Re: What inspite FindInstance
- To: mathgroup at smc.vnet.net
- Subject: [mg122744] Re: What inspite FindInstance
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Wed, 9 Nov 2011 06:25:05 -0500 (EST)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
- References: <201111081215.HAA04920@smc.vnet.net>
On 8 Nov 2011, at 13:15, Artur wrote:
> Dear Mathematica Gurus,
> Matematical problem is following: find pair of integres {d,x} which fill
> equation (d ^2 +
> 270 d Root[1 - 270 #1^3 - 5832 #1^5 + 3645 #1^6 &, 1]^3 -
> 2916 x Root[1 - 270 #1^3 - 5832 #1^5 + 3645 #1^6 &, 1]^5 +
> 3645 Root[1 - 270 #1^3 - 5832 #1^5 + 3645 #1^6 &, 1]^6 ==
> 0)
>
> FindInstance[(d ^2 +
> 270 d Root[1 - 270 #1^3 - 5832 #1^5 + 3645 #1^6 &, 1]^3 -
> 2916 x Root[1 - 270 #1^3 - 5832 #1^5 + 3645 #1^6 &, 1]^5 +
> 3645 Root[1 - 270 #1^3 - 5832 #1^5 + 3645 #1^6 &, 1]^6 ==
> 0) && (x d != 0), {x, d}, Integers]
>
> Mathematica report:
> FindInstance::nsmet: The methods available to FindInstance are
> insufficient to find the requested instances or prove they do not exist. >>
>
> First solution is x=2 and d=-1
> I'm looking for next ones. Who have idea what to do?
>
> Best wishes
> Artur
>
>
Well, the answer is
x=2, d= -1
Checking:
pp = d^2 + 270 d Root[1 - 270 #1^3 - 5832 #1^5 + 3645 #1^6 &, 1]^3 -
2916 x Root[1 - 270 #1^3 - 5832 #1^5 + 3645 #1^6 &, 1]^5 +
3645 Root[1 - 270 #1^3 - 5832 #1^5 + 3645 #1^6 &, 1]^6;
FullSimplify[pp /. {d -> -1, x -> 2}]
0
How did I find it? Well, let
w = Root[3645*#1^6 - 5832*#1^5 - 270*#1^3 + 1 & , 1];
Then w has to be a root of
In[5]:= ff = pp /. w -> z
Out[5]= d^2+270 d z^3-2916 x z^5+3645 z^6
On the other hand,
In[6]:= gg = MinimalPolynomial[w, z]
Out[6]= 3645 z^6-5832 z^5-270 z^3+1
Hence ff has to divide gg. Now
In[7]:= PolynomialQuotientRemainder[ff, gg, z]
Out[7]= {1,d^2+(270 d+270) z^3+(5832-2916 x) z^5-1}
And from this we see that d= -1 and x = 2.
Andrzej Kozlowski
- References:
- What inspite FindInstance
- From: Artur <grafix@csl.pl>
- What inspite FindInstance