[math-fun] ArcCosine Miracles List, and explanation
ArcCosine Miracles List ===Warren D. Smith====July 2015========= Here's a fairly large list of "arccosine miracles" i.e. equations of the form A*arccos(B/C) + D*arccos(E/F) = G*(180 degrees) with A,B,C,D,E,F,G all integers, and min(A,D,G)>=0 and max(|A|,|B|,|C|,|D|,|E|,|F|)<101 and |B|<C<71. I have not proven the exact validity of these, but they are valid to at least 11 decimal places. Various rather trivial infinite families, such as when B=C or E=F or D=0 or B=0 or E=0 or |B/C|=|E/F|, or integer multiples of listed identities, have been omitted from this list to try to keep it short. It occurs to me that if M is a complex number with |M|=C and re(M)=B both integer, while N is a complex number with |N|=F and re(N)=E both integer, then im( (M/C)^A * (N/F)^D ) = 0 where A and D are ordinary nonnegative integers, holds if and only if our arccos identity does. That seems to be a way to rephrase the problem in a way that does not mention any arc-trig functions, and lives solely in the land of complex numbers whose real and imaginary parts are expressible using integers and their square roots only. This rephrasing should allow both speeding up the search, and making it yield rigorous proofs instead of perhaps-merely-approximately-valid identities. And any of our alleged identities may be rigorously proven (or disproven) in this way (I haven't, but you easily could). CLOSE BUT NO CIGAR: 74*acos(9/11)+57*acos(-51/89) =9719.99999999397 12*acos(-15/26)+68*acos(-89/98) =12060.00000001333 45*acos(10/31)+64*acos(-31/71) =10620.00000002814 28*acos(25/33)+67*acos(72/97) =3959.99999997569 76*acos(29/35)+47*acos(-44/65) =8820.00000001841 91*acos(18/43)+71*acos(19/79) =11340.00000000618 37*acos(20/47)+66*acos(1/65) =8279.99999999370 77*acos(47/53)+56*acos(-65/79) =10260.00000000485 7*acos(-19/56)+66*acos(-46/57) =10260.00000002915 8*acos(-29/57)+85*acos(-73/100) =12600.00000001594 66*acos(46/57)+7*acos(19/56) =2879.99999997085 94*acos(-1/58)+25*acos(-19/77) =11160.00000000605 23*acos(-2/59)+8*acos(-67/70) =3420.00000002234 5*acos(59/60)+72*acos(35/74) =4499.99999998777 71*acos(53/62)+42*acos(-89/100) =8639.99999999190 66*acos(-1/65)+37*acos(-20/47) =10260.00000000630 47*acos(44/65)+76*acos(-29/35) =13319.99999998160 8*acos(67/70)+23*acos(2/59) =2159.99999997766 APPARENTLY GENUINE IDENTITIES: 3*acos(-2/3)+1*acos(-22/27) = 540.0 2*acos(-1/3)+1*acos(-7/9) = 360.0 3*acos(1/3)+1*acos(-23/27) = 360.0 2*acos(2/3)+1*acos(1/9) = 180.0 4*acos(-3/4)+1*acos(-31/32) = 720.0 2*acos(-1/4)+1*acos(-7/8) = 360.0 3*acos(1/4)+1*acos(-11/16) = 360.0 2*acos(3/4)+1*acos(-1/8) = 180.0 4*acos(-4/5)+2*acos(-24/25) = 900.0 2*acos(-3/5)+2*acos(-4/5) = 540.0 2*acos(-2/5)+1*acos(-17/25) = 360.0 2*acos(-1/5)+1*acos(-23/25) = 360.0 4*acos(3/5)+2*acos(-24/25) = 540.0 2*acos(4/5)+2*acos(3/5) = 180.0 2*acos(-1/6)+1*acos(-17/18) = 360.0 3*acos(1/6)+1*acos(-13/27) = 360.0 2*acos(5/6)+1*acos(-7/18) = 180.0 2*acos(-3/7)+1*acos(-31/49) = 360.0 2*acos(-2/7)+1*acos(-41/49) = 360.0 3*acos(-1/7)+3*acos(-11/14) = 720.0 3*acos(1/7)+3*acos(-13/14) = 720.0 2*acos(4/7)+1*acos(17/49) = 180.0 2*acos(5/7)+1*acos(-1/49) = 180.0 2*acos(6/7)+1*acos(-23/49) = 180.0 5*acos(-7/8)+2*acos(-61/64) =1080.0 2*acos(-3/8)+1*acos(-23/32) = 360.0 3*acos(-1/8)+2*acos(-9/16) = 540.0 1*acos(1/8)+2*acos(-3/4) = 360.0 2*acos(5/8)+1*acos(7/32) = 180.0 1*acos(7/8)+2*acos(1/4) = 180.0 3*acos(-7/9)+2*acos(-23/27) = 720.0 2*acos(-4/9)+1*acos(-49/81) = 360.0 2*acos(-2/9)+1*acos(-73/81) = 360.0 1*acos(-1/9)+2*acos(-2/3) = 360.0 3*acos(1/9)+2*acos(-22/27) = 540.0 2*acos(5/9)+1*acos(31/81) = 180.0 1*acos(7/9)+2*acos(1/3) = 180.0 2*acos(8/9)+1*acos(-47/81) = 180.0 2*acos(-3/10)+1*acos(-41/50) = 360.0 2*acos(-1/10)+1*acos(-49/50) = 360.0 2*acos(7/10)+1*acos(1/50) = 180.0 2*acos(9/10)+1*acos(-31/50) = 180.0 2*acos(-5/12)+1*acos(-47/72) = 360.0 2*acos(-1/12)+1*acos(-71/72) = 360.0 2*acos(7/12)+1*acos(23/72) = 180.0 2*acos(11/12)+1*acos(-49/72) = 180.0 3*acos(-11/13)+3*acos(-23/26) = 900.0 2*acos(-5/13)+2*acos(-12/13) = 540.0 3*acos(11/13)+3*acos(1/26) = 360.0 2*acos(12/13)+2*acos(5/13) = 180.0 6*acos(-13/14)+3*acos(-47/49) =1440.0 3*acos(-11/14)+3*acos(-13/14) = 900.0 2*acos(-5/14)+1*acos(-73/98) = 360.0 2*acos(-3/14)+1*acos(-89/98) = 360.0 2*acos(-1/14)+1*acos(-97/98) = 360.0 2*acos(9/14)+1*acos(17/98) = 180.0 3*acos(11/14)+3*acos(1/7) = 360.0 3*acos(13/14)+3*acos(-1/7) = 360.0 2*acos(-11/16)+3*acos(-7/8) = 720.0 1*acos(-9/16)+3*acos(-3/4) = 540.0 2*acos(9/16)+3*acos(1/8) = 360.0 1*acos(11/16)+3*acos(-1/4) = 360.0 2*acos(-8/17)+2*acos(-15/17) = 540.0 2*acos(15/17)+2*acos(8/17) = 180.0 1*acos(7/18)+2*acos(-5/6) = 360.0 1*acos(17/18)+2*acos(1/6) = 180.0 3*acos(-13/19)+3*acos(-37/38) = 900.0 3*acos(13/19)+3*acos(11/38) = 360.0 1*acos(-7/25)+2*acos(-3/5) = 360.0 1*acos(7/25)+2*acos(-4/5) = 360.0 1*acos(17/25)+2*acos(2/5) = 180.0 1*acos(23/25)+2*acos(1/5) = 180.0 2*acos(24/25)+4*acos(-3/5) = 540.0 3*acos(-1/26)+3*acos(-11/13) = 720.0 3*acos(1/26)+3*acos(-23/26) = 720.0 3*acos(23/26)+3*acos(11/13) = 180.0 4*acos(-22/27)+3*acos(-79/81) =1080.0 2*acos(-13/27)+3*acos(-17/18) = 720.0 1*acos(-5/27)+3*acos(-5/6) = 540.0 1*acos(13/27)+3*acos(-1/6) = 360.0 1*acos(22/27)+3*acos(2/3) = 180.0 1*acos(23/27)+3*acos(-1/3) = 360.0 2*acos(-20/29)+2*acos(-21/29) = 540.0 2*acos(21/29)+2*acos(20/29) = 180.0 3*acos(-23/31)+3*acos(-59/62) = 900.0 3*acos(23/31)+3*acos(13/62) = 360.0 3*acos(-17/32)+4*acos(-11/16) = 900.0 1*acos(-7/32)+2*acos(-5/8) = 360.0 1*acos(17/32)+4*acos(1/4) = 360.0 1*acos(23/32)+2*acos(3/8) = 180.0 1*acos(31/32)+4*acos(3/4) = 180.0 3*acos(-13/37)+3*acos(-47/74) = 720.0 2*acos(-12/37)+2*acos(-35/37) = 540.0 3*acos(13/37)+3*acos(-73/74) = 720.0 2*acos(35/37)+2*acos(12/37) = 180.0 3*acos(-11/38)+3*acos(-13/19) = 720.0 3*acos(11/38)+3*acos(-37/38) = 720.0 3*acos(37/38)+3*acos(13/19) = 180.0 2*acos(-9/41)+2*acos(-40/41) = 540.0 2*acos(40/41)+2*acos(9/41) = 180.0 3*acos(-11/43)+3*acos(-61/86) = 720.0 3*acos(11/43)+3*acos(-83/86) = 720.0 1*acos(-17/49)+2*acos(-4/7) = 360.0 1*acos(1/49)+2*acos(-5/7) = 360.0 1*acos(23/49)+2*acos(-6/7) = 360.0 1*acos(31/49)+2*acos(3/7) = 180.0 1*acos(41/49)+2*acos(2/7) = 180.0 1*acos(47/49)+2*acos(1/7) = 180.0 1*acos(-1/50)+2*acos(-7/10) = 360.0 1*acos(31/50)+2*acos(-9/10) = 360.0 1*acos(41/50)+2*acos(3/10) = 180.0 1*acos(49/50)+2*acos(1/10) = 180.0 2*acos(-28/53)+2*acos(-45/53) = 540.0 2*acos(45/53)+2*acos(28/53) = 180.0 2*acos(-11/61)+2*acos(-60/61) = 540.0 2*acos(60/61)+2*acos(11/61) = 180.0 3*acos(-13/62)+3*acos(-23/31) = 720.0 3*acos(13/62)+3*acos(-59/62) = 720.0 3*acos(59/62)+3*acos(23/31) = 180.0 4*acos(-57/64)+5*acos(-31/32) =1440.0 1*acos(57/64)+5*acos(-3/4) = 720.0 1*acos(61/64)+5*acos(-1/4) = 540.0 2*acos(-33/65)+2*acos(-56/65) = 540.0 2*acos(-16/65)+2*acos(-63/65) = 540.0 2*acos(56/65)+2*acos(33/65) = 180.0 2*acos(63/65)+2*acos(16/65) = 180.0 end.
It is interesting that some of the most-amazing among the listed miracles involve only powers of 2 in denominators: 5*acos(-7/8)+2*acos(-61/64) =1080.0 4*acos(-57/64)+5*acos(-31/32) =1440.0 1*acos(57/64)+5*acos(-3/4) = 720.0 1*acos(61/64)+5*acos(-1/4) = 540.0 4*acos(-3/4)+1*acos(-31/32) = 720.0 2*acos(-3/8)+1*acos(-23/32) = 360.0 2*acos(5/8)+1*acos(7/32) = 180.0 3*acos(-17/32)+4*acos(-11/16) = 900.0 1*acos(-7/32)+2*acos(-5/8) = 360.0 1*acos(17/32)+4*acos(1/4) = 360.0 1*acos(23/32)+2*acos(3/8) = 180.0 1*acos(31/32)+4*acos(3/4) = 180.0 3*acos(1/4)+1*acos(-11/16) = 360.0 3*acos(-1/8)+2*acos(-9/16) = 540.0 2*acos(-11/16)+3*acos(-7/8) = 720.0 1*acos(-9/16)+3*acos(-3/4) = 540.0 2*acos(9/16)+3*acos(1/8) = 360.0 1*acos(11/16)+3*acos(-1/4) = 360.0 2*acos(-1/4)+1*acos(-7/8) = 360.0 2*acos(3/4)+1*acos(-1/8) = 180.0 1*acos(1/8)+2*acos(-3/4) = 360.0 1*acos(7/8)+2*acos(1/4) = 180.0 There also are examples involving only powers of 3: 2*acos(-4/9)+1*acos(-49/81) = 360.0 2*acos(-2/9)+1*acos(-73/81) = 360.0 2*acos(5/9)+1*acos(31/81) = 180.0 2*acos(8/9)+1*acos(-47/81) = 180.0 4*acos(-22/27)+3*acos(-79/81) =1080.0 1*acos(-31/81)+2*acos(-5/9) = 360.0 1*acos(-17/81)+4*acos(-1/3) = 540.0 1*acos(17/81)+2*acos(-7/9) = 360.0 1*acos(47/81)+2*acos(-8/9) = 360.0 1*acos(49/81)+2*acos(4/9) = 180.0 1*acos(73/81)+2*acos(2/9) = 180.0 1*acos(79/81)+4*acos(-2/3) = 540.0 3*acos(-2/3)+1*acos(-22/27) = 540.0 3*acos(1/3)+1*acos(-23/27) = 360.0 3*acos(-7/9)+2*acos(-23/27) = 720.0 3*acos(1/9)+2*acos(-22/27) = 540.0 1*acos(22/27)+3*acos(2/3) = 180.0 1*acos(23/27)+3*acos(-1/3) = 360.0 2*acos(-1/3)+1*acos(-7/9) = 360.0 2*acos(2/3)+1*acos(1/9) = 180.0 1*acos(-1/9)+2*acos(-2/3) = 360.0 1*acos(7/9)+2*acos(1/3) = 180.0 what is going on here? I will try to rerun the program out to 128 and post the results later. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
On 2015-07-16 08:27, Warren D Smith wrote:
It is interesting that some of the most-amazing among the listed miracles involve only powers of 2 in denominators:
5*acos(-7/8)+2*acos(-61/64) =1080.0 4*acos(-57/64)+5*acos(-31/32) =1440.0 1*acos(57/64)+5*acos(-3/4) = 720.0 1*acos(61/64)+5*acos(-1/4) = 540.0 4*acos(-3/4)+1*acos(-31/32) = 720.0 2*acos(-3/8)+1*acos(-23/32) = 360.0 2*acos(5/8)+1*acos(7/32) = 180.0 3*acos(-17/32)+4*acos(-11/16) = 900.0 ...
Hey, 3-4-5 ! In[905]:= 3*ArcCos@(-17/32) + 4*ArcCos@(-11/16) == 5*Pi In[906]:= FullSimplify[%] Out[906]= 3 \[Pi] == 6 ArcSin[17/32] + 8 ArcSin[11/16] is dumfounded, too. --rwg
ARG!
2*acos(-3/8)+1*acos(-23/32) = 360.0
acos(3/8) = atan( sqrt55 / 3 ) = arg( 3 + i q55) ;using q as sqrt sign (3 + i q55) ^2 = -46 + i 6 q55 the norm of 3+iq55 is 8, and the norm of -46+i6q55 is 64. 46/64 = 23/32. Rich ________________________________________ From: math-fun [math-fun-bounces@mailman.xmission.com] on behalf of rwg [rwg@sdf.org] Sent: Thursday, July 16, 2015 4:56 PM To: math-fun Subject: [EXTERNAL] Re: [math-fun] ArcCosine Miracles List, and explanation On 2015-07-16 08:27, Warren D Smith wrote:
It is interesting that some of the most-amazing among the listed miracles involve only powers of 2 in denominators:
5*acos(-7/8)+2*acos(-61/64) =1080.0 4*acos(-57/64)+5*acos(-31/32) =1440.0 1*acos(57/64)+5*acos(-3/4) = 720.0 1*acos(61/64)+5*acos(-1/4) = 540.0 4*acos(-3/4)+1*acos(-31/32) = 720.0 2*acos(-3/8)+1*acos(-23/32) = 360.0 2*acos(5/8)+1*acos(7/32) = 180.0 3*acos(-17/32)+4*acos(-11/16) = 900.0 ...
Hey, 3-4-5 ! In[905]:= 3*ArcCos@(-17/32) + 4*ArcCos@(-11/16) == 5*Pi In[906]:= FullSimplify[%] Out[906]= 3 \[Pi] == 6 ArcSin[17/32] + 8 ArcSin[11/16] is dumfounded, too. --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
* Warren D Smith <warren.wds@gmail.com> [Jul 17. 2015 08:39]:
It is interesting that some of the most-amazing among the listed miracles involve only powers of 2 in denominators:
5*acos(-7/8)+2*acos(-61/64) =1080.0 4*acos(-57/64)+5*acos(-31/32) =1440.0 1*acos(57/64)+5*acos(-3/4) = 720.0 1*acos(61/64)+5*acos(-1/4) = 540.0 [...]
My comments in the following sequences may be pertinent: https://oeis.org/A038198 https://oeis.org/A168229 I'd hope one could map these relations to arctan relations as given in http://jjj.de/arctan/arctanpage.html Best regards, jj
participants (4)
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Joerg Arndt -
rwg -
Schroeppel, Richard -
Warren D Smith