[Date Index]
[Thread Index]
[Author Index]
How can I make FullSimplify work for user-defined functions?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg52569] How can I make FullSimplify work for user-defined functions?
*From*: gilmar.rodriguez at nwfwmd.state.fl.us (Gilmar Rodr?guez Pierluissi)
*Date*: Thu, 2 Dec 2004 02:21:28 -0500 (EST)
*Sender*: owner-wri-mathgroup at wolfram.com
(First; a necessary short introduction, and then
two questions. Please, bear with me...)
Let N be an even integer greater or equal to 4.
Assume that there exist points (Pi, Qi) on the line
Y = -X + N such that:
(1.) both Pi, and Qi are primes,
(2.) the points (Pi,0) are on the subinterval [2,N/2]
of the X-Axis,
(3.) the points (0,Qi) are on the subinterval [N/2,N-2]
of the Y-Axis.
A Minimal Goldbach Prime Partition Point corresponding
to N,(abbreviated: "MGPPP[N]")is a point (P,Q) among
the points (Pi,Qi) such that:
the distance between the point (P,Q) and the point (N/2,N/2)
= smallest distance among all distances between
the points (Pi, Qi) and the point (N/2, N/2).
If N is of the form N = 2P then MGPPP[N] = (N/2,N/2).
If N is of the form N = P + Q, with P not equal to Q, then
MGPPP[N] = (P,Q).
For geometrical pictures please visit:
http://gilmarlily.netfirms.com/goldbach/goldbach.htm
Please download the following Mathematica notebook
(version 5.0)by double-cliking this link:
http://gilmarlily.netfirms.com/download/FullSimplify.nb
You can use this notebook to duplicate the evaluations
appearing in the discussion below:
Having said the above; the following program
calculates the MGPPP[N]:
MGPPP[n_] := Module[{p, q},
{m = n/2; If[Element[m,
Primes], {p = m, q = m}, {k =PrimePi[m];
Do[If[Element[n - Prime[i], Primes], hit = i;
Break[]], {i, k, 1, -1}],p = Prime[hit],
q = n - p}]}; {p, q}]
Examples:
MGPPP[4]={2,2}
MGPPP[100]={47,53}.
The above program can be easily changed as follows to give
only the "p-value" (instead of the point {p,q}):
pvalue[n_] := Module[{p},
{m = n/2; If[Element[m,
Primes], {p = m}, {k =PrimePi[m];
Do[If[Element[n - Prime[i], Primes], hit = i;
Break[]],{i, k, 1, -1}], p = Prime[hit]}]}; p]
A plot to depict the p-values is given by:
plt1=ListPlot[Table[pvalue[n],{n,4,100,2}],PlotJoined®True,
PlotStyle®Hue[0.1]]
It has been conjectured that the Minimal Goldbach Prime Partition
p-values have a lower bound given by:
Prime[PrimePi[Sqrt[N]].
To visualize this do:
(** oldlb= abbreviation for old lower bound. **)
oldlb[n_]:=Prime[PrimePi[Sqrt[n]]];
plt2=ListPlot[Table[oldlb[n],{n,4,100,2}],
PlotJoined®True,PlotStyle®Hue[0.2]]
Show[plt1,plt2]
Try also:
TableForm[Table[{n,oldlb[n],pvalue[n]},{n,4,100,2}],
TableHeadings->{None,{"n","oldlb[n]","p"}},
TableAlignments->Center]
However; that lower bound can be improved by the following
new lower bound:
Prime[PrimePi[(N+2*Sqrt[N])/4].
The following program makes use of this latest interval:
pval[n_] := Module[{p},
{m = n/2; If[Element[m,
Primes], {p = m}, {k =PrimePi[m];
l=PrimePi[(n+2*Sqrt[n])/4];
Do[If[Element[n - Prime[i], Primes], hit = i;
Break[]], {i, k, l, -1}], p = Prime[hit]}]}; p]
To compare the p-values with the old and new lower bounds do:
plt3=ListPlot[Table[pval[n],{n,4,100,2}],PlotJoined®True,
PlotStyle®Hue[0.1]]
newlb[n_]:=Prime[PrimePi[(n+2*Sqrt[n])/4]]
plt4=ListPlot[Table[newlb[n],{n,4,100,2}],
PlotJoined®True,PlotStyle®Hue[0.6]]
Show[plt3,plt4]
Show[plt2,plt3,plt4]
Try also:
TableForm[Table[{n,newlb[n],pval[n]},{n,4,100,2}],
TableHeadings->{None,{"n","newlb[n]","p"}},
TableAlignments->Center]
Finally; I try the FullSimplify command, to test the validity
of the new bound,via:
FullSimplify[newlb[n]<= pval[n],Element[n,EvenQ]&&n³4]
hoping that this evaluation gives me "True" as a result.
I get instead a cryptic:
"Prime[PrimePi[1/4(2Sqrt[n]+n)]]<= p$214"
as answer.
Then I realize: one would really need to prove Goldbach's Conjecture first,
so that the "True" after the above FullSimplify evaluation is truly valid.
So I try instead:
FullSimplify[newlb[n]<=pval[n],Element[n,EvenQ]&&n³4&&n<10^6]
and this time I get:
"Prime[PrimePi[1/4(2Sqrt[n]+n)]]<= p$358".
My questions are:
(1.) How can I make FullSimplify work for user-defined functions?
(2.) How can I make
FullSimplify[newlb[n]<=pval[n],Element[n,EvenQ]&&n³4&&n<10^6]
"True"?
Thank you for your attention, and your help!
Prev by Date:
**Re: Inserting user material in the HelpBrowser**
Next by Date:
**Fedora3 segmentation fault**
Previous by thread:
**Re: Compiled function with NIntegrate**
Next by thread:
**Re: How can I make FullSimplify work for user-defined functions?**
| |