{11}, \\
&1153125\cdot2^{11},279\cdot2^{11},-7263\cdot2^{12},293715\cdot2^9,1041309\cdot2^9,-279\cdot2^9,28377\cdot2^{12}, \\
&279\cdot2^8,83871\cdot2^{11},429705\cdot2^7,-7263\cdot2^8,-279\cdot2^6,-889173\cdot2^5,-279\cdot2^5, \\
&-7263\cdot2^6,-293715\cdot2^3,83871\cdot2^7,279\cdot2^3,28377\cdot2^6,-2307366, \\
&-13977\cdot2^4,293715,-29052,279,0))=62651\cdot2^49\zeta(5)
\end{align}

\begin{align}
&P(5,2^{60},120,(2783\cdot2^{59},-32699\cdot2^{62},7171925\cdot2^{57},-187547\cdot2^{61}, \\
&-41252441\cdot2^{56},9391097\cdot2^{57},2783\cdot2^{56},52183\cdot2^65,-7171925\cdot2^{54}, \\
&-32699\cdot2^{58},-2783\cdot2^{54},-29483621\cdot2^{53},-2783\cdot2^{53},-32699\cdot2^{56}, \\
&17037475\cdot2^{52},52183\cdot2^{61},2783\cdot2^{51},9391097\cdot2^{51},-2783\cdot2^{50}, \\
&38246123\cdot2^{49},7171925\cdot2^{48},-32699\cdot2^{52},2783\cdot2^{48},41307505\cdot2^{47}, \\
&41252441\cdot2^{46},-32699\cdot2^{50},7171925\cdot2^{45},-187547\cdot2^{49},-2783\cdot2^{45}, \\
&9391097\cdot2^{45},2783\cdot2^{44},52183\cdot2^{53},-7171925\cdot2^{42},-32699\cdot2^{46}, \\
&-41252441\cdot2^{41},-29483621\cdot2^{41},-2783\cdot2^{41},-32699\cdot2^{44},-7171925\cdot2^{39}, \\
&12188517\cdot2^{39},2783\cdot2^{39},9391097\cdot2^{39},-2783\cdot2^{38},-187547\cdot2^{41}, \\
&-17037475\cdot2^{37},-32699\cdot2^{40},2783\cdot2^{36},41307505\cdot2^{35},2783\cdot2^{35}, \\
&-32699\cdot2^{38},7171925\cdot2^{33},-187547\cdot2^{37},-2783\cdot2^{33},9391097\cdot2^{33}, \\
&41252441\cdot2^{31},52183\cdot2^{41},-7171925\cdot2^{30},-32699\cdot2^{34},-2783\cdot2^{30}, \\
&5881627\cdot2^{30},-2783\cdot2^{29},-32699\cdot2^{32},-7171925\cdot2^{27},52183\cdot2^{37}, \\
&41252441\cdot2^{26},9391097\cdot2^{27},-2783\cdot2^{26},-187547\cdot2^{29},7171925\cdot2^{24}, \\
&-32699\cdot2^{28},2783\cdot2^{24},41307505\cdot2^{23},2783\cdot2^{23},-32699\cdot2^{26}, \\
&-17037475\cdot2^{22},-187547\cdot2^{25},-2783\cdot2^{21},9391097\cdot2^{21},2783\cdot2^{20}, \\
&12188517\cdot2^{19},-7171925\cdot2^{18},-32699\cdot2^{22},-2783\cdot2^{18},-29483621\cdot2^{17}, \\
&-41252441\cdot2^{16},-32699\cdot2^{20},-7171925\cdot2^{15},52183\cdot2^{25},2783\cdot2^{15}, \\
&9391097\cdot2^{15},-2783\cdot2^{14},-187547\cdot2^{17},7171925\cdot2^{12},-32699\cdot2^{16}, \\
&41252441\cdot2^{11},41307505\cdot2^{11},2783\cdot2^{11},-32699\cdot2^{14},7171925\cdot2^9, \\
&38246123\cdot2^9,-2783\cdot2^9,9391097\cdot2^9,2783\cdot2^8,52183\cdot2^{17},17037475\cdot2^7, \\
&-32699\cdot2^{10},-2783\cdot2^6,-29483621\cdot2^5,-2783\cdot2^5,-32699\cdot2^8,-7171925\cdot2^3, \\
&52183\cdot2^{13},2783\cdot2^3,9391097\cdot2^3,-82504882,-187547\cdot2^5,7171925, \\
&-32699\cdot2^4,2783,30315))=2021\cdot2^{52}\log^52
\end{align}

\begin{align}
&P(5,2^{60},120,(21345\cdot2^{59},-464511\cdot2^{61},47870835\cdot2^{57},-1312971\cdot2^{61}, \\
&-236170815\cdot2^{56},1579179\cdot2^{62},21345\cdot2^{56},286131\cdot2^65,-47870835\cdot2^{54}, \\
&-464511\cdot2^{57},-21345\cdot2^{54},-173704605\cdot2^{53},-21345\cdot2^{53},-464511\cdot2^{55}, \\
&94128645\cdot2^{52},286131\cdot2^{61},21345\cdot2^{51},1579179\cdot2^{56},-21345\cdot2^{50}, \\
&215120589\cdot2^{49},47870835\cdot2^{48},-464511\cdot2^{51},21345\cdot2^{48},236128125\cdot2^{47}, \\
&236170815\cdot2^{46},-464511\cdot2^{49},47870835\cdot2^{45},-1312971\cdot2^{49},-21345\cdot2^{45}, \\
&1579179\cdot2^{50},21345\cdot2^{44},286131\cdot2^{53},-47870835\cdot2^{42},-464511\cdot2^{45}, \\
&-236170815\cdot2^{41},-173704605\cdot2^{41},-21345\cdot2^{41},-464511\cdot2^{43},-47870835\cdot2^{39}, \\
&56870019\cdot2^{39},21345\cdot2^{39},1579179\cdot2^{44},-21345\cdot2^{38},-1312971\cdot2^{41}, \\
&-94128645\cdot2^{37},-464511\cdot2^{39},21345\cdot2^{36},236128125\cdot2^{35},21345\cdot2^{35}, \\
&-464511\cdot2^{37},47870835\cdot2^{33},-1312971\cdot2^{37},-21345\cdot2^{33},1579179\cdot2^{38}, \\
&236170815\cdot2^{31},286131\cdot2^{41},-47870835\cdot2^{30},-464511\cdot2^{33},-21345\cdot2^{30}, \\
&1950735\cdot2^{34},-21345\cdot2^{29},-464511\cdot2^{31},-47870835\cdot2^{27},286131\cdot2^{37}, \\
&236170815\cdot2^{26},1579179\cdot2^{32},-21345\cdot2^{26},-1312971\cdot2^{29},47870835\cdot2^{24}, \\
&-464511\cdot2^{27},21345\cdot2^{24},236128125\cdot2^{23},21345\cdot2^{23},-464511\cdot2^{25}, \\
&-94128645\cdot2^{22},-1312971\cdot2^{25},-21345\cdot2^{21},1579179\cdot2^{26},21345\cdot2^{20}, \\
&56870019\cdot2^{19},-47870835\cdot2^{18},-464511\cdot2^{21},-21345\cdot2^{18},-173704605\cdot2^{17}, \\
&-236170815\cdot2^{16},-464511\cdot2^{19},-47870835\cdot2^{15},286131\cdot2^{25},21345\cdot2^{15}, \\
&1579179\cdot2^{20},-21345\cdot2^{14},-1312971\cdot2^{17},47870835\cdot2^{12},-464511\cdot2^{15}, \\
&236170815\cdot2^{11},236128125\cdot2^{11},21345\cdot2^{11},-464511\cdot2^{13},47870835\cdot2^9, \\
&215120589\cdot2^9,-21345\cdot2^9,1579179\cdot2^{14},21345\cdot2^8,286131\cdot2^{17},94128645\cdot2^7, \\
&-464511\cdot2^9,-21345\cdot2^6,-173704605\cdot2^5,-21345\cdot2^5,-464511\cdot2^7, \\
&-47870835\cdot2^3,286131\cdot2^{13},21345\cdot2^3,1579179\cdot2^8,-472341630, \\
&-1312971\cdot2^5,47870835,-464511\cdot2^3,21345,0))=2021\cdot2^{53}\pi^2\log^32
\end{align}

\begin{align}
&P(5,2^{60},120,(5157\cdot2^{59},-89127\cdot2^{61},7805295\cdot2^{57},-195183\cdot2^{61}, \\
&-32325939\cdot2^{56},1621107\cdot2^{59},5157\cdot2^{56},37287\cdot2^{65},-7805295\cdot2^{54}, \\
&-89127\cdot2^{57},-5157\cdot2^{54},-24620409\cdot2^{53},-5157\cdot2^{53},-89127\cdot2^{55}, \\
&12255165\cdot2^{52},37287\cdot2^{61},5157\cdot2^{51},1621107\cdot2^{53},-5157\cdot2^{50}, \\
&29192697\cdot2^{49},7805295\cdot2^{48},-89127\cdot2^{51},5157\cdot2^{48},32315625\cdot2^{47}, \\
&32325939\cdot2^{46},-89127\cdot2^{49},7805295\cdot2^{45},-195183\cdot2^{49},-5157\cdot2^{45}, \\
&1621107\cdot2^{47},5157\cdot2^{44},37287\cdot2^{53},-7805295\cdot2^{42},-89127\cdot2^{45}, \\
&-32325939\cdot2^{41},-24620409\cdot2^{41},-5157\cdot2^{41},-89127\cdot2^{43},-7805295\cdot2^{39}, \\
&5866263\cdot2^{39},5157\cdot2^{39},1621107\cdot2^{41},-5157\cdot2^{38},-195183\cdot2^{41}, \\
&-12255165\cdot2^{37},-89127\cdot2^{39},5157\cdot2^{36},32315625\cdot2^{35},5157\cdot2^{35}, \\
&-89127\cdot2^{37},7805295\cdot2^{33},-195183\cdot2^{37},-5157\cdot2^{33},1621107\cdot2^{35}, \\
&32325939\cdot2^{31},37287\cdot2^{41},-7805295\cdot2^{30},-89127\cdot2^{33},-5157\cdot2^{30}, \\
&480951\cdot2^{33},-5157\cdot2^{29},-89127\cdot2^{31},-7805295\cdot2^{27},37287\cdot2^{37}, \\
&32325939\cdot2^{26},1621107\cdot2^{29},-5157\cdot2^{26},-195183\cdot2^{29},7805295\cdot2^{24}, \\
&-89127\cdot2^{27},5157\cdot2^{24},32315625\cdot2^{23},5157\cdot2^{23},-89127\cdot2^{25}, \\
&-12255165\cdot2^{22},-195183\cdot2^{25},-5157\cdot2^{21},1621107\cdot2^{23},5157\cdot2^{20}, \\
&5866263\cdot2^{19},-7805295\cdot2^{18},-89127\cdot2^{21},-5157\cdot2^{18},-24620409\cdot2^{17}, \\
&-32325939\cdot2^{16},-89127\cdot2^{19},-7805295\cdot2^{15},37287\cdot2^{25},5157\cdot2^{15}, \\
&1621107\cdot2^{17},-5157\cdot2^{14},-195183\cdot2^{17},7805295\cdot2^{12},-89127\cdot2^{15}, \\
&32325939\cdot2^{11},32315625\cdot2^{11},5157\cdot2^{11},-89127\cdot2^{13},7805295\cdot2^9, \\
&29192697\cdot2^9,-5157\cdot2^9,1621107\cdot2^{11},5157\cdot2^8,37287\cdot2^{17},12255165\cdot2^7, \\
&-89127\cdot2^9,-5157\cdot2^6,-24620409\cdot2^5,-5157\cdot2^5,-89127\cdot2^7,-7805295\cdot2^3, \\
&37287\cdot2^{13},5157\cdot2^3,1621107\cdot2^5,-64651878,-195183\cdot2^5,7805295, \\
&-89127\cdot2^3,5157,0))=\pi^4\log2
\end{align}

基底が3 編集

P(1,9,2,(1,0))=\frac{3}{2}\log2

P(1,9,2,(9,1))=9\log3

P(1,27,3,(3,-1,0))=6\sqrt{3}\arctan\left(\frac{\sqrt{3}}{7}\right)

P(1,81,4,(9,2,1,0))=\frac{9}{2}\log 10

P(1,81,4,(9,3,1,0))=\frac{27}{4}\log5

P(1,729,6,(405,81,72,9,5,0))=243\log7

P(1,729,6,(567,81,36,9,7,0))=243\log13

P(1,729,12,(81,27,162,-9,27,24,-3,7,6,3,-1,0))=36\pi\sqrt{3}

P(1,729,12,(81,-54,0,-9,0,-12,-3,-2,0,-1,0,0))=9\pi\sqrt{3}

P(1,729,12,(81,189,0,45,27,24,-3,1,0,1,-1,0))=9\pi\sqrt{3}

P(1,59049,10,(85293,10935,9477,1215,648,135,117,15,13,0))=19683\log11

P(2,-27,6,(18,-18,-24,-6,-2,0))=3\sqrt{3}\text{Cl}\left(\frac{1}{3}\pi\right)

P(2,729,12,(243,-405,0,-81,-27,-72,-9,-9,0,-5,1,0))=\frac{27}{2}\pi^2

P(2,729,12,(243,-405,-486,-135,27,0,-9,15,18,5,-1,0))=\frac{27\sqrt{3}}{2}\pi\log3

\begin{align}
&P(2,729,12,(4374,-13122,0,-2106,-486,-1944,-162,-234,0,-162,18,-8)) \\
=&729\log^23
\end{align}

P(3,9,2,(9,1))=\frac{39}{2}\zeta(3)-\frac{13}{2}\pi^2\log3+\frac{13}{2}\log^33

\begin{align}
&P(3,729,12,(243,-405,0,-81,-27,-72,-9,-9,0,-5,1,0)) \\
&=\frac{\pi^2\log3}{18}-\frac{65}{36}\zeta(3)
\end{align}

\begin{align}
&P(3,729,12,(729,243,0,-81,-81,-54,-27,-9,0,3,3,2)) \\
=&1053\zeta(3)+\frac{243}{8}\log^3-\frac{405}{8}\pi^2\log3
\end{align}

\begin{align}
&P(3,729,12,(729,-1215,-1458,-405,81,0,-27,45,54,15,-3,0)) \\
=&\frac{87\sqrt{3}}{8}\pi^3-\frac{81\sqrt{3}}{8}\pi\log^23
\end{align}

P(4,-27,6,(9,-15,-18,-5,1,0))=\frac{1}{\sqrt{3}}\left(11\text{Cl}_4\left(\frac{\pi}{3}\right)-\frac{29}{144}\pi^3\log3+\frac{1}{16}\pi\log^33\right)

\begin{align}
&P(4,729,12,(2187,-10935,0,-1539,-243,-729,-81,-171,0,-135,9,-10)) \\
=&\frac{1143}{64}\pi^4-\frac{729}{32}\pi^2\log^23+\frac{405}{64}\log^43
\end{align}

\begin{align}
&P(5,729,12,(2187,-10935,0,-1539,-243,-1458,-81,-171,0,-135,32,-10)) \\
=&\frac{42471}{16}\zeta(5)-\frac{243}{128}\log^53+\frac{243}{64}\pi^2\log^33-\frac{1143}{128}\pi^4\log3
\end{align}

基底が5 編集

P(1,5,2,(1,0))=\sqrt{5}\log \phi

P(1,25,2,(1,0))=\frac{5}{2}\log\left(\frac{3}{2}\right)

P(1,5^5,5,(0,5,1,0,0))=\frac{25}{2}\log\left(\frac{781}{256}\left(\frac{57-5\sqrt{5}}{57+5\sqrt{5}}\right)^{\sqrt{5}}\right)

\begin{align}
&P(1,5^5,5,(125,-25,5,-1,0)) \\
&=2\cdot5^\frac{13}{4}\left(\frac{1}{\sqrt{\phi}}\arctan\left(\frac{5^\frac{1}{4}(233-329\sqrt{5})}{5938\sqrt{\phi}}\right)+\sqrt{\phi}\arctan\left(\frac{5^\frac{1}{4}(939-281\sqrt{5})}{5938\sqrt{\phi}}\right)\right)
\end{align}

基底が7 編集

P(1,-7,2,(1,0))=\frac{1}{2}\sqrt{7}\arctan \frac{\sqrt{7}}{3}

基底が10 編集

P(1,10,1,(1))=10\log\left(\frac{10}{9}\right)

P(1,10^{10},10,(10^8,10^7,10^6,10^5,10^4,10^3,10^2,10,1,0))=10^8\log\left(\frac{1111111111}{387420489}\right)

複合 編集

P(1,-6,2,(1,0))+P(1,8,6,(4,0,0,0,-1,0))=\frac{\pi\sqrt{6}}{12}

一般の基底 編集

P(1,b^3,3,(b,1,0))=\frac{b^2}{3}\log\left(\frac{b^2+b+1}{b^2-2b+1}\right)

P(1,b^b,b,(b^{b-2},b^{b-3},\cdots,b^2,b,1))=b^{b-2}\log\left(\frac{b^b-1}{(b-1)^b}\right)

P(1,b^2,4,(b,0,-1,0))=b\sqrt{b}\arctan\left(\frac{1}{\sqrt{b^2}}\right)

P(1,-b^3,3,(b,1,0))=\frac{2}{3}b^2\sqrt{3}\arctan\left(\frac{\sqrt{3}}{2b-1}\right)

P(1,b^3,3,(b,-1,0))=\frac{2}{3}b^2\sqrt{3}\arctan\left(\frac{\sqrt{3}}{2b-1}\right)

P(1,-4b^4,8,(2b^2,2b,1,0))=-4b^3\arctan\left(\frac{1}{2b-1}\right)

P(1,-4b^4,8,(2b^2,-2b,1,0))=-4b^3\arctan\left(\frac{1}{2b+1}\right)

P(1,-b^2,4,(b,0,1,0))=\frac{1}{\sqrt{2}}b\sqrt{b}\arctan\left(\frac{\sqrt{2b}}{b-1}\right)

P(1,-27b^6,6,(9b^4,9b^3,6b^2,3b,1,0))=18b^5\sqrt{3}\arctan\left(\frac{1}{(2b-1)\sqrt{3}}\right)

P(1,-27b^6,6,(9b^4,-9b^3,6b^2,-3b,1,0))=18b^5\sqrt{3}\arctan\left(\frac{1}{(2b+1)\sqrt{3}}\right)

P(1,-b^3,6,(b^2,0,2b,0,1,0))=b^2\sqrt{b}\arctan\left(\frac{\sqrt{b}}{b-1}\right)

P(1,-b,1,(1))=b\log\left(\frac{b+1}{b}\right)

P(1,b,1,(1))=b\log\left(\frac{b}{b-1}\right)

P(1,b,2,(1,0))=\frac{\sqrt{b}}{2}\log\left(\frac{\sqrt{b}+1}{\sqrt{b}-1}\right)

P(1,b^3,3,(b^2,b,-2))=b^3\log\left(\frac{b^2+b+1}{b^2}\right)

P(1,-b^3,3,(b^2,-b,-2))=-b^3\log\left(\frac{b^2-b+1}{b^2}\right)

P(1,-4b^4,4,(2b^3,0,-b,-1))=-2b^4\log\left(\frac{2b^2-2b+1}{2b^2}\right)

P(1,-4b^4,4,(2b^3,0,-b,1))=2b^4\log\left(\frac{2b^2+2b+1}{2b^2}\right)

P(1,-b^2,4,(b,0,-1,0))=\frac{b\sqrt{b}}{2\sqrt{2}}\log\left(\frac{b+\sqrt{2b}+1}{b-\sqrt{2b}+1}\right)

P(1,-27b^6,6,(27b^5,-9b^4,0,3b^2,-3b,2))=27b^6\log\left(\frac{3b^2+3b+1}{3b^2}\right)

P(1,-27b^6,6,(27b^5,9b^4,0,-3b^2,-3b,-2))=-27b^6\log\left(\frac{3b^2-3b+1}{3b^2}\right)

P(1,-b^3,6,(b^2,0,0,0,-1,0))=\frac{b^2\sqrt{b}}{2\sqrt{3}}\log\left(\frac{b+\sqrt{3b}+1}{b-\sqrt{3b}+1}\right)

類似する公式 編集

P(2,\phi^5,5,(\phi^3,\phi^4,\phi^3,1,1))=\frac{\pi^2\phi^5}{50}

P(1,\frac{2}{\phi},9,(256\phi,128\phi^3,64\phi^4,32\phi^4,0,-8\phi^6,-6\phi^8,-4\phi^9,0))=\frac{1536\pi\sqrt{\phi}}{5^\frac{5}{4}}

\sum_{k=0}^\infty\frac{D_k}{10^k(k+1)}=\frac{10}{\sqrt{19}}\arccos\left(\frac{9}{10}\right)

ただし、D_0=D_1=1,D_{k+2}=D_{k+1}-5D_k

値が0になる非自明な公式 編集

\begin{align}0=P(1,16,8,(-8,8,4,8,2,2,-1,0))\end{align}

\begin{align}0=P(1,64,6,(16,-24,-8,-6,1,0))\end{align}

\begin{align}
&0=P(1,2^{12},24,(0,0,2^{11},-2^{11},0,-2^9,256,-3\cdot2^8,0,0,-64,-128,0,-32,-32, \\
&-48,0,-24,-4,-8,0,-2,1,0))
\end{align}

\begin{align}
&0=P(1,2^{12},24,(2^{11},-2^{11},-2^{11},0,-2^9,-2^{10},-2^8,0,-2^8,-2^7,2^6,0,-32,32, \\
&32,0,8,16,4,0,4,2,-1,0))
\end{align}

\begin{align}
&0=P(1,2^{12},24,(-2^9,-2^{10},2^{10},7\cdot2^8,256,3\cdot2^8,64,3\cdot2^7,0,0,0,0,8,-32, \\
&-16,12,-4,4,-1,8,0,-1,0,0))
\end{align}

\begin{align}
&0=P(1,2^{12},24,(2^9,-2^{10},-2^9,256,0,256,64,3\cdot2^7,64,0,0,0,-8,-16,8,12,0,4, \\
&-1,2,-1,0,0,0))\end{align}

\begin{align}
&0=P(1,2^{12},24,(3\cdot2^9,-3\cdot2^{10},0,-256,0,0,192,3\cdot2^7,0,0,0,-64,-24,-48, \\
&0,-12,0,0,-3,2,0,0,0,0))
\end{align}

\begin{align}
&0=P(1,2^{12},24,(-2^{10},3\cdot2^9,2^9,256,128,128,-64,-192,0,32,0,32,16,16,-8,0, \\
&-2,-2,1,0,0,0,0,0))
\end{align}

\begin{align}
&0=P(1,2^{20},40,(0,2^{18},-2^{18},2^{17},0,-5\cdot2^{16},2^{16},-5\cdot2^{15},0,-2^{16}, \\
&-2^{14},2^{13},0,-5\cdot2^{12},-2^{14},-5\cdot2^{11},0,2^{10},-2^{10},-2^{11},0,-5\cdot2^8, \\
&256,-5\cdot2^7,0,64,-64,32,0,0,16,-40,0,4,16,2,0,-5,1,0))
\end{align}

\begin{align}
&0=P(1,2^{20},40,(2^{18},-2^{19},0,-2^{17},3\cdot2^{15},2^{16},0,0,2^{14},2^{13},0,-2^{13},-2^{12},2^{12},5\cdot2^{10}, \\
&0,2^{10},-2^{11},0,-2^9,-256,256,0,0,-96,-128,0,-32,-16,-24,0,0,4,-8,-5,-2, \\
&-1,1,0,0))
\end{align}

\begin{align}
&0=P(1,2^{20},40,(-2^{18},3\cdot2^{18},0,-2^{18},-13\cdot2^{15},0,0,5\cdot2^{15},-2^{14},2^{13},0,-2^{14},2^{12},0, \\
&5\cdot2^{10},5\cdot2^{11},-2^{10},3\cdot2^{10},0,3\cdot2^9,256,0,0,5\cdot2^7,13\cdot2^5,192,0,-64,16,40,0,40, \\
&-4,12,-5,-4,1,0,0,0))
\end{align}

\begin{align}
&0=P(1,2^{20},40,(2^{19},-3\cdot2^{19},2^{18},0,2^{19},3\cdot2^{17},-2^{16},0,2^{15},2^{16},2^{14},0,-2^{13}, \\
&3\cdot2^{13},2^{14},0,2^{11},-3\cdot2^{11},2^{10},0,-2^9,3\cdot2^9,-2^8,0,-2^9,-3\cdot2^7,2^6,0,-2^5,-2^6,-2^4, \\
&0,2^3,-3\cdot2^3,-2^4,0,-2,6,-1,0))
\end{align}

\begin{align}
&0=P(2,2^{12},24,(0,2^{10},-3\cdot2^{10},2^9,0,2^{10},0,9\cdot2^7,3\cdot2^7,64,0,128,0,16,48,72,0,16, \\
&0,2,-6,1,0,0))
\end{align}

\begin{align}
&0=P(2,2^{12},24,(-2^{11},0,17\cdot2^{11},-17\cdot2^{10},2^9,-15\cdot2^{10},-256,-63\cdot2^8,-17\cdot2^8, \\
&0,64,-5\cdot2^8,32,0,-17\cdot2^5,-63\cdot2^4,-8,-240,4,-68,68,0,-1,0))
\end{align}

\begin{align}
&0=P(2,2^{20},40,(2^{19},-3\cdot2^{20},-2^{18},13\cdot2^{18},3\cdot2^{20},-3\cdot2^{18},2^{16},-25\cdot2^{16},2^{15}, \\
&-3\cdot2^{16},-2^{14},13\cdot2^{14},-2^{13},-3\cdot2^{14},-3\cdot2^{15},-25\cdot2^{12},2^{11},-3\cdot2^{12},-2^{10}, \\
&-3\cdot2^{12},-2^9,-3\cdot2^{10},256,-25\cdot2^8,-3\cdot2^{10},-3\cdot2^8,-64,13\cdot2^6,-32,-192, \\
&16,-25\cdot2^4,8,-48,96,52,-2,-12,1,0))
\end{align}

\begin{align}
&0=P(3,2^{12},24,(2^{11},-19\cdot2^{11},5\cdot2^{14},-2^{11},-2^9,-23\cdot2^{10},256,-27\cdot2^{10}, \\
&-5\cdot2^{11},-19\cdot2^7,-64,-7\cdot2^9,-32,-19\cdot2^5,-5\cdot2^8,-27\cdot2^6,8,-23\cdot2^4,-4, \\
&-8,160,-38,1,0))
\end{align}

\begin{align}
&0=P(3,2^{60},120,(7\cdot2^{59},-3\cdot5^2\cdot2^{60},1579\cdot2^{57},-29\cdot2^{60},-3\cdot23\cdot31\cdot2^{56}, \\
&3\cdot5\cdot2^{60},7\cdot2^{56},67\cdot2^{59},-1579\cdot2^{54},-3\cdot5^2\cdot2^{56},-7\cdot2^{54}, \\
&-5\cdot11\cdot43\cdot2^53,-7\cdot2^53,-3\cdot5^2\cdot2^{54},3\cdot7\cdot13\cdot2^{52},67\cdot2^{55},7\cdot2^{51}, \\
&3\cdot5\cdot2^{54},-7\cdot2^{50},3\cdot631\cdot2^{49},1579\cdot2^{48},-3\cdot5^2\cdot2^{50},7\cdot2^{48}, \\
&53\cdot17\cdot2^{47},3\cdot23\cdot31\cdot2^{46},-3\cdot5^2\cdot2^{48},1579\cdot2^{45},-29\cdot2^{48},-7\cdot2^{45}, \\
&3\cdot5\cdot2^{48},7\cdot2^{44},67\cdot2^{47},-1579\cdot2^{42},-3\cdot5^2\cdot2^{44},-3\cdot23\cdot31\cdot2^{41}, \\
&-5\cdot11\cdot43\cdot2^{41},-7\cdot2^{41},-3\cdot5^2\cdot2^{42},-1579\cdot2^{39},-34\cdot13\cdot2^{39},7\cdot2^{39}, \\
&3\cdot5\cdot2^{42},-7\cdot2^{38},-29\cdot2^{40},-3\cdot7\cdot13\cdot2^{37},-3\cdot5^2\cdot2^{38},7\cdot2^{36}, \\
&53\cdot17\cdot2^{35},7\cdot2^{35},-3\cdot5^2\cdot2^{36},1579\cdot2^{33},-29\cdot2^{36},-7\cdot2^{33}, \\
&3\cdot5\cdot2^{36},3\cdot23\cdot31\cdot2^{31},67\cdot2^{35},-1579\cdot2^{30},-3\cdot5^2\cdot2^{32},-7\cdot2^{30}, \\
&-3\cdot5\cdot2^{33},-7\cdot2^{29},-3\cdot5^2\cdot2^{30},-1579\cdot2^{27},67\cdot2^{31},3\cdot23\cdot31\cdot2^{26}, \\
&3\cdot5\cdot2^{30},-7\cdot2^{26},-29\cdot2^{28},1579\cdot2^{24},-3\cdot5^2\cdot2^{26},7\cdot2^{24},53\cdot17\cdot2^{23}, \\
&7\cdot2^{23},-3\cdot5^2\cdot2^{24},-3\cdot7\cdot13\cdot2^{22},-29\cdot2^{24},-7\cdot2^{21},3\cdot5\cdot2^{24}, \\
&7\cdot2^{20},-34\cdot13\cdot2^{19},-1579\cdot2^{18},-3\cdot5^2\cdot2^{20},-7\cdot2^{18},-5\cdot11\cdot43\cdot2^{17}, \\
&-3\cdot23\cdot31\cdot2^{16},-3\cdot5^2\cdot2^{18},-1579\cdot2^{15},67\cdot2^{19},7\cdot2^{15},3\cdot5\cdot2^{18}, \\
&-7\cdot2^{14},-29\cdot2^{16},1579\cdot2^{12},-3\cdot5^2\cdot2^{14},3\cdot23\cdot31\cdot2^{11},53\cdot17\cdot2^{11}, \\
&7\cdot2^{11},-3\cdot5^2\cdot2^{12},1579\cdot2^9,3\cdot631\cdot2^9,-7\cdot2^9,3\cdot5\cdot2^{12},7\cdot2^8,67\cdot2^{11}, \\
&3\cdot7\cdot13\cdot2^7,-3\cdot5^2\cdot2^8,-7\cdot2^6,-5\cdot11\cdot43\cdot2^5,-7\cdot2^5,-3\cdot5^2\cdot2^6,-1579\cdot2^3, \\
&67\cdot2^7,7\cdot2^3,3\cdot5\cdot2^6,-3\cdot23\cdot31\cdot2,-29\cdot2^4,1579,-3\cdot5^2\cdot2^2,7,0))
\end{align}

\begin{align}
&0=P(3,2^{60},120,(2^{59},0,-353\cdot2^{57},7\cdot2^{62},3\cdot7\cdot113\cdot2^{56},-33\cdot5\cdot2^{59},2^{56}, \\
&-97\cdot2^{60},353\cdot2^{54},0,-2^{54},5\cdot331\cdot2^53,-2^53,0,-3\cdot337\cdot2^{52},-97\cdot2^{56},2^{51}, \\
&-33\cdot5\cdot2^53,-2^{50},-3^2\cdot239\cdot2^{49},-353\cdot2^{48},0,2^{48},-53\cdot19\cdot2^{47}, \\
&-3\cdot7\cdot113\cdot2^{46},0,-353\cdot2^{45},7\cdot2^{50},-2^{45},-33\cdot5\cdot2^{47},2^{44},-97\cdot2^{48}, \\
&353\cdot2^{42},0,3\cdot7\cdot113\cdot2^{41},5\cdot331\cdot2^{41},-2^{41},0,353\cdot2^{39},-36\cdot2^{39},2^{39}, \\
&-33\cdot5\cdot2^{41},-2^{38},7\cdot2^{42},3\cdot337\cdot2^{37},0,2^{36},-53\cdot19\cdot2^{35},2^{35},0, \\
&-353\cdot2^{33},7\cdot2^{38},-2^{33},-33\cdot5\cdot2^{35},-3\cdot7\cdot113\cdot2^{31},-97\cdot2^{36},353\cdot2^{30}, \\
&0,-2^{30},-3^2\cdot5\cdot2^{33},-2^{29},0,353\cdot2^{27},-97\cdot2^{32},-3\cdot7\cdot113\cdot2^{26}, \\
&-33\cdot5\cdot2^{29},-2^{26},7\cdot2^{30},-353\cdot2^{24},0,2^{24},-53\cdot19\cdot2^{23},2^{23},0,3\cdot337\cdot2^{22}, \\
&7\cdot2^{26},-2^{21},-33\cdot5\cdot2^{23},2^{20},-36\cdot2^{19},353\cdot2^{18},0,-2^{18},5\cdot331\cdot2^{17}, \\
&3\cdot7\cdot113\cdot2^{16},0,353\cdot2^{15},-97\cdot2^{20},2^{15},-33\cdot5\cdot2^{17},-2^{14},7\cdot2^{18}, \\
&-353\cdot2^{12},0,-3\cdot7\cdot113\cdot2^{11},-53\cdot19\cdot2^{11},2^{11},0,-353\cdot2^9,-3^2\cdot239\cdot2^9, \\
&-2^9,-33\cdot5\cdot2^{11},2^8,-97\cdot2^{12},-3\cdot337\cdot2^7,0,-2^6,5\cdot331\cdot2^5,-2^5,0,353\cdot2^3, \\
&-97\cdot2^8,2^3,-33\cdot5\cdot2^5,3\cdot7\cdot113\cdot2,7\cdot2^6,-353,0,1,0))
\end{align}

\begin{align}
&0=P(4,2^{60},120,(-31\cdot2^{59},3\cdot269\cdot2^{60},-5\cdot61\cdot107\cdot2^{57},1553\cdot2^{60}, \\
&3^2\cdot14243\cdot2^{56},-3\cdot1051\cdot2^{60},-31\cdot2^{56},-9319\cdot2^{59},5\cdot61\cdot107\cdot2^{54}, \\
&3\cdot269\cdot2^{56},31\cdot2^{54},31\cdot3187\cdot2^53,31\cdot2^53,3\cdot269\cdot2^{54}, \\
&-3^2\cdot5\cdot1061\cdot2^{52},-9319\cdot2^{55},-31\cdot2^{51},-3\cdot1051\cdot2^{54},31\cdot2^{50}, \\
&-3\cdot38567\cdot2^{49},-5\cdot61\cdot107\cdot2^{48},3\cdot269\cdot2^{50},-31\cdot2^{48},-55\cdot41\cdot2^{47}, \\
&-3^2\cdot14243\cdot2^{46},3\cdot269\cdot2^{48},-5\cdot61\cdot107\cdot2^{45},1553\cdot2^{48},31\cdot2^{45}, \\
&-3\cdot1051\cdot2^{48},-31\cdot2^{44},-9319\cdot2^{47},5\cdot61\cdot107\cdot2^{42},3\cdot269\cdot2^{44}, \\
&3^2\cdot14243\cdot2^{41},31\cdot3187\cdot2^{41},31\cdot2^{41},3\cdot269\cdot2^{42},5\cdot61\cdot107\cdot2^{39}, \\
&-34\cdot7\cdot37\cdot2^{39},-31\cdot2^{39},-3\cdot1051\cdot2^{42},31\cdot2^{38},1553\cdot2^{40}, \\
&3^2\cdot5\cdot1061\cdot2^{37},3\cdot269\cdot2^{38},-31\cdot2^{36},-55\cdot41\cdot2^{35},-31\cdot2^{35}, \\
&3\cdot269\cdot2^{36},-5\cdot61\cdot107\cdot2^{33},1553\cdot2^{36},31\cdot2^{33},-3\cdot1051\cdot2^{36}, \\
&-3^2\cdot14243\cdot2^{31},-9319\cdot2^{35},5\cdot61\cdot107\cdot2^{30},3\cdot269\cdot2^{32},31\cdot2^{30}, \\
&-3\cdot13\cdot47\cdot2^{33},31\cdot2^{29},3\cdot269\cdot2^{30},5\cdot61\cdot107\cdot2^{27},-9319\cdot2^{31}, \\
&-3^2\cdot14243\cdot2^{26},-3\cdot1051\cdot2^{30},31\cdot2^{26},1553\cdot2^{28},-5\cdot61\cdot107\cdot2^{24}, \\
&3\cdot269\cdot2^{26},-31\cdot2^{24},-55\cdot41\cdot2^{23},-31\cdot2^{23},3\cdot269\cdot2^{24}, \\
&3^2\cdot5\cdot1061\cdot2^{22},1553\cdot2^{24},31\cdot2^{21},-3\cdot1051\cdot2^{24},-31\cdot2^{20}, \\
&-34\cdot7\cdot37\cdot2^{19},5\cdot61\cdot107\cdot2^{18},3\cdot269\cdot2^{20},31\cdot2^{18},31\cdot3187\cdot2^{17}, \\
&3^2\cdot14243\cdot2^{16},3\cdot269\cdot2^{18},5\cdot61\cdot107\cdot2^{15},-9319\cdot2^{19},-31\cdot2^{15}, \\
&-3\cdot1051\cdot2^{18},31\cdot2^{14},1553\cdot2^{16},-5\cdot61\cdot107\cdot2^{12},3\cdot269\cdot2^{14}, \\
&-3^2\cdot14243\cdot2^{11},-55\cdot41\cdot2^{11},-31\cdot2^{11},3\cdot269\cdot2^{12},-5\cdot61\cdot107\cdot2^9, \\
&-3\cdot38567\cdot2^9,31\cdot2^9,-3\cdot1051\cdot2^{12},-31\cdot2^8,-9319\cdot2^{11},-3^2\cdot5\cdot1061\cdot2^7, \\
&3\cdot269\cdot2^8,31\cdot2^6,31\cdot3187\cdot2^5,31\cdot2^5,3\cdot269\cdot2^6,5\cdot61\cdot107\cdot2^3, \\
&-9319\cdot2^7,-31\cdot2^3,-3\cdot1051\cdot2^6,3^2\cdot14243\cdot2,1553\cdot2^4,-61\cdot107\cdot5, \\
&3\cdot269\cdot2^2,-31,0))
\end{align}

\begin{align}0=P(1,729,12,(0,81,-162,0,27,36,0,9,6,4,-1,0))\end{align}

\begin{align}0=P(1,729,12,(243,-324,-162,-81,0,-36,-9,0,6,-1,0,0))\end{align}

性質 編集

より効率のいい公式を作るためには、値が0になる公式を調べることも重要である。また、コンピュータでの計算を考える場合、基底が2であるものがもっとも望ましい。

関連項目 編集

出典 編集

  1. http://mathworld.wolfram.com/BBP-TypeFormula.html
  2. http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/bbp-formulas.pdf
  3. http://www.pi314.net/eng/index.php

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