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are you interested in number theory AND programming

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  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

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#10908 — are you interested in number theory AND programming

I am an electronics engineer with a keen interest in mathematics. I would like to be a member of a group where programming for solving mathematical problems (e g number theory)  is pursued. 

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#10909

syedanwarowais5@gmail.com <syedanwarowais5@gmail.com> wrote:
> I am an electronics engineer with a keen interest in mathematics. I would like to be a member
> of a group where programming for solving mathematical problems (e g number theory)  is pursued. 

I don't know of any such usenet groups however there are several texts which use Maple
or Mathematica for such research. I would specifically recommend two books by
Borwein and Bailey:
MathematicsbyExperiment-PlausibleReasoninginthe21stCentury-J.Borwein,D.Bailey2004
ExperimentationinMathematics-ComputationalPathstoDiscovery-J.Borwein,etal.,2004

Should you be interested in pursuing efforts with Mathematica I would
also recommend that you assemble a Raspberry Pi system. The Pi part
of such a system costs less than $50 and comes with Mathematica included..

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#10918

Actually,my situation is this:i have several(original work) projects in mathematics on the anvil and they require computer programming to take them forward. 
My programming skills are not yet well developed! 
If someone is interested in such projects as a programmer in such a collaboration, it would be beneficial to both of us.
The lines of code for the programs will be about those as in a college project. 

I will be glad to share an example right away!

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#10919

syedanwarowais5@gmail.com writes:

> Actually,my situation is this:i have several(original work) projects
> in mathematics on the anvil and they require computer programming to
> take them forward.
> My programming skills are not yet well developed! 
> If someone is interested in such projects as a programmer in such a
> collaboration, it would be beneficial to both of us.  The lines of
> code for the programs will be about those as in a college project.
>
> I will be glad to share an example right away!

Number theory often gives rise to interesting programming puzzles so
post away...

-- 
Ben.

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#10920

As part of a (hopefully growing) exploration in number theory, I need to find solutions to the following diophantine equation :
ab(a+b)=cd(c+d) (in positive integers)
The first solution set is a=1,b=5 and c=2,d=3.
I know there are several more solution sets, though I don't know if there are an infinite number of solutions sets to this diophantine equation. 
Would you like to write a program to solve this puzzle?  
Once you respond, I will go deeper into the mathematics this connects to. 

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#10921

On Saturday, 13 June 2020 13:07:38 UTC+2, syedanw...@gmail.com  wrote:
> As part of a (hopefully growing) exploration in number theory, I need to find
> solutions to the following diophantine equation :
> ab(a+b)=cd(c+d) (in positive integers)
> The first solution set is a=1,b=5 and c=2,d=3.

So you show that there is at least one solution.

> I know there are several more solution sets, though I don't know if there are
> an infinite number of solutions sets to this diophantine equation. 

Since for every solution {a,b,c,d} also e.g. {2a,2b,2c,2d} is a solution (as
can be easily verified)...

> Would you like to write a program to solve this puzzle?

Writing a program that systematically checks every possible quadruple of
positive integers is almost trivial (below I show one in Prolog), but, while
inspecting generated solutions may often give clues as to the overall pattern,
it won't per se prove anything about the solution set itself, except that it
is not empty.  We need some higher order "logic" to get to proving e.g. that
{(a,b,c,d) | a,b,c,d in |Z+, ab(a+b)=cd(c+d)} is *not finite*.

> Once you respond, I will go deeper into the mathematics this connects to.

OK.

Julio

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Bounded for termination (it's just an example), gen not optimised:

?- MaxInt = 30,
   forall(
   (   combination(L, 4, MaxInt),  % L is a comb. of 4 items in [1..Maxint]
       permutation(L, M),          % M is a perm. of L
       M = [A,B,C,D],
       A =< B, C =< D, A =< C,     % avoiding equivalent solutions
       X is A*B*(A+B),
       Y is C*D*(C+D),
       X =:= Y
   ), format('~w  (~d=~d)~n', [M,X,Y])).
[1,5,2,3]  (30=30)
[1,15,2,10]  (240=240)
[1,14,3,7]  (210=210)
[1,21,3,11]  (462=462)
[1,15,4,6]  (240=240)
[1,20,5,7]  (420=420)
[2,13,3,10]  (390=390)
[2,10,4,6]  (240=240)
[2,30,4,20]  (1920=1920)
[2,20,5,11]  (880=880)
[2,28,5,16]  (1680=1680)
[2,28,6,14]  (1680=1680)
[2,30,8,12]  (1920=1920)
[3,25,4,21]  (2100=2100)
[3,17,5,12]  (1020=1020)
[3,30,5,22]  (2970=2970)
[3,15,6,9]  (810=810)
[4,26,6,20]  (3120=3120)
[4,14,7,9]  (1008=1008)
[4,20,8,12]  (1920=1920)
[5,16,6,14]  (1680=1680)
[5,27,6,24]  (4320=4320)
[5,25,10,15]  (3750=3750)
[6,30,12,18]  (6480=6480)
[6,29,14,15]  (6090=6090)
[7,15,10,11]  (2310=2310)
[8,28,14,18]  (8064=8064)
[10,24,15,17]  (8160=8160)
[10,28,16,19]  (10640=10640)
[12,22,16,17]  (8976=8976)
[13,27,15,24]  (14040=14040)
[14,30,20,22]  (18480=18480)
MaxInt = 30.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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#10922

On Saturday, 13 June 2020 16:20:32 UTC+2, Julio Di Egidio  wrote:
<snip>
> Since for every solution {a,b,c,d} also e.g. {2a,2b,2c,2d} is a solution (as
> can be easily verified)...

Where I should have written (a,b,c,d) as it is a 4-tuple (order matters), not
just a set.

Julio

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#10923

On Saturday, 13 June 2020 16:20:32 UTC+2, Julio Di Egidio  wrote:

>    (   combination(L, 4, MaxInt),  % L is a comb. of 4 items in [1..Maxint]

Plus there is bug there since those are combinations without repetition, e.g.
(1,1,1,1) is also a solution but I am missing it: sorry...

Julio

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#10924

syedanwarowais5@gmail.com writes:

> As part of a (hopefully growing) exploration in number theory, I need
> to find solutions to the following diophantine equation :
> ab(a+b)=cd(c+d) (in positive integers)
> The first solution set is a=1,b=5 and c=2,d=3.
> I know there are several more solution sets, though I don't know if
> there are an infinite number of solutions sets to this diophantine
> equation.
> Would you like to write a program to solve this puzzle?

It's quite interesting.  Here's a first stab using Haskell (I always use
Haskell for this sort of investigation):

  import System.Environment
   
  psum (a, b) = a * b * (a + b)
   
  get_b_from product a =
    (floor (sqrt(fromIntegral (4*a*product + a^4))) - a^2) `div` (2*a)
   
  possible_pairs_from product =
    takeWhile (\(a, b) -> a <= b) $ map (\a -> (a, get_b_from product a)) [1..]
   
  actual_pairs_from product =
    filter ((==product).psum) $ possible_pairs_from product
   
  solution_list = filter ((>1).length) $ map actual_pairs_from [2,4..]
   
  main = do
    [n] <- getArgs
    mapM (putStrLn.showSolution) $ take (read n) $ solution_list
    where showSolution s = foldl1 (\s1 s2 -> s1 ++ " " ++ s2) $ map show s

The first 10 solution sets this program finds are:

(1,5) (2,3)
(1,14) (3,7)
(1,15) (2,10) (4,6)
(2,13) (3,10)
(1,20) (5,7)
(1,21) (3,11)
(3,15) (6,9)
(2,20) (5,11)
(4,14) (7,9)
(3,17) (5,12)

and there are a few 6-pair solutions that come up:

(1,384) (6,154) (8,132) (14,96) (28,60) (40,44)
(9,231) (12,198) (21,144) (22,140) (42,90) (60,66)
(6,385) (15,238) (23,187) (46,119) (51,110) (69,85)

> Once you respond, I will go deeper into the mathematics this connects
> to.

Probably best not to hold back like this.  Just make it fun with as much
information as you can.  That will encourage more responses.

-- 
Ben.

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#10927

let a*3 +b*3 +(c+d)*3=(a+b)*3+c*3+d*3
Implies cd(c+d)=ab(a+b)
Well, solutions to the second equation give solutions to the first equation. 

I have deeper explorations on the anvil here,so tell me what you think of the little thing above! 

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#10928

This solves the following problem, for example :what is the smallest integer that can be written as a sum of three POSITIVE cubes in two different ways.

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#10932

On Sunday, June 14, 2020 at 1:23:56 PM UTC+7, syedanw...@gmail.com wrote:
> This solves the following problem, for example :what is the smallest integer that can be written as a sum of three POSITIVE cubes in two different ways.

I see that your puzzle does solve this problem, involving "smallest."
But why?  It misses the *second* smallest such integer:
   1009 = 9^3 + 6^3 + 4^3 = 10^3 + 2^3 + 1^3
cannot be expressed with your a,b,c,d formula.

By the way, too bad you wrote "positive" instead of "non-negative." :-)
We miss out on the famous
    0 + 0 + 6^3 = 5^3 + 4^3 + 3^3


On Saturday, June 13, 2020 at 11:02:49 PM UTC+7, Ben Bacarisse wrote:
> The first 10 solution sets this program finds are:
> 
> (1,5) (2,3)
> (1,14) (3,7)
> (1,15) (2,10) (4,6)
...
The first triplet Ben found is closely related to the smallest number
which has *three* expressions as three positive cubes:
   5104 = 15^3 + 10^3 + 9^3 = 15^3 + 12^3 + 1^3 = 16^3 + 10^3 + 2^3
But the "Why?" of this isn't obvious to me.

Nitpick on terminology:  The quadruplets we are talking about are
SOLUTIONS, not "solution sets."  The Solution Set would be the
set of all possible solution quadruplets.

Jamesdowallen at Gmail

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#10933

On 17/06/2020 08:55, James Dow Allen wrote:
> 
> By the way, too bad you wrote "positive" instead of "non-negative." :-)
> We miss out on the famous
>      0 + 0 + 6^3 = 5^3 + 4^3 + 3^3

Nitpick on terminology:

That should be 0^3 + 0^3 + 6^3 = 5^3 + 4^3 + 3^3

;-)

(Those with a decent keyboard will of course use 0³ + 0³ + 6³ etc, in 
flagrant violation of Usenet's ASCII-only protocol.)

-- 
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

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#10935

On Wednesday, June 17, 2020 at 3:30:12 PM UTC+7, Richard Heathfield wrote:
> On 17/06/2020 08:55, James Dow Allen wrote:
> > 
> > By the way, too bad you wrote "positive" instead of "non-negative." :-)
> > We miss out on the famous
> >      0 + 0 + 6^3 = 5^3 + 4^3 + 3^3
> 
> Nitpick on terminology:
> 
> That should be 0^3 + 0^3 + 6^3 = 5^3 + 4^3 + 3^3
> 
> ;-)
> 
> (Those with a decent keyboard will of course use 0³ + 0³ + 6³ etc, in 
> flagrant violation of Usenet's ASCII-only protocol.)

Wow!  First time I visit comp,programming in YEARS and here
is ... the inimitable Richard Heathfield. And replying within minutes.
Bravo!  How are you doing?, Richard!

I'll try to incorporate more fancy characters in future posts, if any. :-)

I am so sad. :-(  My long-unused account at Save the Dodoes Foundation
has been turned off, perhaps, and I had to post with Google Groups.
How embarrassing.  (Though not even in the Top Ten list of recent sadnesses.  :-( )

Good to see you, Richard!

James D. Allen

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#10951

On 17/06/2020 10:32, James Dow Allen wrote:
> On Wednesday, June 17, 2020 at 3:30:12 PM UTC+7, Richard Heathfield wrote:
>> On 17/06/2020 08:55, James Dow Allen wrote:
>>>
>>> By the way, too bad you wrote "positive" instead of "non-negative." :-)
>>> We miss out on the famous
>>>       0 + 0 + 6^3 = 5^3 + 4^3 + 3^3
>>
>> Nitpick on terminology:
>>
>> That should be 0^3 + 0^3 + 6^3 = 5^3 + 4^3 + 3^3
>>
>> ;-)
>>
>> (Those with a decent keyboard will of course use 0³ + 0³ + 6³ etc, in
>> flagrant violation of Usenet's ASCII-only protocol.)
> 
> Wow!  First time I visit comp,programming in YEARS and here
> is ... the inimitable Richard Heathfield.

My blushes, Watson!

> And replying within minutes.

It was either that or weave another wheelbarrow entirely out of lockdown 
hair.

> Bravo!  How are you doing?, Richard!

Well... Jane's a mod.

(Solve.)

<snip>

> Good to see you, Richard!

And you, sir.

-- 
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

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#10952

On Saturday, June 20, 2020 at 2:26:54 PM UTC+7, Richard Heathfield wrote:
> 
> Well... Jane's a mod.

I'm glad you mentioned Jane.  I always admired 

Her dear child-faith.

James

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#10955

On 20/06/2020 11:58, James Dow Allen wrote:
> On Saturday, June 20, 2020 at 2:26:54 PM UTC+7, Richard Heathfield wrote:
>>
>> Well... Jane's a mod.
> 
> I'm glad you mentioned Jane.  I always admired
> 
> Her dear child-faith.

:-)

-- 
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

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#10950

James Dow Allen <jdallen2000@yahoo.com> writes:
<cut>
> On Saturday, June 13, 2020 at 11:02:49 PM UTC+7, Ben Bacarisse wrote:
>> The first 10 solution sets this program finds are:
<cut>
> Nitpick on terminology:  The quadruplets we are talking about are
> SOLUTIONS, not "solution sets."  The Solution Set would be the
> set of all possible solution quadruplets.

My program does not look for these solution quadruplets.  It looks for
solutions to ab(a+b) and groups them by equal value.  That's why I
called them "solution sets".  There is a filter in main that removes all
those "sets" that have only one member.

-- 
Ben.

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#10953

I like the way this is building up. My original post is addressed to people into interested in number theory AND programming!
My puzzle gives a sufficient condition for a number too be a sum of three (positive) cubes in two different ways. It is  clearly not the necessary condition. I said I will discuss the "deeper maths"...here is a question :what is the necessary condition? 

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#10954

On Saturday, 20 June 2020 13:45:02 UTC+2, syed owais  wrote:

[You should quote enough context.]

> My puzzle gives a sufficient condition for a number too be a sum of three
> (positive) cubes in two different ways. It is  clearly not the necessary
> condition.

Notice that a^3+b^3+(c+d)^3=(a+b)^3+c^3+d^3 <=> ab(a+b)=cd(c+d), i.e. the two
are in fact equivalent, not just the former implies the latter.

(You wrote a*3+b*3+ etc. but I take the above to be what you meant or the
derivation is simply false.)

> I said I will discuss the "deeper maths"...here is a question
> :what is the necessary condition?

That the universe is rational...?  Still waiting for the "deeper maths".

Julio

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