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From: mukesh tiwari on 20 Apr 2010 10:41
Hello everyone, I am trying to solve this problem but i am not getting
efficient solution. Here is my analysis.

A=(x^2+3x)/(1-x-x^2)
x=-(A+1)+Sqrt(5A^2+14A+1)/2(A+3) since x is rational so 5A^2+14A+1
must be a perfect square.
5A^2+14A+1=D^2
A=-14+Sqrt(20D^2+176)/10.. This is the value of A which we have to add
upto 30 numbers. Now i am iteration D from 1 to infinity and got some
values but it tool almost 30 mins and produce only 21 values.
2
5
21
42
152
296
1050
2037
7205
13970
49392
95760
338546
656357
2320437
4498746
15904520
30834872
109011210
211345365
747173957
kindly give me some more insights.
Thank you
From: mukesh tiwari on 20 Apr 2010 11:56
On Apr 20, 7:41 pm, mukesh tiwari <mukeshtiwari.ii...(a)gmail.com>
wrote:
> Hello everyone, I am trying to solve this problem but i am not getting
> efficient solution. Here is my analysis.
>
> A=(x^2+3x)/(1-x-x^2)
> x=-(A+1)+Sqrt(5A^2+14A+1)/2(A+3) since x is rational so 5A^2+14A+1
> must be a perfect square.
> 5A^2+14A+1=D^2
> A=-14+Sqrt(20D^2+176)/10.. This is the value of A which we have to add
> upto 30 numbers. Now i am iteration D from 1 to infinity and got some
> values but it tool almost 30 mins and produce only 21 values.
> 2
> 5
> 21
> 42
> 152
> 296
> 1050
> 2037
> 7205
> 13970
> 49392
> 95760
> 338546
> 656357
> 2320437
> 4498746
> 15904520
> 30834872
> 109011210
> 211345365
> 747173957
> kindly give me some more insights.
> Thank you

I for got to post the link of problem.http://projecteuler.net/
index.php?section=problems&id=140
From: Robert Israel on 20 Apr 2010 12:32
mukesh tiwari <mukeshtiwari.iiitm(a)gmail.com> writes:

> Hello everyone, I am trying to solve this problem but i am not getting
> efficient solution. Here is my analysis.
>
> A=(x^2+3x)/(1-x-x^2)
> x=-(A+1)+Sqrt(5A^2+14A+1)/2(A+3) since x is rational so 5A^2+14A+1
> must be a perfect square.
> 5A^2+14A+1=D^2
> A=-14+Sqrt(20D^2+176)/10.. This is the value of A which we have to add
> upto 30 numbers. Now i am iteration D from 1 to infinity and got some
> values but it tool almost 30 mins and produce only 21 values.
> 2
> 5
> 21
> 42
> 152
> 296
> 1050
> 2037
> 7205
> 13970
> 49392
> 95760
> 338546
> 656357
> 2320437
> 4498746
> 15904520
> 30834872
> 109011210
> 211345365
> 747173957
> kindly give me some more insights.
> Thank you

Hint: Look up Pell's equation.
--
Robert Israel israel(a)math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
From: mukesh tiwari on 20 Apr 2010 12:41
On Apr 20, 9:32 pm, Robert Israel
<isr...(a)math.MyUniversitysInitials.ca> wrote:
> mukesh tiwari <mukeshtiwari.ii...(a)gmail.com> writes:
> > Hello everyone, I am trying to solve this problem but i am not getting
> > efficient solution. Here is my analysis.
>
> > A=(x^2+3x)/(1-x-x^2)
> > x=-(A+1)+Sqrt(5A^2+14A+1)/2(A+3) since x is rational so 5A^2+14A+1
> > must be a perfect square.
> > 5A^2+14A+1=D^2
> > A=-14+Sqrt(20D^2+176)/10.. This is the value of A which we have to add
> > upto 30 numbers. Now i am iteration D from 1 to infinity and got some
> > values but it tool almost 30 mins and produce only 21 values.
> > 2
> > 5
> > 21
> > 42
> > 152
> > 296
> > 1050
> > 2037
> > 7205
> > 13970
> > 49392
> > 95760
> > 338546
> > 656357
> > 2320437
> > 4498746
> > 15904520
> > 30834872
> > 109011210
> > 211345365
> > 747173957
> > kindly give me some more insights.
> > Thank you
>
> Hint: Look up Pell's equation.
> --
> Robert Israel              isr...(a)math.MyUniversitysInitials.ca
> Department of Mathematics        http://www.math.ubc.ca/~israel
> University of British Columbia            Vancouver, BC, Canada

It's a great hint. Thank you. Very Much
From: Mensanator on 20 Apr 2010 12:54
On Apr 20, 9:41 am, mukesh tiwari <mukeshtiwari.ii...(a)gmail.com>
wrote:
> Hello everyone, I am trying to solve this problem but i am not getting
> efficient solution. Here is my analysis.
>
> A=(x^2+3x)/(1-x-x^2)
> x=-(A+1)+Sqrt(5A^2+14A+1)/2(A+3) since x is rational so 5A^2+14A+1
> must be a perfect square.
> 5A^2+14A+1=D^2
> A=-14+Sqrt(20D^2+176)/10.. This is the value of A which we have to add
> upto 30 numbers. Now i am iteration D from 1 to infinity and got some
> values but it tool almost 30 mins and produce only 21 values.

What are you using? An abacus?


> 2
> 5
> 21
> 42
> 152
> 296
> 1050
> 2037
> 7205
> 13970
> 49392
> 95760
> 338546
> 656357
> 2320437
> 4498746
> 15904520
> 30834872
> 109011210
> 211345365
> 747173957
> kindly give me some more insights.
> Thank you

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