Pell's equation

Jombo G · 2022-10-26, 17:59

Hey there!

Attached is a Pell's equation solver program, in Pari-GP

It uses very basic operations but seems to be quick enough.
Warning though, if Y^2 - N*X^2 does not go to 1 quickly, the numbers tend to become
quite prohibitively large. Nothing wrong with that perhaps, but before (and/or if) you run
it in Pari, please set the precision to 200-odd digits using the command:

\p 200

in the terminal window. At least this gives it a good chance of finding answers for larger

values of N - though obviously even (relatively) small N values can still have large-ish smallest

solutions, if you see what I mean?!!

Thanks for reading.

PS: The code is a bit of a mess atm, with not all variables being used etc. I'll clean it up when I get the time.
Cheers.

sweety439 · 2022-10-26, 20:11

Quote:

Originally Posted by Jombo G

Attached is a Pell's equation solver program, in Pari-GP

Well, I have a data, for the smallest solution of x^2-d*y^2 = 0, +-1, +-2, +-3, +-4 (if exists, the +-4 case only list primitive solutions, i.e. not a double of the x and y for the +-1 case), for d <= 25000

R. Gerbicz · 2022-10-26, 20:19

That is too slow, there is a pretty nice stuff for Pell equation:
http://library.msri.org/books/Book44.../01lenstra.pdf
See section 7, that is really working, I've coded that, not public.

Jombo G · 2022-10-26, 21:29

Thanks very much. I was looking for a resource like this to test against online. I have a Pell's Equation book, but it only has a few results in it.

sweety439 · 2022-10-28, 03:13

PARI/GP code for Pell equation: http://www.kaynet.or.jp/~kay/misc/program/pell.gp

Many other computer program codes for Pell equation: https://rosettagit.org/drafts/pells-equation/

Data for solutions of Pell equations: http://www.kaynet.or.jp/~kay/misc/pell2.html and https://raw.githubusercontent.com/xa...20equation.txt and https://raw.githubusercontent.com/xa...le%2025000.txt

Online Pell equation solvers:

1. http://www.jakebakermaths.org.uk/mat...ntegerv10.html

2. http://math.fau.edu/richman/pell-m.htm

3. http://www.numbertheory.org/php/pell.html

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2022-10-26, 20:19	#3
R. Gerbicz "Robert Gerbicz" Oct 2005 Hungary 3²×179 Posts	That is too slow, there is a pretty nice stuff for Pell equation: http://library.msri.org/books/Book44.../01lenstra.pdf See section 7, that is really working, I've coded that, not public.

2022-10-26, 21:29	#4
Jombo G Aug 2019 2·3 Posts	Thanks very much. I was looking for a resource like this to test against online. I have a Pell's Equation book, but it only has a few results in it.

2022-10-28, 03:13	#5
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2×7×263 Posts	PARI/GP code for Pell equation: http://www.kaynet.or.jp/~kay/misc/program/pell.gp Many other computer program codes for Pell equation: https://rosettagit.org/drafts/pells-equation/ Data for solutions of Pell equations: http://www.kaynet.or.jp/~kay/misc/pell2.html and https://raw.githubusercontent.com/xa...20equation.txt and https://raw.githubusercontent.com/xa...le%2025000.txt Online Pell equation solvers: 1. http://www.jakebakermaths.org.uk/mat...ntegerv10.html 2. http://math.fau.edu/richman/pell-m.htm 3. http://www.numbertheory.org/php/pell.html