[math-fun] Machinoid arcsin π formulæ
These must have been done to death. In[526]:= FullSimplify[Sin[-ArcSin[23/361] + 2*ArcSin[11/38]]] Out[526]= 1/2 In[528]:= FullSimplify[Sin[ArcSin[97/961] + 2*ArcSin[13/62]]] Out[528]= 1/2 In[532]:= FullSimplify[Sin[2*ArcSin[11/43] + ArcSin[23/3698]]] Out[532]= 1/2 In[517]:= FullSimplify[Sin[ArcSin[23/98] + 2*ArcSin[1/7]]] Out[517]= 1/2 I.e., ArcSin[97/961] + 2*ArcSin[13/62] = π/6, but Wolfram's nascent artificial intelligence is too stupid to notice. I don't know if this scheme can beat Machin--I only looked at 2 arcsin a + arcsin b. The apparent messiness of the arcsin series vs arctan is illusory if we replace clunky old summation with matrix products. (gosper.org/pathi.pdf). --rwg
The atan formulas can be understood as being about atan(N/D) = imag(log D+iN), and factoring D+iN into complex primes. You combine a few relations, and eventually wind up with log(i) or imag(log 1+i). You can go after new formulas by factoring complex numbers A+Bi with large A and small B. "Fortuitously", 239+i has norm 57122 = 2 * 13^4, while 5+i has norm 26 = 2 * 13, leading to pi/4 = 4 atan(1/5) - atan(1/239). ("Fortuitously" comes from 169, a square, turning up in the solutions of x2-2y2 = +-1 as (239,169). Maybe something similar could be made of 144 being a Fib, and being part of x2-5y2 = +-4 as (321,144)?) Can you do something similar with the arcsins? I notice that the implied base leg for the 11/43 case is sqrt(43^2-11^2) -> sqrt(32 * 54) -> 24 sqrt3. This could also be useful in simplifying your arcsin expressions, if you regard "factoring" as a simplification. Rich ----- Quoting Bill Gosper <billgosper@gmail.com>:
These must have been done to death. In[526]:= FullSimplify[Sin[-ArcSin[23/361] + 2*ArcSin[11/38]]]
Out[526]= 1/2
In[528]:= FullSimplify[Sin[ArcSin[97/961] + 2*ArcSin[13/62]]]
Out[528]= 1/2
In[532]:= FullSimplify[Sin[2*ArcSin[11/43] + ArcSin[23/3698]]]
Out[532]= 1/2
In[517]:= FullSimplify[Sin[ArcSin[23/98] + 2*ArcSin[1/7]]]
Out[517]= 1/2
I.e., ArcSin[97/961] + 2*ArcSin[13/62] = ?/6, but Wolfram's nascent artificial intelligence is too stupid to notice.
I don't know if this scheme can beat Machin--I only looked at 2 arcsin a + arcsin b. The apparent messiness of the arcsin series vs arctan is illusory if we replace clunky old summation with matrix products. (gosper.org/pathi.pdf). --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
And there's this: Manipulate[ Plot[{x/(-Im[ArcSin[x]] - Log[5.91095]) - PrimePi[x]}, {x, 2, max}], {{max, 100, "plot range"}, 20, 50000, 1}] with a bit of a puzzle for you being "why 5.91095?". On Tue, Sep 23, 2014 at 1:59 PM, <rcs@xmission.com> wrote:
The atan formulas can be understood as being about atan(N/D) = imag(log D+iN), and factoring D+iN into complex primes. You combine a few relations, and eventually wind up with log(i) or imag(log 1+i). You can go after new formulas by factoring complex numbers A+Bi with large A and small B. "Fortuitously", 239+i has norm 57122 = 2 * 13^4, while 5+i has norm 26 = 2 * 13, leading to pi/4 = 4 atan(1/5) - atan(1/239).
("Fortuitously" comes from 169, a square, turning up in the solutions of x2-2y2 = +-1 as (239,169). Maybe something similar could be made of 144 being a Fib, and being part of x2-5y2 = +-4 as (321,144)?)
Can you do something similar with the arcsins? I notice that the implied base leg for the 11/43 case is sqrt(43^2-11^2) -> sqrt(32 * 54) -> 24 sqrt3. This could also be useful in simplifying your arcsin expressions, if you regard "factoring" as a simplification.
Rich
----- Quoting Bill Gosper <billgosper@gmail.com>:
These must have been done to death. In[526]:= FullSimplify[Sin[-ArcSin[23/361] + 2*ArcSin[11/38]]]
Out[526]= 1/2
In[528]:= FullSimplify[Sin[ArcSin[97/961] + 2*ArcSin[13/62]]]
Out[528]= 1/2
In[532]:= FullSimplify[Sin[2*ArcSin[11/43] + ArcSin[23/3698]]]
Out[532]= 1/2
In[517]:= FullSimplify[Sin[ArcSin[23/98] + 2*ArcSin[1/7]]]
Out[517]= 1/2
I.e., ArcSin[97/961] + 2*ArcSin[13/62] = ?/6, but Wolfram's nascent artificial intelligence is too stupid to notice.
I don't know if this scheme can beat Machin--I only looked at 2 arcsin a + arcsin b. The apparent messiness of the arcsin series vs arctan is illusory if we replace clunky old summation with matrix products. (gosper.org/pathi.pdf). --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
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Bill Gosper -
Jeff Caldwell -
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