Going by my previous posts on FLT, the FLT for its non-numerical proof poses the relation
arcsin(sqrt(x^n) + arcsin(sqrt(1-x^n) < pi/2
where
0<x<1
x=a/c
where a c and n have their usual representations in FLT, that is
a^n + b^n cannot be c^n where a b c and n have integer values higher than 2.

Now, this is a challenge.

Cheers,
Arindam Banerjee

Shooting down Putin's plane makes the entire Earth a much safer place.

Not sure: labeling Putin as a war criminal diminishes the chances of shooting down his plane// Science Council Rules Earth.

Arindam is FLT, Flight 909 out of Moscow?
On Tuesday, April 5, 2022 at 6:10:25 PM UTC-5, banerjee...@gmail.com wrote:
> Now, this is a challenge.
>
> Cheers,
> Arindam Banerjee

Shooting down Putin's plane makes the entire Earth a much safer place.

Not sure: labeling Putin as a war criminal diminishes the chances of shooting down his plane// Science Council Rules Earth.

On Tuesday, April 5, 2022 at 5:05:23 PM UTC-5, Clutterfreak wrote:
> On 4/5/2022 1:05 PM, Clutterfreak wrote:
> > I like the way Chechens grab that Nazi's little head from behind and
> > shake his head like it is a cantaloupe :-))) Hehe :)
> >

Not sure: labeling Putin as a war criminal diminishes the chances of shooting down his plane// Science Council Rules Earth.

Heard on TV last night of a expert on War Crimes saying Putin is to be tried as a "War Criminal Aggressor" as the easiest legal crime court.

But the trouble with labeling Putin as war criminal is then, he will just hang around Russia and never get out of Russia. Never fly nor take trains or cars out of Russia. And the chances to shot down his plane and blame it on engine failure, is diminished.

We do not need to see Putin in some court of law docket. We need to see him gone completely, as the insane oaf he is.

Never forget this moment in human history, where one insane oaf has the power to push nuclear war buttons and cause the extinction of not only the human species but the entire life on Earth. Because humanity needs to colonize Europa for our Sun has gone Red Giant phase. We have no time for a petty dictator Putin or Xi wanting to be Alexander the Great. We have to colonize Europa in the next 1,000 years, not clean up after the insane oafs of Putin and Xi.

On Wednesday, 6 April 2022 at 09:30:07 UTC+10, Archimedes Plutonium wrote:
> Shooting down Putin's plane makes the entire Earth a much safer place.
Increasing Archie's IQ by 150 points, from its present negative value, would make Usenet a better place.

On Wednesday, 6 April 2022 at 09:10:25 UTC+10, banerjee...@gmail.com wrote:
> Going by my previous posts on FLT, the FLT for its non-numerical proof poses the relation
> arcsin(sqrt(x^n) + arcsin(sqrt(1-x^n) < pi/2
> where
> 0<x<1
> x=a/c
> where a c and n have their usual representations in FLT, that is
> a^n + b^n cannot be c^n where a b c and n have integer values higher than 2.
>
> Now, this is a challenge.
>
> Cheers,
> Arindam Banerjee

There could be a set of equations, giving the maximum value of the sum of theta and phi as a function of n.
Find the derivative of
arcsin(sqrt(x^n) + arcsin(sqrt(1-x^n)
set it to zero and solve x for a given n for the derived function, to get the maximum value of x.
Set that value of x in the above function, to get the sum of theta and phi which should be less than pi/2
Thus there is a graph for n vs max(theta+phi)
Which will solve FLT for all values of n, and show some pattern.

On Thursday, April 7, 2022 at 6:50:08 AM UTC-7, banerjee...@gmail.com wrote:
> There could be a set of equations, giving the maximum value of the sum of theta and phi as a function of n.
> Find the derivative of
> arcsin(sqrt(x^n) + arcsin(sqrt(1-x^n)
> set it to zero and solve x for a given n for the derived function, to get the maximum value of x.

If I haven't made a mistake, the derivative of that sum equals zero for any positive integer n.
Please check that carefully.

Sure, freedom of speech to the Arsewipedam Banerjee, who spray paints bullshit everyday of the year for decades to sci.physics and sci.math. This arsehole Banerjee never understood physics units with his arsewipe E = 1/2 mv^2 and wants to include a factor of k with his arsewipe "rail gun" con art.
Then there is the time the arsewipe thinks Sun and Earth cores are near absolute zero temperature--- I mean well-- has India checked for asylum escapees??????????????

On Friday, 8 April 2022 at 00:53:32 UTC+10, bwr fml wrote:
> On Thursday, April 7, 2022 at 6:50:08 AM UTC-7, banerjee...@gmail.com wrote:
> > There could be a set of equations, giving the maximum value of the sum of theta and phi as a function of n.
> > Find the derivative of
> > arcsin(sqrt(x^n) + arcsin(sqrt(1-x^n)
> > set it to zero and solve x for a given n for the derived function, to get the maximum value of x.
> If I haven't made a mistake, the derivative of that sum equals zero for any positive integer n.
> Please check that carefully.
You have made a mistake, for numerical results indicate otherwise, as one may easily see using an Excel sheet, as I did in my one page depiction. They show that as expected the value of the maxima comes very near pi divided by two.

On Thursday, April 7, 2022 at 1:56:13 PM UTC-7, banerjee...@gmail.com wrote:
> On Friday, 8 April 2022 at 00:53:32 UTC+10, bwr fml wrote:
> > On Thursday, April 7, 2022 at 6:50:08 AM UTC-7, banerjee...@gmail.com wrote:
> > > There could be a set of equations, giving the maximum value of the sum of theta and phi as a function of n.
> > > Find the derivative of
> > > arcsin(sqrt(x^n) + arcsin(sqrt(1-x^n)
> > > set it to zero and solve x for a given n for the derived function, to get the maximum value of x.
> > If I haven't made a mistake, the derivative of that sum equals zero for any positive integer n.
> > Please check that carefully.
> You have made a mistake, for numerical results indicate otherwise, as one may easily see using an Excel sheet, as I did in my one page depiction. They show that as expected the value of the maxima comes very near pi divided by two.

I apologize for any mistake.

Here is an attempt to use computation to display a plot check this

https://www.wolframalpha.com/input?i=plot+arcsin%28sqrt%28x%5E3%29%29+%2B+arcsin%28sqrt%281-x%5E3%29%29+for+x%3D0+to+2+pi

and again plot with a smaller range near pi/2 to see if anything is hiding

https://www.wolframalpha.com/input?i=plot+arcsin%28sqrt%28x%5E3%29%29+%2B+arcsin%28sqrt%281-x%5E3%29%29+for+x%3Dpi%2F2-1%2F10+to+pi%2F2%2B1%2F10

Can you please explain the mistake I've made so I can correct it?

Thank you

On Friday, 8 April 2022 at 08:47:26 UTC+10, bwr fml wrote:
> On Thursday, April 7, 2022 at 1:56:13 PM UTC-7, banerjee...@gmail.com wrote:
> > On Friday, 8 April 2022 at 00:53:32 UTC+10, bwr fml wrote:
> > > On Thursday, April 7, 2022 at 6:50:08 AM UTC-7, banerjee...@gmail.com wrote:
> > > > There could be a set of equations, giving the maximum value of the sum of theta and phi as a function of n.
> > > > Find the derivative of
> > > > arcsin(sqrt(x^n) + arcsin(sqrt(1-x^n)
> > > > set it to zero and solve x for a given n for the derived function, to get the maximum value of x.
> > > If I haven't made a mistake, the derivative of that sum equals zero for any positive integer n.
> > > Please check that carefully.
> > You have made a mistake, for numerical results indicate otherwise, as one may easily see using an Excel sheet, as I did in my one page depiction. They show that as expected the value of the maxima comes very near pi divided by two.
> I apologize for any mistake.
>
> Here is an attempt to use computation to display a plot check this
>
> https://www.wolframalpha.com/input?i=plot+arcsin%28sqrt%28x%5E3%29%29+%2B+arcsin%28sqrt%281-x%5E3%29%29+for+x%3D0+to+2+pi
>
> and again plot with a smaller range near pi/2 to see if anything is hiding
>
> https://www.wolframalpha.com/input?i=plot+arcsin%28sqrt%28x%5E3%29%29+%2B+arcsin%28sqrt%281-x%5E3%29%29+for+x%3Dpi%2F2-1%2F10+to+pi%2F2%2B1%2F10
>
> Can you please explain the mistake I've made so I can correct it?
>
> Thank you

I used to use Wolfram, but now I don't have the license.
So I used an Excel sheet. Simple and easy. See results, below.

The columns are x ranging from 0.001 increasing by steps of .01
Then two columns for the arcsin function that is there in Excel.
A fourth column for the sum
See my facebook timeline for the results, on the solution
https://www.facebook.com/arindam.banerjee.31149359/
There I have shown that
for x=a/c=,781, theta + phi = 1.569097521
x=0.791 it is 1.569155609 which is the highest in the spread, maximum value, and less than pi/2 which up to 4 places is 1.5708
x=0.801 it is 1.569036116
So you see there is a maximum around x=0.791

Which means that the derivative will having a zero value around there.
Point is how to get that analytically.
The derivative of an arcsin has to be known.
Then solve for x for any integer n>2
That should be some value around 0.791
If substituted in the sum of arcsins function it should be less than pi/2, as numerical calculations suggest.
Do this for n=4,5, 101, 10001, etc.
You'll prove FLT for any n.

Cheers,
Arindam Banerjee

x theta phi sum
0.001 3.16228E-05 1.316691467 1.31672309
0.011 0.00115369 1.316689831 1.31784352
0.021 0.003043194 1.316680073 1.319723267
0.031 0.00545814 1.316654812 1.322112952
0.041 0.008301963 1.316606666 1.324908628
0.051 0.011517676 1.316528254 1.328045931
0.061 0.015066459 1.316412202 1.33147866
0.071 0.018919665 1.316251136 1.3351708
0.081 0.023055047 1.316037691 1.339092738
0.091 0.027454697 1.315764514 1.34321921
0.101 0.032103817 1.315424259 1.347528075
0.111 0.03698993 1.315009599 1.351999529
0.121 0.042102353 1.314513225 1.356615578
0.131 0.047431821 1.313927849 1.361359671
0.141 0.052970221 1.313246213 1.366216434
0.151 0.058710387 1.312461087 1.371171474
0.161 0.064645954 1.311565279 1.376211233
0.171 0.070771232 1.310551637 1.381322869
0.181 0.077081118 1.309413057 1.386494175
0.191 0.083571019 1.308142487 1.391713505
0.201 0.090236789 1.306732932 1.396969721
0.211 0.097074686 1.305177465 1.40225215
0.221 0.104081324 1.303469225 1.407550548
0.231 0.111253644 1.301601432 1.412855075
0.241 0.118588883 1.299567389 1.418156272
0.251 0.126084552 1.297360491 1.423445043
0.261 0.133738412 1.29497423 1.428712643
0.271 0.141548463 1.292402205 1.433950668
0.281 0.149512923 1.289638124 1.439151047
0.291 0.15763022 1.286675817 1.444306037
0.301 0.16589898 1.28350924 1.44940822
0.311 0.17431802 1.28013248 1.4544505
0.321 0.182886335 1.276539767 1.459426102
0.331 0.191603101 1.272725473 1.464328575
0.341 0.200467663 1.268684127 1.469151789
0.351 0.209479531 1.264410411 1.473889943
0.361 0.218638384 1.259899176 1.47853756
0.371 0.227944059 1.255145438 1.483089497
0.381 0.237396556 1.250144386 1.487540942
0.391 0.246996034 1.244891387 1.491887421
0.401 0.256742815 1.239381987 1.496124801
0.411 0.266637379 1.233611913 1.500249292
0.421 0.276680371 1.227577078 1.504257449
0.431 0.286872603 1.221273574 1.508146177
0.441 0.297215053 1.214697681 1.511912734
0.451 0.307708874 1.207845855 1.515554728
0.461 0.318355393 1.200714733 1.519070125
0.471 0.329156122 1.193301123 1.522457245
0.481 0.34011276 1.185602002 1.525714762
0.491 0.351227204 1.177614503 1.528841707
0.501 0.362501551 1.169335911 1.531837462
0.511 0.373938114 1.160763646 1.53470176
0.521 0.385539427 1.151895255 1.537434682
0.531 0.397308258 1.142728389 1.540036647
0.541 0.40924762 1.133260793 1.542508412
0.551 0.421360786 1.123490276 1.544851062
0.561 0.433651306 1.113414693 1.547065999
0.571 0.446123017 1.103031917 1.549154935
0.581 0.458780069 1.092339807 1.551119876
0.591 0.471626939 1.081336174 1.552963113
0.601 0.484668457 1.070018743 1.5546872
0.611 0.497909827 1.058385113 1.55629494
0.621 0.511356656 1.046432704 1.55778936
0.631 0.525014981 1.034158712 1.559173693
0.641 0.538891303 1.021560042 1.560451345
0.651 0.55299262 1.008633249 1.561625869
0.661 0.567326464 0.995374466 1.56270093
0.671 0.581900946 0.981779317 1.563680263
0.681 0.596724799 0.967842832 1.564567631
0.691 0.61180743 0.953559346 1.565366776
0.701 0.627158971 0.938922382 1.566081353
0.711 0.642790342 0.923924526 1.566714868
0.721 0.658713316 0.90855728 1.567270596
0.731 0.674940587 0.892810896 1.567751484
0.741 0.691485852 0.87667419 1.568160042
0.751 0.708363893 0.860134316 1.568498209
0.761 0.725590669 0.84317652 1.568767189
0.771 0.743183421 0.825783838 1.568967259
0.781 0.761160779 0.807936742 1.569097521
0.791 0.779542882 0.789612728 1.569155609
0.801 0.798351511 0.770785804 1.569137315
0.811 0.817610232 0.751425884 1.569036116
0.821 0.837344549 0.731498026 1.568842575
0.831 0.857582075 0.710961494 1.568543569
0.841 0.878352717 0.689768568 1.568121285
0.851 0.899688875 0.667863019 1.567551895
0.861 0.921625662 0.645178119 1.566803781
0.871 0.944201138 0.621634002 1.56583514
0.881 0.967456569 0.597134091 1.56459066
0.891 0.991436708 0.571560133 1.562996841
0.901 1.016190096 0.544765148 1.560955244
0.911 1.041769392 0.516563053 1.558332446
0.921 1.068231732 0.486712867 1.554944599
0.931 1.095639111 0.454893512 1.550532623
0.941 1.124058808 0.420661378 1.544720186
0.951 1.153563835 0.383373496 1.536937331
0.961 1.184233431 0.342034643 1.526268074
0.971 1.216153593 0.294949151 1.511102744
0.981 1.24941765 0.238743812 1.488161462
0.991 1.284126884 0.164316321 1.448443204

On Friday, 8 April 2022 at 10:17:50 UTC+10, banerjee...@gmail.com wrote:
> On Friday, 8 April 2022 at 08:47:26 UTC+10, bwr fml wrote:
> > On Thursday, April 7, 2022 at 1:56:13 PM UTC-7, banerjee...@gmail.com wrote:
> > > On Friday, 8 April 2022 at 00:53:32 UTC+10, bwr fml wrote:
> > > > On Thursday, April 7, 2022 at 6:50:08 AM UTC-7, banerjee...@gmail.com wrote:
> > > > > There could be a set of equations, giving the maximum value of the sum of theta and phi as a function of n.
> > > > > Find the derivative of
> > > > > arcsin(sqrt(x^n) + arcsin(sqrt(1-x^n)
> > > > > set it to zero and solve x for a given n for the derived function, to get the maximum value of x.
> > > > If I haven't made a mistake, the derivative of that sum equals zero for any positive integer n.
> > > > Please check that carefully.
> > > You have made a mistake, for numerical results indicate otherwise, as one may easily see using an Excel sheet, as I did in my one page depiction. They show that as expected the value of the maxima comes very near pi divided by two.
> > I apologize for any mistake.
> >
> > Here is an attempt to use computation to display a plot check this
> >
> > https://www.wolframalpha.com/input?i=plot+arcsin%28sqrt%28x%5E3%29%29+%2B+arcsin%28sqrt%281-x%5E3%29%29+for+x%3D0+to+2+pi
> >
> > and again plot with a smaller range near pi/2 to see if anything is hiding
> >
> > https://www.wolframalpha.com/input?i=plot+arcsin%28sqrt%28x%5E3%29%29+%2B+arcsin%28sqrt%281-x%5E3%29%29+for+x%3Dpi%2F2-1%2F10+to+pi%2F2%2B1%2F10
> >
> > Can you please explain the mistake I've made so I can correct it?
> >
> > Thank you
> I used to use Wolfram, but now I don't have the license.
> So I used an Excel sheet. Simple and easy. See results, below.
>
> The columns are x ranging from 0.001 increasing by steps of .01
> Then two columns for the arcsin function that is there in Excel.
> A fourth column for the sum
> See my facebook timeline for the results, on the solution
> https://www.facebook.com/arindam.banerjee.31149359/
> There I have shown that
> for x=a/c=,781, theta + phi = 1.569097521
> x=0.791 it is 1.569155609 which is the highest in the spread, maximum value, and less than pi/2 which up to 4 places is 1.5708
> x=0.801 it is 1.569036116
> So you see there is a maximum around x=0.791
>
> Which means that the derivative will having a zero value around there.
> Point is how to get that analytically.
> The derivative of an arcsin has to be known.
> Then solve for x for any integer n>2
> That should be some value around 0.791
> If substituted in the sum of arcsins function it should be less than pi/2, as numerical calculations suggest.
> Do this for n=4,5, 101, 10001, etc.
> You'll prove FLT for any n.
>
> Cheers,
> Arindam Banerjee
>
>
> x theta phi sum
> 0.001 3.16228E-05 1.316691467 1.31672309
> 0.011 0.00115369 1.316689831 1.31784352
> 0.021 0.003043194 1.316680073 1.319723267
> 0.031 0.00545814 1.316654812 1.322112952
> 0.041 0.008301963 1.316606666 1.324908628
> 0.051 0.011517676 1.316528254 1.328045931
> 0.061 0.015066459 1.316412202 1.33147866
> 0.071 0.018919665 1.316251136 1.3351708
> 0.081 0.023055047 1.316037691 1.339092738
> 0.091 0.027454697 1.315764514 1.34321921
> 0.101 0.032103817 1.315424259 1.347528075
> 0.111 0.03698993 1.315009599 1.351999529
> 0.121 0.042102353 1.314513225 1.356615578
> 0.131 0.047431821 1.313927849 1.361359671
> 0.141 0.052970221 1.313246213 1.366216434
> 0.151 0.058710387 1.312461087 1.371171474
> 0.161 0.064645954 1.311565279 1.376211233
> 0.171 0.070771232 1.310551637 1.381322869
> 0.181 0.077081118 1.309413057 1.386494175
> 0.191 0.083571019 1.308142487 1.391713505
> 0.201 0.090236789 1.306732932 1.396969721
> 0.211 0.097074686 1.305177465 1.40225215
> 0.221 0.104081324 1.303469225 1.407550548
> 0.231 0.111253644 1.301601432 1.412855075
> 0.241 0.118588883 1.299567389 1.418156272
> 0.251 0.126084552 1.297360491 1.423445043
> 0.261 0.133738412 1.29497423 1.428712643
> 0.271 0.141548463 1.292402205 1.433950668
> 0.281 0.149512923 1.289638124 1.439151047
> 0.291 0.15763022 1.286675817 1.444306037
> 0.301 0.16589898 1.28350924 1.44940822
> 0.311 0.17431802 1.28013248 1.4544505
> 0.321 0.182886335 1.276539767 1.459426102
> 0.331 0.191603101 1.272725473 1.464328575
> 0.341 0.200467663 1.268684127 1.469151789
> 0.351 0.209479531 1.264410411 1.473889943
> 0.361 0.218638384 1.259899176 1.47853756
> 0.371 0.227944059 1.255145438 1.483089497
> 0.381 0.237396556 1.250144386 1.487540942
> 0.391 0.246996034 1.244891387 1.491887421
> 0.401 0.256742815 1.239381987 1.496124801
> 0.411 0.266637379 1.233611913 1.500249292
> 0.421 0.276680371 1.227577078 1.504257449
> 0.431 0.286872603 1.221273574 1.508146177
> 0.441 0.297215053 1.214697681 1.511912734
> 0.451 0.307708874 1.207845855 1.515554728
> 0.461 0.318355393 1.200714733 1.519070125
> 0.471 0.329156122 1.193301123 1.522457245
> 0.481 0.34011276 1.185602002 1.525714762
> 0.491 0.351227204 1.177614503 1.528841707
> 0.501 0.362501551 1.169335911 1.531837462
> 0.511 0.373938114 1.160763646 1.53470176
> 0.521 0.385539427 1.151895255 1.537434682
> 0.531 0.397308258 1.142728389 1.540036647
> 0.541 0.40924762 1.133260793 1.542508412
> 0.551 0.421360786 1.123490276 1.544851062
> 0.561 0.433651306 1.113414693 1.547065999
> 0.571 0.446123017 1.103031917 1.549154935
> 0.581 0.458780069 1.092339807 1.551119876
> 0.591 0.471626939 1.081336174 1.552963113
> 0.601 0.484668457 1.070018743 1.5546872
> 0.611 0.497909827 1.058385113 1.55629494
> 0.621 0.511356656 1.046432704 1.55778936
> 0.631 0.525014981 1.034158712 1.559173693
> 0.641 0.538891303 1.021560042 1.560451345
> 0.651 0.55299262 1.008633249 1.561625869
> 0.661 0.567326464 0.995374466 1.56270093
> 0.671 0.581900946 0.981779317 1.563680263
> 0.681 0.596724799 0.967842832 1.564567631
> 0.691 0.61180743 0.953559346 1.565366776
> 0.701 0.627158971 0.938922382 1.566081353
> 0.711 0.642790342 0.923924526 1.566714868
> 0.721 0.658713316 0.90855728 1.567270596
> 0.731 0.674940587 0.892810896 1.567751484
> 0.741 0.691485852 0.87667419 1.568160042
> 0.751 0.708363893 0.860134316 1.568498209
> 0.761 0.725590669 0.84317652 1.568767189
> 0.771 0.743183421 0.825783838 1.568967259
> 0.781 0.761160779 0.807936742 1.569097521
> 0.791 0.779542882 0.789612728 1.569155609
> 0.801 0.798351511 0.770785804 1.569137315
> 0.811 0.817610232 0.751425884 1.569036116
> 0.821 0.837344549 0.731498026 1.568842575
> 0.831 0.857582075 0.710961494 1.568543569
> 0.841 0.878352717 0.689768568 1.568121285
> 0.851 0.899688875 0.667863019 1.567551895
> 0.861 0.921625662 0.645178119 1.566803781
> 0.871 0.944201138 0.621634002 1.56583514
> 0.881 0.967456569 0.597134091 1.56459066
> 0.891 0.991436708 0.571560133 1.562996841
> 0.901 1.016190096 0.544765148 1.560955244
> 0.911 1.041769392 0.516563053 1.558332446
> 0.921 1.068231732 0.486712867 1.554944599
> 0.931 1.095639111 0.454893512 1.550532623
> 0.941 1.124058808 0.420661378 1.544720186
> 0.951 1.153563835 0.383373496 1.536937331
> 0.961 1.184233431 0.342034643 1.526268074
> 0.971 1.216153593 0.294949151 1.511102744
> 0.981 1.24941765 0.238743812 1.488161462
> 0.991 1.284126884 0.164316321 1.448443204

So you see, the point is that for n>3 the sum of the included angles never reaches, let alone exceed pi/2.
If it did exceed, then at one point there would be a matching set of integers a b c which would disprove FLT!
But that is not there. Near, though.
My bet is that for n=4 the max will be even less.
And keep on decreasing with greater values of n.
That graph should be interesting.
Can be easily found out numerically.
Someone might give it a go and post here.

Cheers,
Arindam Banerjee

On Thursday, April 7, 2022 at 5:17:50 PM UTC-7, banerjee...@gmail.com wrote:
> I used to use Wolfram, but now I don't have the license.

The two links I provided were for the free WolframAlpha web page, no license required.
Each of those just plotted arcsin(sqrt(x^n)) + arcsin(sqrt(1-x^n)) with n=3
the first for 0<x<2pi and the second for pi/2-1/10<x<pi/2+1/10. Nothing more.
Those showed that the sum appears to be exactly pi/2.
Doing algebraic/trig simplification gives the same result.

I realize you are deep into what you are doing and it may all be transparent to you.
But I don't know where your a and c and theta and phi are coming from or what they mean.

I suspect my not knowing what to do with all your variables is the reason for my mistake.

I think you asked about the derivative

d/dx(sin^(-1)(sqrt(x^n))) = (n x^(n - 1))/(2 sqrt(-x^n (x^n - 1)))
and
d/dx(sin^(-1)(sqrt(1 - x^n))) = -(n x^(n - 1))/(2 sqrt(-x^n (x^n - 1)))

and if you look closely you will see the second = -1*the first
so the sum of those two is exactly zero.

BUT all that is just assuming there is some ordinary variable x and some
variable n and there aren't deeper things going on with a and b
and c and theta and phi and ...

If you can help me get up to speed then I can try to understand what you
are doing and possibly even answer a few of your simpler questions.

No obligation and let me know when you have had enough of this.
Thank you

On Friday, 8 April 2022 at 14:05:27 UTC+10, bwr fml wrote:
> On Thursday, April 7, 2022 at 5:17:50 PM UTC-7, banerjee...@gmail.com wrote:
> > I used to use Wolfram, but now I don't have the license.
> The two links I provided were for the free WolframAlpha web page, no license required.
> Each of those just plotted arcsin(sqrt(x^n)) + arcsin(sqrt(1-x^n)) with n=3
> the first for 0<x<2pi and the second for pi/2-1/10<x<pi/2+1/10. Nothing more.

If you go to my original proof on my facebook timeline
https://www.facebook.com/photo?fbid=5686846731343077&set=pcb.5686846781343072
you might get a better idea of the brief proof, and what I have been talking about.

> Those showed that the sum appears to be exactly pi/2.

Nothing can be exactly pi/2 !
I have no clue about what you have done. So I cannot help you.
What I suggested one may do to get an analytic solution you have deleted here.

> Doing algebraic/trig simplification gives the same result.

You can do what you like, so long as you ignore what I have done it makes no sense, unless it is a kind of putdown. Which is understandable.
If we are being honest you should understand what *I* have claimed, what I have done, and repeat what I have done for verifying my claim.
What *I* have done wrong is up for discussion, not what wrong you have done.
Anything else is political. Which is also understandable.
> I realize you are deep into what you are doing and it may all be transparent to you.

It should be so to any intelligent high school student. I worked on it for a couple of hours or so. I have much else to do. On the other hand, maths in engineering has been my living since 1978.

> But I don't know where your a and c and theta and phi are coming from or what they mean.

see above link to my facebook photo. Check out my proof of the FLT and then you may understand how I have transformed it into a function
>
> I suspect my not knowing what to do with all your variables is the reason for my mistake.
>
> I think you asked about the derivative
>
> d/dx(sin^(-1)(sqrt(x^n))) = (n x^(n - 1))/(2 sqrt(-x^n (x^n - 1)))
________?
> and
> d/dx(sin^(-1)(sqrt(1 - x^n))) = -(n x^(n - 1))/(2 sqrt(-x^n (x^n - 1)))
>
> and if you look closely you will see the second = -1*the first
> so the sum of those two is exactly zero.

I don't see how the first one that is derived after differentiation can be correct.
(x^n-1) term, how that can come? Quite impossible!
The two terms after differentiation cannot be the same! Mistake somewhere.
As evidenced by numerical calculation, see above.
You will get another f(x) which does not sum to zero.
But you will set it to zero and then solve for x.
Take that value of x and put it in the original function.
That should be near to pi/2
>
> BUT all that is just assuming there is some ordinary variable x and some
> variable n and there aren't deeper things going on with a and b
> and c and theta and phi and ...

x<0<1 as a b c are the sides of a right angle triangle where c is the hypotenuse and x=a/c so 0<x<1
theta and phi are the interior angles which has to sum up to pi/2 at a certain value of x
If it does not do so, for any out of all x, or if their sum do not exceed pi/2 there is no right angle triangle; and FLT is proved to be correct for that n, and for any n using the same method.
To understand the logic better, consult the facebook photo with my handwritten notes, 17th century style.
>
> If you can help me get up to speed then I can try to understand what you
> are doing and possibly even answer a few of your simpler questions.

That is nice.
I would be obliged if you or anyone go through my proof and try to find flaws there, in the proper scientific spirit.
Try making an Excel worksheet, to verify any analytically obtained results.
I have done it for n=3, repeat for n=4.
Use Taylor expansion of arcsine upto at least 5 terms; more than that should not affect the result.
I hope you can use Excel.
>
> No obligation and let me know when you have had enough of this.
> Thank you

Welcome. Your politeness is a refreshing change from the usual toxic stink thrown up by Archie Poo & Co.

Cheers,
Arindam Banerjee

On Friday, 8 April 2022 at 08:47:26 UTC+10, bwr fml wrote:
> On Thursday, April 7, 2022 at 1:56:13 PM UTC-7, banerjee...@gmail.com wrote:
> > On Friday, 8 April 2022 at 00:53:32 UTC+10, bwr fml wrote:
> > > On Thursday, April 7, 2022 at 6:50:08 AM UTC-7, banerjee...@gmail.com wrote:
> > > > There could be a set of equations, giving the maximum value of the sum of theta and phi as a function of n.
> > > > Find the derivative of
> > > > arcsin(sqrt(x^n) + arcsin(sqrt(1-x^n)
> > > > set it to zero and solve x for a given n for the derived function, to get the maximum value of x.
> > > If I haven't made a mistake, the derivative of that sum equals zero for any positive integer n.
> > > Please check that carefully.
> > You have made a mistake, for numerical results indicate otherwise, as one may easily see using an Excel sheet, as I did in my one page depiction. They show that as expected the value of the maxima comes very near pi divided by two.
> I apologize for any mistake.
>
> Here is an attempt to use computation to display a plot check this
>
> https://www.wolframalpha.com/input?i=plot+arcsin%28sqrt%28x%5E3%29%29+%2B+arcsin%28sqrt%281-x%5E3%29%29+for+x%3D0+to+2+pi

Tried that for 0<x<1, nothing happened.
>
> and again plot with a smaller range near pi/2 to see if anything is hiding
>
> https://www.wolframalpha.com/input?i=plot+arcsin%28sqrt%28x%5E3%29%29+%2B+arcsin%28sqrt%281-x%5E3%29%29+for+x%3Dpi%2F2-1%2F10+to+pi%2F2%2B1%2F10
>
> Can you please explain the mistake I've made so I can correct it?
>
> Thank you

On Thursday, April 7, 2022 at 5:17:50 PM UTC-7, banerjee...@gmail.com wrote:
> On Friday, 8 April 2022 at 08:47:26 UTC+10, bwr fml wrote:
> > I apologize for any mistake.
....
> > No obligation and let me know when you have had enough of this.
> > Thank you
> Welcome. Your politeness is a refreshing change from the usual toxic stink thrown up by Archie Poo & Co.

I'm deeply sorry, but the gap between us appears to be too large for me
and this looks like it could easily just lead to confusion and frustration
for both of us. So I apologize again and hope that your idea works out for you.

On Saturday, 9 April 2022 at 00:42:59 UTC+10, bwr fml wrote:
> On Thursday, April 7, 2022 at 5:17:50 PM UTC-7, banerjee...@gmail.com wrote:
> > On Friday, 8 April 2022 at 08:47:26 UTC+10, bwr fml wrote:
> > > I apologize for any mistake.
> ...
> > > No obligation and let me know when you have had enough of this.
> > > Thank you
> > Welcome. Your politeness is a refreshing change from the usual toxic stink thrown up by Archie Poo & Co.
> I'm deeply sorry, but the gap between us appears to be too large for me
> and this looks like it could easily just lead to confusion and frustration
> for both of us. So I apologize again and hope that your idea works out for you.

No worries, my proof of FLT has been approved by my Indian colleagues, who are highly trained and discerning scientists and engineers.
I have no doubt that in due course it will be universally accepted.
After all, I have made it all so simple, that clever high school students can easily understand.

https://www.facebook.com/photo?fbid=5686846731343077&set=pcb.5686846781343072

I would very much like this bit of paper to be sold!
Would help my work in Internal Force Engines.
Cheers,
Arindam Banerjee

On 4/8/2022 1:50 AM, banerjee...@gmail.com wrote:
> On Friday, 8 April 2022 at 14:05:27 UTC+10, bwr fml wrote:
>> On Thursday, April 7, 2022 at 5:17:50 PM UTC-7, banerjee...@gmail.com wrote:
>>> I used to use Wolfram, but now I don't have the license.
>> The two links I provided were for the free WolframAlpha web page, no license required.
>> Each of those just plotted arcsin(sqrt(x^n)) + arcsin(sqrt(1-x^n)) with n=3
>> the first for 0<x<2pi and the second for pi/2-1/10<x<pi/2+1/10. Nothing more.
>
> If you go to my original proof on my facebook timeline
> https://www.facebook.com/photo?fbid=5686846731343077&set=pcb.5686846781343072
> you might get a better idea of the brief proof, and what I have been talking about.
>
>> Those showed that the sum appears to be exactly pi/2.
>
> Nothing can be exactly pi/2 !
[...]

90 degrees?

On Saturday, 9 April 2022 at 10:34:01 UTC+10, Chris M. Thomasson wrote:
> On 4/8/2022 1:50 AM, banerjee...@gmail.com wrote:
> > On Friday, 8 April 2022 at 14:05:27 UTC+10, bwr fml wrote:
> >> On Thursday, April 7, 2022 at 5:17:50 PM UTC-7, banerjee...@gmail.com wrote:
> >>> I used to use Wolfram, but now I don't have the license.
> >> The two links I provided were for the free WolframAlpha web page, no license required.
> >> Each of those just plotted arcsin(sqrt(x^n)) + arcsin(sqrt(1-x^n)) with n=3
> >> the first for 0<x<2pi and the second for pi/2-1/10<x<pi/2+1/10. Nothing more.
> >
> > If you go to my original proof on my facebook timeline
> > https://www.facebook.com/photo?fbid=5686846731343077&set=pcb.5686846781343072
> > you might get a better idea of the brief proof, and what I have been talking about.
> >
> >> Those showed that the sum appears to be exactly pi/2.
> >
> > Nothing can be exactly pi/2 !
> [...]
>
> 90 degrees?
Point.

On Thursday, 7 April 2022 at 23:50:08 UTC+10, banerjee...@gmail.com wrote:
> On Wednesday, 6 April 2022 at 09:10:25 UTC+10, banerjee...@gmail.com wrote:
> > Going by my previous posts on FLT, the FLT for its non-numerical proof poses the relation
> > arcsin(sqrt(x^n) + arcsin(sqrt(1-x^n) < pi/2
> > where
> > 0<x<1
> > x=a/c
> > where a c and n have their usual representations in FLT, that is
> > a^n + b^n cannot be c^n where a b c and n have integer values higher than 2.
> >
> > Now, this is a challenge.
> >
> > Cheers,
> > Arindam Banerjee
> There could be a set of equations, giving the maximum value of the sum of theta and phi as a function of n.
> Find the derivative of
> arcsin(sqrt(x^n) + arcsin(sqrt(1-x^n)

Now the derivative of arcsin(x) is 1/sqrt(1-x^2)
Take the first time, call it y1, n is constant
Let z=sqrt(x^n)
dz/dx=(n/2)(x^(n/2 - 1)
and dy1/dz=1/sqrt(1-z^2)
So dy1/dx=dy/dz * dz/dx = (1/sqrt(1-x^n))*(n/2)(x^(n/2-1))

Similarly we can find dy2/dx for the second term.
Which turns out to be quite large!
Set them to zero for a given n>2 to find x.
Very difficult to get an analytic expression as a solution for x!

The numerical proof using an Excel sheet was simple and straightforward.

> set it to zero and solve x for a given n for the derived function, to get the maximum value of x.
> Set that value of x in the above function, to get the sum of theta and phi which should be less than pi/2
> Thus there is a graph for n vs max(theta+phi)
> Which will solve FLT for all values of n, and show some pattern.

On Sunday, 10 April 2022 at 20:02:13 UTC+10, banerjee...@gmail.com wrote:
> On Thursday, 7 April 2022 at 23:50:08 UTC+10, banerjee...@gmail.com wrote:
> > On Wednesday, 6 April 2022 at 09:10:25 UTC+10, banerjee...@gmail.com wrote:
> > > Going by my previous posts on FLT, the FLT for its non-numerical proof poses the relation
> > > arcsin(sqrt(x^n) + arcsin(sqrt(1-x^n) < pi/2
> > > where
> > > 0<x<1
> > > x=a/c
> > > where a c and n have their usual representations in FLT, that is
> > > a^n + b^n cannot be c^n where a b c and n have integer values higher than 2.
> > >
> > > Now, this is a challenge.
> > >
> > > Cheers,
> > > Arindam Banerjee
> > There could be a set of equations, giving the maximum value of the sum of theta and phi as a function of n.
> > Find the derivative of
> > arcsin(sqrt(x^n) + arcsin(sqrt(1-x^n)
> Now the derivative of arcsin(x) is 1/sqrt(1-x^2)
> Take the first time, call it y1, n is constant
> Let z=sqrt(x^n)
> dz/dx=(n/2)(x^(n/2 - 1)
> and dy1/dz=1/sqrt(1-z^2)
> So dy1/dx=dy/dz * dz/dx = (1/sqrt(1-x^n))*(n/2)(x^(n/2-1))
>
> Similarly we can find dy2/dx for the second term.
> Which turns out to be quite large!
> Set them to zero for a given n>2 to find x.
> Very difficult to get an analytic expression as a solution for x!

However when we put n=2 we get a polynomial in terms of x, which means there are multiple values of x to satisfy, that will create the right angle.
As it should!
Anything higher than 2, is just not possible.
Thus FLT can also be proved analytically but some more steps are involved, that I have skipped.
>
> The numerical proof using an Excel sheet was simple and straightforward.
> > set it to zero and solve x for a given n for the derived function, to get the maximum value of x.
> > Set that value of x in the above function, to get the sum of theta and phi which should be less than pi/2
> > Thus there is a graph for n vs max(theta+phi)
> > Which will solve FLT for all values of n, and show some pattern.

On Wednesday, 6 April 2022 at 09:10:25 UTC+10, banerjee...@gmail.com wrote:
> Going by my previous posts on FLT, the FLT for its non-numerical proof poses the relation
> arcsin(sqrt(x^n) + arcsin(sqrt(1-x^n) < pi/2
> where
> 0<x<1
> x=a/c
> where a c and n have their usual representations in FLT, that is
> a^n + b^n cannot be c^n where a b c and n have integer values higher than 2.
>
> Now, this is a challenge.
>
> Cheers,
> Arindam Banerjee

https://www.facebook.com/photo.php?fbid=5686846731343077&set=a.3856470274380741&type=3&comment_id=5715352665159150&reply_comment_id=5717829131578170&force_theater=true&notif_id=1650063854713120&notif_t=photo_comment&ref=notif

shows some comments, clarifying the FLT trig proof, indicates future work...

On Monday, 18 April 2022 at 11:58:10 UTC+10, banerjee...@gmail.com wrote:
> On Wednesday, 6 April 2022 at 09:10:25 UTC+10, banerjee...@gmail.com wrote:
> > Going by my previous posts on FLT, the FLT for its non-numerical proof poses the relation
> > arcsin(sqrt(x^n) + arcsin(sqrt(1-x^n) < pi/2
> > where
> > 0<x<1
> > x=a/c
> > where a c and n have their usual representations in FLT, that is
> > a^n + b^n cannot be c^n where a b c and n have integer values higher than 2.
> >
> > Now, this is a challenge.
> >
> > Cheers,
> > Arindam Banerjee
> https://www.facebook.com/photo.php?fbid=5686846731343077&set=a.3856470274380741&type=3&comment_id=5715352665159150&reply_comment_id=5717829131578170&force_theater=true&notif_id=1650063854713120&notif_t=photo_comment&ref=notif
>
> shows some comments, clarifying the FLT trig proof, indicates future work....

Amazing, it shows the relation arcsin(sqrt(x^n)) + arcsin(sqrt(1 - x^n)) = pi/2 going by the Excel asin function, numerically. In my earlier work I had used only the first 5 terms of the Taylor expansion to get wrong results, should have used the Excel asin. This is thanks to Don Travis, who commented on the above link. 0<x<1 and n is any integer. Don had used the asin function and found that it always added to pi/2 contrary to what I had mentioned, thinking thus to have proved FLT. As things are, this looks like there are many solutions to a^n+b^n=c^n where a b c n could be extremely large integers! So large that they have never been found!

Is this something new, this trigonometric identity above, or well known?