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MathGroup Archive 2004

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Re: Simplifying with assumptions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg48986] Re: [mg48949] Simplifying with assumptions
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Fri, 25 Jun 2004 17:52:33 -0400 (EDT)
  • References: <200406250658.CAA12398@smc.vnet.net> <9198ADAD-C6AD-11D8-8C1E-000A95B4967A@mimuw.edu.pl>
  • Sender: owner-wri-mathgroup at wolfram.com

In my first reply I did not describe  how I obtained my answer  x = 43; 
n = 14, because Mathematica played only a fairly minor role in it, but 
on second thoughts I decided it might be interesting to do so. So here 
is the mathematical argument with a bit of Mathematica at the end.

We are looking for solutions of the Diophantine equation

48 - n^2 + 8*x == m^2

This can be re-written in the form

48+8x ==8(6+x) ==m^2+n^2

There is a following well known theorem:

A positive integer is a sum of two squares if an only if all of its 
prime divisors of the form 4k+3 appear in its prime factorization with 
even exponents. (e.g. K. Chandrasekharan "Introduction to Analytic 
Number Theory"). This means that 6+x must have this property. I simply 
took 6+x = 7^2 ==49 so x =43. So now we need to find integers n and m 
such that 8*49= n^2 + m^2. Now we can use Matheamtica. Load in the 
package

<< NumberTheory`NumberTheoryFunctions`

and evaluate:


SumOfSquaresRepresentations[2, 8*49]


{{-14, -14}, {-14, 14}, {14, -14}, {14, 14}}


You can also see at one that there are infinitely many solutions that 
can be obtained with this method.

Andrzej Kozlowski

On 25 Jun 2004, at 22:42, Andrzej Kozlowski wrote:

> On 25 Jun 2004, at 15:58, Mietek Bak wrote:
>
>> Hello,
>>
>> I'm a complete newcomer to Mathematica, so please excuse this possibly
>> silly question.
>>
>> I'm trying to determine if a formula will ever give an integer result,
>> assuming that all variables used in it are integer.  I've been 
>> searching
>> through the built-in documentation, but my best guess didn't really do
>> anything:
>>
>> Simplify[Element[Sqrt[48 - n^2 + 8*x],Integers],Element[{n, 
>> x},Integers]]
>>
>> It would be best if I could somehow determine the set of combinations 
>> of
>> variables that would give an integer result -- if there are any.  Is
>> there a way to do that in Mathematica?
>>
>> Thanks in advance,
>> Mietek Bak.
>>
>>
>> -- 
>>     desp;
>> }
>>
>
> If you want to find just a single solution it is easy: take
>  x = 43; n = 14;
>  then
>
> 48 - n^2 + 8*x
>
> 196
>
> which is just 14^2. I can also prove that there are infinitely many 
> such solutions. However, I don't think there is any way to solve such 
> problems in general with Mathematica or any other computer program.
>
>
> Andrzej Kozlowski
> Chiba, Japan
> http://www.mimuw.edu.pl/~akoz/
>


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