Groups | Search | Server Info | Keyboard shortcuts | Login | Register [http] [https] [nntp] [nntps]
Groups > comp.programming > #11044
| Newsgroups | comp.programming |
|---|---|
| Date | 2020-07-10 05:32 -0700 |
| References | (18 earlier) <86k0zslhbp.fsf@example.com> <rd7psp$q8c$1@dont-email.me> <86sgegjw9d.fsf@example.com> <4a95c227-9988-417f-9d66-66942147bfdco@googlegroups.com> <7d2718b1-ad38-4210-96ef-428894ae05a2o@googlegroups.com> |
| Message-ID | <2cd5c048-96cf-45a6-9308-d4e4577d623do@googlegroups.com> (permalink) |
| Subject | Re: are you interested in number theory AND programming |
| From | James Dow Allen <jdallen2000@yahoo.com> |
On Friday, July 10, 2020 at 9:51:48 AM UTC+7, syed owais wrote:
> I am exploring this diophantine equation in the set of complex (Gaussian) integers.
> What about a computer program that would list (all?) the solution sets ...
PLEASE! You may as well get the terminology right.
A SOLUTION is a triplet of Gaussian integers (a, b, c).
A SOLUTION SET would be a set like
{ (a1,b1,c1), (a2,b2,c2), ... }
which contains all such solution triplets.
YOU may prefer your terminology. But it will make you look the fool.
As to your question, I've worked almost zero with complex numbers
and don't intend to start now. But I'll offer three comments.
Only the first is relevant to programming, but I hope the others
-- related to the history of proving a^3 + b^3 = c^3 and
a^4 + b^4 = c^4 impossible in rational integers -- may be of interest also.
(1) Doing complex arithmetic in a programming language that
doesn't support them is a nuisance. The ability to add such
abstractions as needed is an argument for languages like C++.
HOWEVER, [here comes a plug for my favorite language!]
Gnu's C language has supported complex numbers for DECADES.
You can solve an instance of your equation in C with simply
_Complex double a, b, c;
...
c = csqrt(a*a + b*b);
(2) Carl Gauss (the 3rd greatest mathematician ever according
to a popular webpage :-) ) is on record as implying that Fermat's
Last Theorem was a frivolous time-waster. HOWEVER, by investigating
Gaussian integers rather than ordinary integers, he found a simpler
proof for the n=3 impossibility. This proof of the complex case
was simpler than that of the real case, IIUC! This discovery
advanced the use of complex numbers.
(3) The n=4 case of Fermat's Last Theorem is *much* easier to prove
than the n=3 case. In fact it was proved way back in the 13th century
by the great mathematician Leonardo of Pisa. It is a shame that
that great mathematician is linked to one minor discovery via his
nickname 'Fibonacci' -- he did much much more than that.
The fact that Leonardo 'Fibonacci' proved the impossibility
of a^4 + b^4 = c^4 is widely ignored or disputed.
(And I had my Edit reverted
when I noted this fact at Wikipedia.) Leonardo didn't quite
connect all the dots as would be needed for publication today.
(And he actually proved the slightly stronger result that
a^4 + b^4 = c^2 is impossible in positive integers.)
Best ever,
James Dow Allen
Back to comp.programming | Previous | Next — Previous in thread | Next in thread | Find similar | Unroll thread
are you interested in number theory AND programming syedanwarowais5@gmail.com - 2020-06-12 02:25 -0700
Re: are you interested in number theory AND programming root <NoEMail@home.org> - 2020-06-12 14:26 +0000
Re: are you interested in number theory AND programming syedanwarowais5@gmail.com - 2020-06-12 19:34 -0700
Re: are you interested in number theory AND programming Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-06-13 11:16 +0100
Re: are you interested in number theory AND programming syedanwarowais5@gmail.com - 2020-06-13 04:07 -0700
Re: are you interested in number theory AND programming Julio Di Egidio <julio@diegidio.name> - 2020-06-13 07:20 -0700
Re: are you interested in number theory AND programming Julio Di Egidio <julio@diegidio.name> - 2020-06-13 07:47 -0700
Re: are you interested in number theory AND programming Julio Di Egidio <julio@diegidio.name> - 2020-06-13 08:29 -0700
Re: are you interested in number theory AND programming Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-06-13 17:02 +0100
Re: are you interested in number theory AND programming syedanwarowais5@gmail.com - 2020-06-13 23:11 -0700
Re: are you interested in number theory AND programming syedanwarowais5@gmail.com - 2020-06-13 23:23 -0700
Re: are you interested in number theory AND programming James Dow Allen <jdallen2000@yahoo.com> - 2020-06-17 00:55 -0700
Re: are you interested in number theory AND programming Richard Heathfield <rjh@cpax.org.uk> - 2020-06-17 09:30 +0100
Re: are you interested in number theory AND programming James Dow Allen <jdallen2000@yahoo.com> - 2020-06-17 02:32 -0700
Re: are you interested in number theory AND programming Richard Heathfield <rjh@cpax.org.uk> - 2020-06-20 08:26 +0100
Re: are you interested in number theory AND programming James Dow Allen <jdallen2000@yahoo.com> - 2020-06-20 03:58 -0700
Re: are you interested in number theory AND programming Richard Heathfield <rjh@cpax.org.uk> - 2020-06-20 14:55 +0100
Re: are you interested in number theory AND programming Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-06-17 23:11 +0100
Re: are you interested in number theory AND programming syed owais <syedanwarowais5@gmail.com> - 2020-06-20 04:44 -0700
Re: are you interested in number theory AND programming Julio Di Egidio <julio@diegidio.name> - 2020-06-20 06:27 -0700
Re: are you interested in number theory AND programming syed owais <syedanwarowais5@gmail.com> - 2020-06-21 17:31 -0700
Re: are you interested in number theory AND programming James Dow Allen <jdallen2000@yahoo.com> - 2020-06-22 02:00 -0700
Re: are you interested in number theory AND programming syed owais <syedanwarowais5@gmail.com> - 2020-06-22 05:34 -0700
Re: are you interested in number theory AND programming root <NoEMail@home.org> - 2020-06-22 14:05 +0000
Re: are you interested in number theory AND programming syed owais <syedanwarowais5@gmail.com> - 2020-06-23 04:50 -0700
Re: are you interested in number theory AND programming David Brown <david.brown@hesbynett.no> - 2020-06-23 15:50 +0200
Re: are you interested in number theory AND programming Chad <cdalten@gmail.com> - 2020-06-23 14:34 -0700
Re: are you interested in number theory AND programming syed owais <syedanwarowais5@gmail.com> - 2020-06-24 01:39 -0700
Re: are you interested in number theory AND programming David Brown <david.brown@hesbynett.no> - 2020-06-24 15:48 +0200
Re: are you interested in number theory AND programming Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-06-24 13:52 +0100
Re: are you interested in number theory AND programming David Brown <david.brown@hesbynett.no> - 2020-06-24 16:00 +0200
Re: are you interested in number theory AND programming Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-06-24 17:03 +0100
Re: are you interested in number theory AND programming David Brown <david.brown@hesbynett.no> - 2020-06-24 20:32 +0200
Re: are you interested in number theory AND programming Chad <cdalten@gmail.com> - 2020-06-24 14:19 -0700
Re: are you interested in number theory AND programming Julio Di Egidio <julio@diegidio.name> - 2020-06-26 06:24 -0700
Re: are you interested in number theory AND programming David Brown <david.brown@hesbynett.no> - 2020-06-26 16:28 +0200
Re: are you interested in number theory AND programming Julio Di Egidio <julio@diegidio.name> - 2020-06-26 06:11 -0700
Re: are you interested in number theory AND programming James Dow Allen <jdallen2000@yahoo.com> - 2020-06-22 09:39 -0700
Re: are you interested in number theory AND programming syed owais <syedanwarowais5@gmail.com> - 2020-06-23 00:50 -0700
Re: are you interested in number theory AND programming James Dow Allen <jdallen2000@yahoo.com> - 2020-06-23 05:14 -0700
Re: are you interested in number theory AND programming syed owais <syedanwarowais5@gmail.com> - 2020-06-24 02:39 -0700
Re: are you interested in number theory AND programming Richard Heathfield <rjh@cpax.org.uk> - 2020-06-25 20:48 +0100
Re: are you interested in number theory AND programming David Brown <david.brown@hesbynett.no> - 2020-06-26 08:48 +0200
Re: are you interested in number theory AND programming syed owais <syedanwarowais5@gmail.com> - 2020-06-26 01:25 -0700
Re: are you interested in number theory AND programming David Brown <david.brown@hesbynett.no> - 2020-06-26 11:10 +0200
Re: are you interested in number theory AND programming James Dow Allen <jdallen2000@yahoo.com> - 2020-06-26 03:56 -0700
Re: are you interested in number theory AND programming David Brown <david.brown@hesbynett.no> - 2020-06-26 13:34 +0200
Re: are you interested in number theory AND programming Boris Dorestand <bdorestand@example.com> - 2020-06-27 11:22 -0300
Re: are you interested in number theory AND programming David Brown <david.brown@hesbynett.no> - 2020-06-27 17:50 +0200
Re: are you interested in number theory AND programming Boris Dorestand <bdorestand@example.com> - 2020-06-27 13:42 -0300
Re: are you interested in number theory AND programming syed owais <syedanwarowais5@gmail.com> - 2020-07-04 18:44 -0700
Re: are you interested in number theory AND programming syed owais <syedanwarowais5@gmail.com> - 2020-07-09 19:51 -0700
Re: are you interested in number theory AND programming James Dow Allen <jdallen2000@yahoo.com> - 2020-07-10 05:32 -0700
Re: are you interested in number theory AND programming syed owais <syedanwarowais5@gmail.com> - 2020-07-10 09:03 -0700
Re: are you interested in number theory AND programming Julio Di Egidio <julio@diegidio.name> - 2020-06-26 06:02 -0700
Re: are you interested in number theory AND programming James Dow Allen <jdallen2000@yahoo.com> - 2020-06-26 01:58 -0700
Re: are you interested in number theory AND programming David Brown <david.brown@hesbynett.no> - 2020-06-26 11:14 +0200
Re: are you interested in number theory AND programming Richard Damon <Richard@Damon-Family.org> - 2020-06-26 13:10 -0400
Re: are you interested in number theory AND programming Julio Di Egidio <julio@diegidio.name> - 2020-06-26 06:12 -0700
Re: are you interested in number theory AND programming David Brown <david.brown@hesbynett.no> - 2020-06-13 23:28 +0200
csiph-web