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tech / sci.math / New set of triogonmetic identities from the disproof - and that, some - of FLT

SubjectAuthor
* New set of triogonmetic identities from the disproof - and that, somebanerjee...@gmail.com
+* Re: New set of triogonmetic identities from the disproof - and that, some - of FBarry Schwarz
|+* Re: New set of triogonmetic identities from the disproof - and that,banerjee...@gmail.com
||`- Re: New set of triogonmetic identities from the disproof - and that,banerjee...@gmail.com
|`* Re: New set of triogonmetic identities from the disproof - and that, some - of FJames Waldby
| `* Re: New set of triogonmetic identities from the disproof - and that,banerjee...@gmail.com
|  `* Re: New set of triogonmetic identities from the disproof - and that,Chris M. Thomasson
|   `- Re: New set of triogonmetic identities from the disproof - and that,banerjee...@gmail.com
+- Re: New set of triogonmetic identities from the disproof - and that,banerjee...@gmail.com
`* Re: New set of triogonmetic identities from the disproof - and that,Dave
 `* Re: New set of triogonmetic identities from the disproof - and that,banerjee...@gmail.com
  `* Re: New set of triogonmetic identities from the disproof - and that,Eric
   `- Re: New set of triogonmetic identities from the disproof - and that,banerjee...@gmail.com

1
New set of triogonmetic identities from the disproof - and that, some - of FLT

<6ecf238e-cfb5-483d-ad6a-50a3978169aan@googlegroups.com>

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Subject: New set of triogonmetic identities from the disproof - and that, some
- of FLT
From: banerjee...@gmail.com (banerjee...@gmail.com)
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 by: banerjee...@gmail.co - Fri, 21 Apr 2023 06:12 UTC

One of them is

For any x < 1, and integer n>2
*****
arcsin(sqrt(x^n)) + arcsin(sqrt(1-x^n))=pi/2
*****
as a result of the disproof of Fermat's Last Theorem
where n is any integer greater than 3.

n= 3

x theta phi sum
0.001 3.16228E-05 1.570764704 1.570796327
0.011 0.00115369 1.569642637 1.570796327
0.021 0.003043194 1.567753133 1.570796327
0.031 0.00545814 1.565338186 1.570796327
0.041 0.008301963 1.562494364 1.570796327
0.051 0.011517676 1.559278651 1.570796327
0.061 0.015066459 1.555729868 1.570796327
0.071 0.018919665 1.551876662 1.570796327
0.081 0.023055047 1.54774128 1.570796327
0.091 0.027454697 1.54334163 1.570796327
*****************
n=20

x theta phi sum
0.001 1E-30 1.570796327 1.570796327
0.011 2.59374E-20 1.570796327 1.570796327
0.021 1.66799E-17 1.570796327 1.570796327
0.031 8.19628E-16 1.570796327 1.570796327
0.041 1.34227E-14 1.570796327 1.570796327
0.051 1.19042E-13 1.570796327 1.570796327
0.061 7.13343E-13 1.570796327 1.570796327
0.071 3.25524E-12 1.570796327 1.570796327
0.081 1.21577E-11 1.570796327 1.570796327
0.091 3.89416E-11 1.570796327 1.570796327

Cheers,
Arindam Banerjee

Note: About a year ago, I thought I had found a proof for FLT but as things are with the use of the arcsin function in Excel I found I had been wrong.

The disproof (on paper) is in my facebook page.

Cheers,
Arindam Banerjee

Re: New set of triogonmetic identities from the disproof - and that, some - of FLT

<snh44i5eu5dr1a2p4v4elk1lj2obec7d2n@4ax.com>

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From: schwa...@delq.com (Barry Schwarz)
Newsgroups: sci.math
Subject: Re: New set of triogonmetic identities from the disproof - and that, some - of FLT
Date: Fri, 21 Apr 2023 01:20:23 -0700
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 by: Barry Schwarz - Fri, 21 Apr 2023 08:20 UTC

On Thu, 20 Apr 2023 23:12:30 -0700 (PDT), "banerjee...@gmail.com"
<banerjeeadda1234@gmail.com> wrote:

>One of them is
>
>For any x < 1, and integer n>2
>*****
>arcsin(sqrt(x^n)) + arcsin(sqrt(1-x^n))=pi/2
>*****

Not really. When x is .5 and n is 3, the sum is off by over 15%.

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Re: New set of triogonmetic identities from the disproof - and that, some - of FLT

<b4b90658-9a80-4634-b141-d5b6e5725ff5n@googlegroups.com>

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Subject: Re: New set of triogonmetic identities from the disproof - and that,
some - of FLT
From: banerjee...@gmail.com (banerjee...@gmail.com)
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 by: banerjee...@gmail.co - Fri, 21 Apr 2023 08:51 UTC

On Friday, 21 April 2023 at 18:20:38 UTC+10, Barry Schwarz wrote:
> On Thu, 20 Apr 2023 23:12:30 -0700 (PDT), "banerjee...@gmail.com"
> <banerjee...@gmail.com> wrote:
>
> >One of them is
> >
> >For any x < 1, and integer n>2
> >*****
> >arcsin(sqrt(x^n)) + arcsin(sqrt(1-x^n))=pi/2
> >*****
> Not really. When x is .5 and n is 3, the sum is off by over 15%.

NO.
n= 3

x theta phi sum
0.001 3.16228E-05 1.570764704 1.570796327
0.011 0.00115369 1.569642637 1.570796327
0.021 0.003043194 1.567753133 1.570796327
0.031 0.00545814 1.565338186 1.570796327
0.041 0.008301963 1.562494364 1.570796327
0.051 0.011517676 1.559278651 1.570796327
0.061 0.015066459 1.555729868 1.570796327
0.071 0.018919665 1.551876662 1.570796327
0.081 0.023055047 1.54774128 1.570796327
0.091 0.027454697 1.54334163 1.570796327
0.101 0.032103817 1.53869251 1.570796327
0.111 0.03698993 1.533806397 1.570796327
0.121 0.042102353 1.528693974 1.570796327
0.131 0.047431821 1.523364506 1.570796327
0.141 0.052970221 1.517826106 1.570796327
0.151 0.058710387 1.51208594 1.570796327
0.161 0.064645954 1.506150373 1.570796327
0.171 0.070771232 1.500025094 1.570796327
0.181 0.077081118 1.493715208 1.570796327
0.191 0.083571019 1.487225308 1.570796327
0.201 0.090236789 1.480559538 1.570796327
0.211 0.097074686 1.473721641 1.570796327
0.221 0.104081324 1.466715003 1.570796327
0.231 0.111253644 1.459542683 1.570796327
0.241 0.118588883 1.452207444 1.570796327
0.251 0.126084552 1.444711775 1.570796327
0.261 0.133738412 1.437057915 1.570796327
0.271 0.141548463 1.429247864 1.570796327
0.281 0.149512923 1.421283404 1.570796327
0.291 0.15763022 1.413166107 1.570796327
0.301 0.16589898 1.404897346 1.570796327
0.311 0.17431802 1.396478307 1.570796327
0.321 0.182886335 1.387909991 1.570796327
0.331 0.191603102 1.379193225 1.570796327
0.341 0.200467663 1.370328664 1.570796327
0.351 0.209479532 1.361316795 1.570796327
0.361 0.218638385 1.352157942 1.570796327
0.371 0.227944061 1.342852266 1.570796327
0.381 0.237396559 1.333399768 1.570796327
0.391 0.246996039 1.323800288 1.570796327
0.401 0.256742821 1.314053506 1.570796327
0.411 0.266637389 1.304158938 1.570796327
0.421 0.276680386 1.294115941 1.570796327
0.431 0.286872625 1.283923701 1.570796327
0.441 0.297215086 1.273581241 1.570796327
0.451 0.307708921 1.263087406 1.570796327
0.461 0.318355461 1.252440866 1.570796327
0.471 0.32915622 1.241640107 1.570796327
0.481 0.3401129 1.230683427 1.570796327
0.491 0.351227401 1.219568926 1.570796327
0.501 0.362501828 1.208294499 1.570796327
0.511 0.3739385 1.196857827 1.570796327
0.521 0.385539962 1.185256365 1.570796327
0.531 0.397308995 1.173487332 1.570796327
0.541 0.40924863 1.161547696 1.570796327
0.551 0.421362165 1.149434162 1.570796327
0.561 0.433653176 1.13714315 1.570796327
0.571 0.446125543 1.124670784 1.570796327
0.581 0.458783464 1.112012863 1.570796327
0.591 0.471631481 1.099164845 1.570796327
0.601 0.484674508 1.086121818 1.570796327
0.611 0.497917856 1.07287847 1.570796327
0.621 0.511367268 1.059429059 1.570796327
0.631 0.525028953 1.045767374 1.570796327
0.641 0.538909633 1.031886694 1.570796327
0.651 0.553016582 1.017779744 1.570796327
0.661 0.567357687 1.003438639 1.570796327
0.671 0.581941502 0.988854825 1.570796327
0.681 0.596777318 0.974019009 1.570796327
0.691 0.611875243 0.958921083 1.570796327
0.701 0.627246294 0.943550033 1.570796327
0.711 0.642902495 0.927893832 1.570796327
0.721 0.658857003 0.911939324 1.570796327
0.731 0.675124247 0.895672079 1.570796327
0.741 0.691720092 0.879076234 1.570796327
0.751 0.708662032 0.862134295 1.570796327
0.761 0.725969417 0.84482691 1.570796327
0.771 0.74366373 0.827132597 1.570796327
0.781 0.76176891 0.809027417 1.570796327
0.791 0.780311747 0.79048458 1.570796327
0.801 0.799322364 0.771473963 1.570796327
0.811 0.81883481 0.751961516 1.570796327
0.821 0.838887794 0.731908533 1.570796327
0.831 0.859525604 0.711270723 1.570796327
0.841 0.880799286 0.689997041 1.570796327
0.851 0.902768152 0.668028175 1.570796327
0.861 0.925501758 0.645294569 1.570796327
0.871 0.949082531 0.621713796 1.570796327
0.881 0.973609347 0.59718698 1.570796327
0.891 0.99920248 0.571593847 1.570796327
0.901 1.026010666 0.544785661 1.570796327
0.911 1.054221477 0.51657485 1.570796327
0.921 1.084077135 0.486719192 1.570796327
0.931 1.115899712 0.454896615 1.570796327
0.941 1.150133595 0.420662732 1.570796327
0.951 1.187422327 0.383373999 1.570796327
0.961 1.228761536 0.342034791 1.570796327
0.971 1.275847146 0.294949181 1.570796327
0.981 1.332052511 0.238743815 1.570796327
0.991 1.406480006 0.164316321 1.570796327

Same pi/2 all the way.

For n-20, below

n=20

x theta phi sum
0.001 1E-30 1.570796327 1.570796327
0.011 2.59374E-20 1.570796327 1.570796327
0.021 1.66799E-17 1.570796327 1.570796327
0.031 8.19628E-16 1.570796327 1.570796327
0.041 1.34227E-14 1.570796327 1.570796327
0.051 1.19042E-13 1.570796327 1.570796327
0.061 7.13343E-13 1.570796327 1.570796327
0.071 3.25524E-12 1.570796327 1.570796327
0.081 1.21577E-11 1.570796327 1.570796327
0.091 3.89416E-11 1.570796327 1.570796327
0.101 1.10462E-10 1.570796327 1.570796327
0.111 2.83942E-10 1.570796327 1.570796327
0.121 6.7275E-10 1.570796327 1.570796327
0.131 1.48838E-09 1.570796327 1.570796328
0.141 3.10593E-09 1.570796327 1.57079633
0.151 6.16268E-09 1.570796327 1.570796333
0.161 1.1702E-08 1.570796327 1.570796338
0.171 2.13777E-08 1.570796306 1.570796327
0.181 3.77386E-08 1.57079629 1.570796328
0.191 6.4615E-08 1.570796262 1.570796326
0.201 1.07637E-07 1.570796219 1.570796327
0.211 1.74914E-07 1.570796152 1.570796327
0.221 2.77922E-07 1.570796049 1.570796327
0.231 4.32633E-07 1.570795894 1.570796327
0.241 6.60953E-07 1.570795666 1.570796327
0.251 9.92515E-07 1.570795334 1.570796327
0.261 1.46692E-06 1.57079486 1.570796327
0.271 2.13645E-06 1.57079419 1.570796327
0.281 3.06947E-06 1.570793257 1.570796327
0.291 4.35442E-06 1.570791972 1.570796327
0.301 6.10471E-06 1.570790222 1.570796327
0.311 8.46455E-06 1.570787862 1.570796327
0.321 1.16158E-05 1.570784711 1.570796327
0.331 1.57863E-05 1.570780541 1.570796327
0.341 2.1259E-05 1.570775068 1.570796327
0.351 2.83838E-05 1.570767943 1.570796327
0.361 3.759E-05 1.570758737 1.570796327
0.371 4.94014E-05 1.570746925 1.570796327
0.381 6.4454E-05 1.570731873 1.570796327
0.391 8.35156E-05 1.570712811 1.570796327
0.401 0.000107509 1.570688818 1.570796327
0.411 0.000137537 1.57065879 1.570796327
0.421 0.000174913 1.570621414 1.570796327
0.431 0.000221194 1.570575133 1.570796327
0.441 0.000278218 1.570518108 1.570796327
0.451 0.000348149 1.570448178 1.570796327
0.461 0.00043352 1.570362807 1.570796327
0.471 0.00053729 1.570259036 1.570796327
0.481 0.000662904 1.570133423 1.570796327
0.491 0.000814357 1.56998197 1.570796327
0.501 0.000996271 1.569800056 1.570796327
0.511 0.001213973 1.569582354 1.570796327
0.521 0.001473592 1.569322734 1.570796327
0.531 0.001782155 1.569014172 1.570796327
0.541 0.002147697 1.56864863 1.570796327
0.551 0.002579387 1.56821694 1.570796327
0.561 0.003087659 1.567708668 1.570796327
0.571 0.00368436 1.567111967 1.570796327
0.581 0.004382912 1.566413415 1.570796327
0.591 0.005198485 1.565597842 1.570796327
0.601 0.006148192 1.564648134 1.570796327
0.611 0.007251301 1.563545026 1.570796327
0.621 0.008529455 1.562266872 1.570796327
0.631 0.01000693 1.560789397 1.570796327
0.641 0.011710899 1.559085428 1.570796327
0.651 0.013671728 1.557124599 1.570796327
0.661 0.015923295 1.554873032 1.570796327
0.671 0.018503334 1.552292993 1.570796327
0.681 0.02145381 1.549342517 1.570796327
0.691 0.024821331 1.545974996 1.570796327
0.701 0.028657587 1.542138739 1.570796327
0.711 0.033019837 1.537776489 1.570796327
0.721 0.037971435 1.532824891 1.570796327
0.731 0.043582412 1.527213915 1.570796327
0.741 0.049930113 1.520866214 1.570796327
0.751 0.057099906 1.513696421 1.570796327
0.761 0.065185977 1.50561035 1.570796327
0.771 0.074292224 1.496504103 1.570796327
0.781 0.084533276 1.486263051 1.570796327
0.791 0.096035679 1.474760648 1.570796327
0.801 0.108939285 1.461857042 1.570796327
0.811 0.1233989 1.447397426 1.570796327
0.821 0.139586305 1.431210022 1.570796327
0.831 0.157692728 1.413103599 1.570796327
0.841 0.177931972 1.392864354 1.570796327
0.851 0.200544425 1.370251901 1.570796327
0.861 0.225802292 1.344994035 1.570796327
0.871 0.254016583 1.316779743 1.570796327
0.881 0.285546634 1.285249693 1.570796327
0.891 0.320813365 1.249982962 1.570796327
0.901 0.36031826 1.210478067 1.570796327
0.911 0.40467131 1.166125016 1.570796327
0.921 0.454633676 1.116162651 1.570796327
0.931 0.511185648 1.059610679 1.570796327
0.941 0.575640964 0.995155363 1.570796327
0.951 0.649852995 0.920943331 1.570796327
0.961 0.736623359 0.834172968 1.570796327
0.971 0.840627495 0.730168832 1.570796327
0.981 0.970996586 0.599799741 1.570796327
0.991 1.151950315 0.418846012 1.570796327

>
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Re: New set of triogonmetic identities from the disproof - and that, some - of FLT

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Date: Fri, 21 Apr 2023 01:59:14 -0700 (PDT)
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Subject: Re: New set of triogonmetic identities from the disproof - and that,
some - of FLT
From: banerjee...@gmail.com (banerjee...@gmail.com)
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 by: banerjee...@gmail.co - Fri, 21 Apr 2023 08:59 UTC

On Friday, 21 April 2023 at 16:12:34 UTC+10, banerjee...@gmail.com wrote:
> One of them is
>
> For any x < 1, and integer n>2
> *****
> arcsin(sqrt(x^n)) + arcsin(sqrt(1-x^n))=pi/2
> *****
> as a result of the disproof of Fermat's Last Theorem
> where n is any integer greater than 3.
>
> n= 3
>
> x theta phi sum
> 0.001 3.16228E-05 1.570764704 1.570796327
> 0.011 0.00115369 1.569642637 1.570796327
> 0.021 0.003043194 1.567753133 1.570796327
> 0.031 0.00545814 1.565338186 1.570796327
> 0.041 0.008301963 1.562494364 1.570796327
> 0.051 0.011517676 1.559278651 1.570796327
> 0.061 0.015066459 1.555729868 1.570796327
> 0.071 0.018919665 1.551876662 1.570796327
> 0.081 0.023055047 1.54774128 1.570796327
> 0.091 0.027454697 1.54334163 1.570796327
> *****************
> n=20
>
> x theta phi sum
> 0.001 1E-30 1.570796327 1.570796327
> 0.011 2.59374E-20 1.570796327 1.570796327
> 0.021 1.66799E-17 1.570796327 1.570796327
> 0.031 8.19628E-16 1.570796327 1.570796327
> 0.041 1.34227E-14 1.570796327 1.570796327
> 0.051 1.19042E-13 1.570796327 1.570796327
> 0.061 7.13343E-13 1.570796327 1.570796327
> 0.071 3.25524E-12 1.570796327 1.570796327
> 0.081 1.21577E-11 1.570796327 1.570796327
> 0.091 3.89416E-11 1.570796327 1.570796327
>
>
> Cheers,
> Arindam Banerjee
>
> Note: About a year ago, I thought I had found a proof for FLT but as things are with the use of the arcsin function in Excel I found I had been wrong.
>
> The disproof (on paper) is in my facebook page.
>
> Cheers,
> Arindam Banerjee

Should not take long to do the verification in Excel, using the asin() function.

Of course, the identities could be extended to the squares, cubes, etc. of the LHS and RHS to get more complex identities.
And arccos() is also there.

Re: New set of triogonmetic identities from the disproof - and that, some - of FLT

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From: no...@no.no (James Waldby)
Newsgroups: sci.math
Subject: Re: New set of triogonmetic identities from the disproof - and that, some - of FLT
Date: Fri, 21 Apr 2023 19:28:08 -0000 (UTC)
Organization: A noiseless patient Spider
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 by: James Waldby - Fri, 21 Apr 2023 19:28 UTC

Barry Schwarz <schwarzb@delq.com> wrote:
> On Thu, 20 Apr 2023 23:12:30 -0700 (PDT), "banerjee...@gmail.com"
> <banerjeeadda1234@gmail.com> wrote:
>
>>One of them is
>>
>>For any x < 1, and integer n>2
>>*****
>>arcsin(sqrt(x^n)) + arcsin(sqrt(1-x^n))=pi/2
>>*****
>
> Not really. When x is .5 and n is 3, the sum is off by over 15%.

Perhaps recalculate that?

The identity shown is a simple consequence of sin^2(t) + cos^2(t) = 1,
slightly obfuscated by appearance of the exponent n.

For x in [0,1], for any non-negative n, we will also have x^n in
[0,1]. If we let u = x^n, the identity is asin(sqrt(u))+asin(sqrt(1-u))
= pi/2, which may be a well-known identity, I'm not sure.

Anyhow, to see it, note that as u is in [0,1], there is a t in [0,pi/2]
with u = (sin t)^2. Now asin(sqrt(u)) = asin(sin t) = t and
asin(sqrt(1-u)) = asin(cos t) = asin(sin pi/2-t) = pi/2 - t, whence
the result. Obviously the powers of n are irrelevant. - jiw

Re: New set of triogonmetic identities from the disproof - and that, some - of FLT

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From: efft...@mail.net (Dave)
Newsgroups: sci.math
Subject: Re: New set of triogonmetic identities from the disproof - and that,
some - of FLT
Date: Fri, 21 Apr 2023 12:36:31 -0700
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 by: Dave - Fri, 21 Apr 2023 19:36 UTC

On 4/20/2023 11:12 PM, banerjee...@gmail.com wrote:
> One of them is

Shut up idiot.

Re: New set of triogonmetic identities from the disproof - and that, some - of FLT

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Subject: Re: New set of triogonmetic identities from the disproof - and that,
some - of FLT
From: banerjee...@gmail.com (banerjee...@gmail.com)
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 by: banerjee...@gmail.co - Fri, 21 Apr 2023 22:44 UTC

On Saturday, 22 April 2023 at 05:28:18 UTC+10, James Waldby wrote:
> Barry Schwarz <schw...@delq.com> wrote:
> > On Thu, 20 Apr 2023 23:12:30 -0700 (PDT), "banerjee...@gmail.com"
> > <banerjee...@gmail.com> wrote:
> >
> >>One of them is
> >>
> >>For any x < 1, and integer n>2
> >>*****
> >>arcsin(sqrt(x^n)) + arcsin(sqrt(1-x^n))=pi/2
> >>*****
> >
> > Not really. When x is .5 and n is 3, the sum is off by over 15%.
> Perhaps recalculate that?
>
> The identity shown is a simple consequence of sin^2(t) + cos^2(t) = 1,
> slightly obfuscated by appearance of the exponent n.
>
> For x in [0,1], for any non-negative n, we will also have x^n in
> [0,1]. If we let u = x^n, the identity is asin(sqrt(u))+asin(sqrt(1-u))
> = pi/2, which may be a well-known identity, I'm not sure.

Well, there seems no doubt that such an identity now exists.
Someone may have found it in the past.
This is a disproof of Fermat's Last Theorem.
There are very very large numbers beyond what we can manipulate now, and that is why the identity works.
It is the mathematical proof of infinity.

Cheers,
Arindam Banerjee
>
> Anyhow, to see it, note that as u is in [0,1], there is a t in [0,pi/2]
> with u = (sin t)^2. Now asin(sqrt(u)) = asin(sin t) = t and
> asin(sqrt(1-u)) = asin(cos t) = asin(sin pi/2-t) = pi/2 - t, whence
> the result. Obviously the powers of n are irrelevant. - jiw

Yes, but pi/2 is not 1, so this is a different identity to the famous one you showed.

Re: New set of triogonmetic identities from the disproof - and that, some - of FLT

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Subject: Re: New set of triogonmetic identities from the disproof - and that,
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 by: banerjee...@gmail.co - Fri, 21 Apr 2023 22:45 UTC

On Saturday, 22 April 2023 at 05:36:40 UTC+10, Dave wrote:
> On 4/20/2023 11:12 PM, banerjee...@gmail.com wrote:
> > One of them is
> Shut up idiot.
tch tch, go to a moderated group, wretch.

Re: New set of triogonmetic identities from the disproof - and that, some - of FLT

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From: effta...@mail.net (Eric)
Newsgroups: sci.math
Subject: Re: New set of triogonmetic identities from the disproof - and that,
some - of FLT
Date: Fri, 21 Apr 2023 17:09:49 -0700
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 by: Eric - Sat, 22 Apr 2023 00:09 UTC

On 4/21/2023 3:45 PM, banerjee...@gmail.com wrote:
> On Saturday, 22 April 2023 at 05:36:40 UTC+10, Dave wrote:
>> On 4/20/2023 11:12 PM, banerjee...@gmail.com wrote:
>>> One of them is
>> Shut up idiot.
> tch tch, go to a moderated group, wretch.

Shut up imbecile.

Re: New set of triogonmetic identities from the disproof - and that, some - of FLT

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From: chris.m....@gmail.com (Chris M. Thomasson)
Newsgroups: sci.math
Subject: Re: New set of triogonmetic identities from the disproof - and that,
some - of FLT
Date: Fri, 21 Apr 2023 17:25:38 -0700
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 by: Chris M. Thomasson - Sat, 22 Apr 2023 00:25 UTC

On 4/21/2023 3:44 PM, banerjee...@gmail.com wrote:
> On Saturday, 22 April 2023 at 05:28:18 UTC+10, James Waldby wrote:
>> Barry Schwarz <schw...@delq.com> wrote:
>>> On Thu, 20 Apr 2023 23:12:30 -0700 (PDT), "banerjee...@gmail.com"
>>> <banerjee...@gmail.com> wrote:
>>>
>>>> One of them is
>>>>
>>>> For any x < 1, and integer n>2
>>>> *****
>>>> arcsin(sqrt(x^n)) + arcsin(sqrt(1-x^n))=pi/2
>>>> *****
>>>
>>> Not really. When x is .5 and n is 3, the sum is off by over 15%.
>> Perhaps recalculate that?
>>
>> The identity shown is a simple consequence of sin^2(t) + cos^2(t) = 1,
>> slightly obfuscated by appearance of the exponent n.
>>
>> For x in [0,1], for any non-negative n, we will also have x^n in
>> [0,1]. If we let u = x^n, the identity is asin(sqrt(u))+asin(sqrt(1-u))
>> = pi/2, which may be a well-known identity, I'm not sure.
>
> Well, there seems no doubt that such an identity now exists.
> Someone may have found it in the past.
> This is a disproof of Fermat's Last Theorem.
> There are very very large numbers beyond what we can manipulate now, and that is why the identity works.
> It is the mathematical proof of infinity.
>
> Cheers,
> Arindam Banerjee
>>
>> Anyhow, to see it, note that as u is in [0,1], there is a t in [0,pi/2]
>> with u = (sin t)^2. Now asin(sqrt(u)) = asin(sin t) = t and
>> asin(sqrt(1-u)) = asin(cos t) = asin(sin pi/2-t) = pi/2 - t, whence
>> the result. Obviously the powers of n are irrelevant. - jiw
>
> Yes, but pi/2 is not 1, so this is a different identity to the famous one you showed.
>

cos(0) = sin(pi/2) = 1

;^)

Re: New set of triogonmetic identities from the disproof - and that, some - of FLT

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Subject: Re: New set of triogonmetic identities from the disproof - and that,
some - of FLT
From: banerjee...@gmail.com (banerjee...@gmail.com)
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 by: banerjee...@gmail.co - Sat, 22 Apr 2023 02:01 UTC

On Saturday, 22 April 2023 at 10:25:48 UTC+10, Chris M. Thomasson wrote:
> On 4/21/2023 3:44 PM, banerjee...@gmail.com wrote:
> > On Saturday, 22 April 2023 at 05:28:18 UTC+10, James Waldby wrote:
> >> Barry Schwarz <schw...@delq.com> wrote:
> >>> On Thu, 20 Apr 2023 23:12:30 -0700 (PDT), "banerjee...@gmail.com"
> >>> <banerjee...@gmail.com> wrote:
> >>>
> >>>> One of them is
> >>>>
> >>>> For any x < 1, and integer n>2
> >>>> *****
> >>>> arcsin(sqrt(x^n)) + arcsin(sqrt(1-x^n))=pi/2
> >>>> *****
> >>>
> >>> Not really. When x is .5 and n is 3, the sum is off by over 15%.
> >> Perhaps recalculate that?
> >>
> >> The identity shown is a simple consequence of sin^2(t) + cos^2(t) = 1,
> >> slightly obfuscated by appearance of the exponent n.
> >>
> >> For x in [0,1], for any non-negative n, we will also have x^n in
> >> [0,1]. If we let u = x^n, the identity is asin(sqrt(u))+asin(sqrt(1-u))
> >> = pi/2, which may be a well-known identity, I'm not sure.
> >
> > Well, there seems no doubt that such an identity now exists.
> > Someone may have found it in the past.
> > This is a disproof of Fermat's Last Theorem.
> > There are very very large numbers beyond what we can manipulate now, and that is why the identity works.
> > It is the mathematical proof of infinity.
> >
> > Cheers,
> > Arindam Banerjee
> >>
> >> Anyhow, to see it, note that as u is in [0,1], there is a t in [0,pi/2]
> >> with u = (sin t)^2. Now asin(sqrt(u)) = asin(sin t) = t and
> >> asin(sqrt(1-u)) = asin(cos t) = asin(sin pi/2-t) = pi/2 - t, whence
> >> the result. Obviously the powers of n are irrelevant. - jiw
> >
> > Yes, but pi/2 is not 1, so this is a different identity to the famous one you showed.
> >
> cos(0) = sin(pi/2) = 1

Irrelevant

1 and pi/2 are not the same.

So we can have trigonometric identities as:
arcsin(sqrt(x^n)) + arcsin(sqrt(1-x^n))=pi/2(sin(y)^2 + cos(y)^2)

Re: New set of triogonmetic identities from the disproof - and that, some - of FLT

<3985de18-cab1-4a71-b6d3-d79fa36640f9n@googlegroups.com>

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Subject: Re: New set of triogonmetic identities from the disproof - and that,
some - of FLT
From: banerjee...@gmail.com (banerjee...@gmail.com)
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 by: banerjee...@gmail.co - Sat, 22 Apr 2023 02:01 UTC

On Saturday, 22 April 2023 at 10:09:58 UTC+10, Eric wrote:
> On 4/21/2023 3:45 PM, banerjee...@gmail.com wrote:
> > On Saturday, 22 April 2023 at 05:36:40 UTC+10, Dave wrote:
> >> On 4/20/2023 11:12 PM, banerjee...@gmail.com wrote:
> >>> One of them is
> >> Shut up idiot.
> > tch tch, go to a moderated group, wretch.
> Shut up imbecile.
Go to hell, creep.

Re: New set of triogonmetic identities from the disproof - and that, some - of FLT

<4305678b-1f3f-4f31-a01f-549a3e61583dn@googlegroups.com>

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<snh44i5eu5dr1a2p4v4elk1lj2obec7d2n@4ax.com> <b4b90658-9a80-4634-b141-d5b6e5725ff5n@googlegroups.com>
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Subject: Re: New set of triogonmetic identities from the disproof - and that,
some - of FLT
From: banerjee...@gmail.com (banerjee...@gmail.com)
Injection-Date: Sun, 23 Apr 2023 04:13:04 +0000
Content-Type: text/plain; charset="UTF-8"
 by: banerjee...@gmail.co - Sun, 23 Apr 2023 04:13 UTC

On Friday, 21 April 2023 at 18:51:58 UTC+10, banerjee...@gmail.com wrote:
> On Friday, 21 April 2023 at 18:20:38 UTC+10, Barry Schwarz wrote:
> > On Thu, 20 Apr 2023 23:12:30 -0700 (PDT), "banerjee...@gmail.com"
> > <banerjee...@gmail.com> wrote:
> >
> > >One of them is
> > >
> > >For any x < 1, and integer n>2
> > >*****
> > >arcsin(sqrt(x^n)) + arcsin(sqrt(1-x^n))=pi/2
> > >*****
> > Not really. When x is .5 and n is 3, the sum is off by over 15%.
> NO.
> n= 3
>
> x theta phi sum
> 0.001 3.16228E-05 1.570764704 1.570796327
> 0.011 0.00115369 1.569642637 1.570796327
> 0.021 0.003043194 1.567753133 1.570796327
> 0.031 0.00545814 1.565338186 1.570796327
> 0.041 0.008301963 1.562494364 1.570796327
> 0.051 0.011517676 1.559278651 1.570796327
> 0.061 0.015066459 1.555729868 1.570796327
> 0.071 0.018919665 1.551876662 1.570796327
> 0.081 0.023055047 1.54774128 1.570796327
> 0.091 0.027454697 1.54334163 1.570796327
> 0.101 0.032103817 1.53869251 1.570796327
> 0.111 0.03698993 1.533806397 1.570796327
> 0.121 0.042102353 1.528693974 1.570796327
> 0.131 0.047431821 1.523364506 1.570796327
> 0.141 0.052970221 1.517826106 1.570796327
> 0.151 0.058710387 1.51208594 1.570796327
> 0.161 0.064645954 1.506150373 1.570796327
> 0.171 0.070771232 1.500025094 1.570796327
> 0.181 0.077081118 1.493715208 1.570796327
> 0.191 0.083571019 1.487225308 1.570796327
> 0.201 0.090236789 1.480559538 1.570796327
> 0.211 0.097074686 1.473721641 1.570796327
> 0.221 0.104081324 1.466715003 1.570796327
> 0.231 0.111253644 1.459542683 1.570796327
> 0.241 0.118588883 1.452207444 1.570796327
> 0.251 0.126084552 1.444711775 1.570796327
> 0.261 0.133738412 1.437057915 1.570796327
> 0.271 0.141548463 1.429247864 1.570796327
> 0.281 0.149512923 1.421283404 1.570796327
> 0.291 0.15763022 1.413166107 1.570796327
> 0.301 0.16589898 1.404897346 1.570796327
> 0.311 0.17431802 1.396478307 1.570796327
> 0.321 0.182886335 1.387909991 1.570796327
> 0.331 0.191603102 1.379193225 1.570796327
> 0.341 0.200467663 1.370328664 1.570796327
> 0.351 0.209479532 1.361316795 1.570796327
> 0.361 0.218638385 1.352157942 1.570796327
> 0.371 0.227944061 1.342852266 1.570796327
> 0.381 0.237396559 1.333399768 1.570796327
> 0.391 0.246996039 1.323800288 1.570796327
> 0.401 0.256742821 1.314053506 1.570796327
> 0.411 0.266637389 1.304158938 1.570796327
> 0.421 0.276680386 1.294115941 1.570796327
> 0.431 0.286872625 1.283923701 1.570796327
> 0.441 0.297215086 1.273581241 1.570796327
> 0.451 0.307708921 1.263087406 1.570796327
> 0.461 0.318355461 1.252440866 1.570796327
> 0.471 0.32915622 1.241640107 1.570796327
> 0.481 0.3401129 1.230683427 1.570796327
> 0.491 0.351227401 1.219568926 1.570796327
> 0.501 0.362501828 1.208294499 1.570796327
> 0.511 0.3739385 1.196857827 1.570796327
> 0.521 0.385539962 1.185256365 1.570796327
> 0.531 0.397308995 1.173487332 1.570796327
> 0.541 0.40924863 1.161547696 1.570796327
> 0.551 0.421362165 1.149434162 1.570796327
> 0.561 0.433653176 1.13714315 1.570796327
> 0.571 0.446125543 1.124670784 1.570796327
> 0.581 0.458783464 1.112012863 1.570796327
> 0.591 0.471631481 1.099164845 1.570796327
> 0.601 0.484674508 1.086121818 1.570796327
> 0.611 0.497917856 1.07287847 1.570796327
> 0.621 0.511367268 1.059429059 1.570796327
> 0.631 0.525028953 1.045767374 1.570796327
> 0.641 0.538909633 1.031886694 1.570796327
> 0.651 0.553016582 1.017779744 1.570796327
> 0.661 0.567357687 1.003438639 1.570796327
> 0.671 0.581941502 0.988854825 1.570796327
> 0.681 0.596777318 0.974019009 1.570796327
> 0.691 0.611875243 0.958921083 1.570796327
> 0.701 0.627246294 0.943550033 1.570796327
> 0.711 0.642902495 0.927893832 1.570796327
> 0.721 0.658857003 0.911939324 1.570796327
> 0.731 0.675124247 0.895672079 1.570796327
> 0.741 0.691720092 0.879076234 1.570796327
> 0.751 0.708662032 0.862134295 1.570796327
> 0.761 0.725969417 0.84482691 1.570796327
> 0.771 0.74366373 0.827132597 1.570796327
> 0.781 0.76176891 0.809027417 1.570796327
> 0.791 0.780311747 0.79048458 1.570796327
> 0.801 0.799322364 0.771473963 1.570796327
> 0.811 0.81883481 0.751961516 1.570796327
> 0.821 0.838887794 0.731908533 1.570796327
> 0.831 0.859525604 0.711270723 1.570796327
> 0.841 0.880799286 0.689997041 1.570796327
> 0.851 0.902768152 0.668028175 1.570796327
> 0.861 0.925501758 0.645294569 1.570796327
> 0.871 0.949082531 0.621713796 1.570796327
> 0.881 0.973609347 0.59718698 1.570796327
> 0.891 0.99920248 0.571593847 1.570796327
> 0.901 1.026010666 0.544785661 1.570796327
> 0.911 1.054221477 0.51657485 1.570796327
> 0.921 1.084077135 0.486719192 1.570796327
> 0.931 1.115899712 0.454896615 1.570796327
> 0.941 1.150133595 0.420662732 1.570796327
> 0.951 1.187422327 0.383373999 1.570796327
> 0.961 1.228761536 0.342034791 1.570796327
> 0.971 1.275847146 0.294949181 1.570796327
> 0.981 1.332052511 0.238743815 1.570796327
> 0.991 1.406480006 0.164316321 1.570796327
>
> Same pi/2 all the way.
>
> For n-20, below
> n=20
>
> x theta phi sum
> 0.001 1E-30 1.570796327 1.570796327
> 0.011 2.59374E-20 1.570796327 1.570796327
> 0.021 1.66799E-17 1.570796327 1.570796327
> 0.031 8.19628E-16 1.570796327 1.570796327
> 0.041 1.34227E-14 1.570796327 1.570796327
> 0.051 1.19042E-13 1.570796327 1.570796327
> 0.061 7.13343E-13 1.570796327 1.570796327
> 0.071 3.25524E-12 1.570796327 1.570796327
> 0.081 1.21577E-11 1.570796327 1.570796327
> 0.091 3.89416E-11 1.570796327 1.570796327
> 0.101 1.10462E-10 1.570796327 1.570796327
> 0.111 2.83942E-10 1.570796327 1.570796327
> 0.121 6.7275E-10 1.570796327 1.570796327
> 0.131 1.48838E-09 1.570796327 1.570796328
> 0.141 3.10593E-09 1.570796327 1.57079633
> 0.151 6.16268E-09 1.570796327 1.570796333
> 0.161 1.1702E-08 1.570796327 1.570796338
> 0.171 2.13777E-08 1.570796306 1.570796327
> 0.181 3.77386E-08 1.57079629 1.570796328
> 0.191 6.4615E-08 1.570796262 1.570796326
> 0.201 1.07637E-07 1.570796219 1.570796327
> 0.211 1.74914E-07 1.570796152 1.570796327
> 0.221 2.77922E-07 1.570796049 1.570796327
> 0.231 4.32633E-07 1.570795894 1.570796327
> 0.241 6.60953E-07 1.570795666 1.570796327
> 0.251 9.92515E-07 1.570795334 1.570796327
> 0.261 1.46692E-06 1.57079486 1.570796327
> 0.271 2.13645E-06 1.57079419 1.570796327
> 0.281 3.06947E-06 1.570793257 1.570796327
> 0.291 4.35442E-06 1.570791972 1.570796327
> 0.301 6.10471E-06 1.570790222 1.570796327
> 0.311 8.46455E-06 1.570787862 1.570796327
> 0.321 1.16158E-05 1.570784711 1.570796327
> 0.331 1.57863E-05 1.570780541 1.570796327
> 0.341 2.1259E-05 1.570775068 1.570796327
> 0.351 2.83838E-05 1.570767943 1.570796327
> 0.361 3.759E-05 1.570758737 1.570796327
> 0.371 4.94014E-05 1.570746925 1.570796327
> 0.381 6.4454E-05 1.570731873 1.570796327
> 0.391 8.35156E-05 1.570712811 1.570796327
> 0.401 0.000107509 1.570688818 1.570796327
> 0.411 0.000137537 1.57065879 1.570796327
> 0.421 0.000174913 1.570621414 1.570796327
> 0.431 0.000221194 1.570575133 1.570796327
> 0.441 0.000278218 1.570518108 1.570796327
> 0.451 0.000348149 1.570448178 1.570796327
> 0.461 0.00043352 1.570362807 1.570796327
> 0.471 0.00053729 1.570259036 1.570796327
> 0.481 0.000662904 1.570133423 1.570796327
> 0.491 0.000814357 1.56998197 1.570796327
> 0.501 0.000996271 1.569800056 1.570796327
> 0.511 0.001213973 1.569582354 1.570796327
> 0.521 0.001473592 1.569322734 1.570796327
> 0.531 0.001782155 1.569014172 1.570796327
> 0.541 0.002147697 1.56864863 1.570796327
> 0.551 0.002579387 1.56821694 1.570796327
> 0.561 0.003087659 1.567708668 1.570796327
> 0.571 0.00368436 1.567111967 1.570796327
> 0.581 0.004382912 1.566413415 1.570796327
> 0.591 0.005198485 1.565597842 1.570796327
> 0.601 0.006148192 1.564648134 1.570796327
> 0.611 0.007251301 1.563545026 1.570796327
> 0.621 0.008529455 1.562266872 1.570796327
> 0.631 0.01000693 1.560789397 1.570796327
> 0.641 0.011710899 1.559085428 1.570796327
> 0.651 0.013671728 1.557124599 1.570796327
> 0.661 0.015923295 1.554873032 1.570796327
> 0.671 0.018503334 1.552292993 1.570796327
> 0.681 0.02145381 1.549342517 1.570796327
> 0.691 0.024821331 1.545974996 1.570796327
> 0.701 0.028657587 1.542138739 1.570796327
> 0.711 0.033019837 1.537776489 1.570796327
> 0.721 0.037971435 1.532824891 1.570796327
> 0.731 0.043582412 1.527213915 1.570796327
> 0.741 0.049930113 1.520866214 1.570796327
> 0.751 0.057099906 1.513696421 1.570796327
> 0.761 0.065185977 1.50561035 1.570796327
> 0.771 0.074292224 1.496504103 1.570796327
> 0.781 0.084533276 1.486263051 1.570796327
> 0.791 0.096035679 1.474760648 1.570796327
> 0.801 0.108939285 1.461857042 1.570796327
> 0.811 0.1233989 1.447397426 1.570796327
> 0.821 0.139586305 1.431210022 1.570796327
> 0.831 0.157692728 1.413103599 1.570796327
> 0.841 0.177931972 1.392864354 1.570796327
> 0.851 0.200544425 1.370251901 1.570796327
> 0.861 0.225802292 1.344994035 1.570796327
> 0.871 0.254016583 1.316779743 1.570796327
> 0.881 0.285546634 1.285249693 1.570796327
> 0.891 0.320813365 1.249982962 1.570796327
> 0.901 0.36031826 1.210478067 1.570796327
> 0.911 0.40467131 1.166125016 1.570796327
> 0.921 0.454633676 1.116162651 1.570796327
> 0.931 0.511185648 1.059610679 1.570796327
> 0.941 0.575640964 0.995155363 1.570796327
> 0.951 0.649852995 0.920943331 1.570796327
> 0.961 0.736623359 0.834172968 1.570796327
> 0.971 0.840627495 0.730168832 1.570796327
> 0.981 0.970996586 0.599799741 1.570796327
> 0.991 1.151950315 0.418846012 1.570796327
> >
> > --
> > Remove del for email


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