[Date Index]
[Thread Index]
[Author Index]
Re: Simplifying with assumptions
*To*: mathgroup at smc.vnet.net
*Subject*: [mg49057] Re: Simplifying with assumptions
*From*: Paul Abbott <paul at physics.uwa.edu.au>
*Date*: Tue, 29 Jun 2004 04:50:49 -0400 (EDT)
*Organization*: The University of Western Australia
*References*: <200406250658.CAA12398@smc.vnet.net> <cbi7jv$lep$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
In article <cbi7jv$lep$1 at smc.vnet.net>,
Daniel Lichtblau <danl at wolfram.com> wrote:
> 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.
>
>
> I'm not sure what would be a good way to do this sort of thing in
> general. For relatively simple examples you can make progress if you
> assign integer values to one parameter, as below.
>
> In[7]:= InputForm[Table[
> Reduce[{48-n^2+8*x==k^2, Element[{x,k},Integers]}, {x,k}], {n,10}]]
>
> Out[7]//InputForm=
> {False, (C[1] \[Element] Integers && x == -5 + 4*C[1] + 8*C[1]^2 &&
> k == 2 + 8*C[1]) || (C[1] \[Element] Integers &&
> x == -1 + 12*C[1] + 8*C[1]^2 && k == 6 + 8*C[1]), False,
> (C[1] \[Element] Integers && x == -4 + 8*C[1]^2 && k == 8*C[1]) ||
> (C[1] \[Element] Integers && x == -2 + 8*C[1] + 8*C[1]^2 &&
> k == 4 + 8*C[1]), False, (C[1] \[Element] Integers &&
> x == -1 + 4*C[1] + 8*C[1]^2 && k == 2 + 8*C[1]) ||
> (C[1] \[Element] Integers && x == 3 + 12*C[1] + 8*C[1]^2 &&
> k == 6 + 8*C[1]), False, (C[1] \[Element] Integers && x == 2 +
> 8*C[1]^2 &&
> k == 8*C[1]) || (C[1] \[Element] Integers && x == 4 + 8*C[1] +
> 8*C[1]^2 &&
> k == 4 + 8*C[1]), False, (C[1] \[Element] Integers &&
> x == 7 + 4*C[1] + 8*C[1]^2 && k == 2 + 8*C[1]) ||
> (C[1] \[Element] Integers && x == 11 + 12*C[1] + 8*C[1]^2 &&
> k == 6 + 8*C[1])}
There is a pattern in the coefficients, easily determined using
InterpolatingPolynomial. After some manipulation I get that the general
solution can be expressed in the following form:
sol[m_,p_] = {{m,p} \[Element] Integers &&
n == 4p && x == 8m^2 + 2p^2 - 6 &&
x == 8m^2 + 8m + 2p^2 - 4
||
n == 4p + 2 && x == 8m^2 + 4m + 2p^2 + 2p - 5 &&
x == 8m^2 + 12m + 2p^2 + 2p - 1}
Tabulating values for various m and p,
vals = Table[ToRules /@ sol[m, p], {m, -5, 5}, {p, -7, 7}]
we check that Sqrt[48 - n^2 + 8 x] is an integer:
Sqrt[48 - n^2 + 8 x] /. vals
Cheers,
Paul
--
Paul Abbott Phone: +61 8 9380 2734
School of Physics, M013 Fax: +61 8 9380 1014
The University of Western Australia (CRICOS Provider No 00126G)
35 Stirling Highway
Crawley WA 6009 mailto:paul at physics.uwa.edu.au
AUSTRALIA http://physics.uwa.edu.au/~paul
Prev by Date:
**RE: one question about draw graph in frame**
Next by Date:
**Re: Re: Re: Net/Link: Problem with DLL (2)**
Previous by thread:
**Re: Simplifying with assumptions**
Next by thread:
**RE: Simplifying with assumptions**
| |