mersenneforum.org  

Go Back   mersenneforum.org > Fun Stuff > Puzzles

Reply
 
Thread Tools
Old 2007-05-24, 17:50   #1
m_f_h
 
m_f_h's Avatar
 
Feb 2007

6608 Posts
Default squares or not squares

Find integers m>1 such that n(m^2-1)+1 is a square, for n=614, 662 and/or 719.
Give a method for finding the lowest possible m
(a) when n is prime,
(b) when n=2^k p with p prime
(c) n square-free
(d) in the general case.


PS: partial answers are accepted...
m_f_h is offline   Reply With Quote
Old 2007-05-24, 17:56   #2
R.D. Silverman
 
R.D. Silverman's Avatar
 
Nov 2003

164448 Posts
Default

Quote:
Originally Posted by m_f_h View Post
Find integers m>1 such that n(m^2-1)+1 is a square, for n=614, 662 and/or 719.
Give a method for finding the lowest possible m
(a) when n is prime,
(b) when n=2^k p with p prime
(c) n square-free
(d) in the general case.


PS: partial answers are accepted...
(1) Find the fundamental units of Q(sqrt(n)).
(2) Factor (1-n) over the ring of integers of Q(sqrt(n)).

Solutions are the norms of the (factors times powers of the fundamental
units._
R.D. Silverman is offline   Reply With Quote
Old 2007-05-24, 19:34   #3
m_f_h
 
m_f_h's Avatar
 
Feb 2007

24·33 Posts
Thumbs up

Quote:
Originally Posted by R.D. Silverman View Post
(1) Find the fundamental units of Q(sqrt(n)).
(2) Factor (1-n) over the ring of integers of Q(sqrt(n)).
Solutions are the norms of the (factors times powers of the fundamental
units._
Congrats : You win *at least* the prize for the fastest answer...

Maybe there's a candidate for the most elegant PARI implementation....

In case MFGoode would wander by here, I won't let him miss the following citation,
supposedly due to Hermann Hankel, who states that the chakravala method is
"the finest thing achieved in the theory of numbers before Lagrange."

Last fiddled with by m_f_h on 2007-05-24 at 19:57 Reason: +WP link
m_f_h is offline   Reply With Quote
Old 2007-05-25, 00:12   #4
maxal
 
maxal's Avatar
 
Feb 2005

22·32·7 Posts
Default

Use Dario Alpern's applet:
http://www.alpertron.com.ar/QUAD.HTM

E.g.:
for n=614, the smallest m is 334235297891.
for n=662, the smallest m is 1651326551.
for n=719, the smallest m is 388433033911.

Last fiddled with by maxal on 2007-05-25 at 00:19
maxal is offline   Reply With Quote
Old 2007-05-25, 09:41   #5
mfgoode
Bronze Medalist
 
mfgoode's Avatar
 
Jan 2004
Mumbai,India

22×33×19 Posts
Thumbs up citation!

Quote:
Originally Posted by m_f_h View Post
Congrats : You win *at least* the prize for the fastest answer...

Maybe there's a candidate for the most elegant PARI implementation....

In case MFGoode would wander by here, I won't let him miss the following citation,
supposedly due to Hermann Hankel, who states that the chakravala method is
"the finest thing achieved in the theory of numbers before
Lagrange."
Thank you m_f_h for even thinking of me.

Recently an Indian born mathematician Varadhan a professor at the Courant
Institute and 67 yrs old received the Abel prize for work on probability theory.
Say where are all the news hogs in our forum to have missed out on this one!

"before Lagrange" ??

Hey m_f_h ! Wait for my paper. It even predates Pythagoras ! Ha! Ha! And I am a year older than Varadhan.

So wish me luck!

Mally
mfgoode is offline   Reply With Quote
Old 2007-05-25, 19:06   #6
m_f_h
 
m_f_h's Avatar
 
Feb 2007

43210 Posts
Default

Quote:
Originally Posted by mfgoode View Post
"before Lagrange" ??
Hey m_f_h ! Wait for my paper. It even predates Pythagoras !
well,
"the finest thing acheived before Lagrange"
is a stronger compliment than
"the finest thing acheived before Pythagoras".
(?)
m_f_h is offline   Reply With Quote
Old 2007-05-25, 19:10   #7
m_f_h
 
m_f_h's Avatar
 
Feb 2007

24·33 Posts
Default

Quote:
Originally Posted by maxal View Post
Use Dario Alpern's applet: http://www.alpertron.com.ar/QUAD.HTM
E.g.:
for n=614, the smallest m is 334235297891.
for n=662, the smallest m is 1651326551.
for n=719, the smallest m is 388433033911.
Thanks for the link.
Well, for computers its not difficult to find.
(it's sufficient to type isolve(719*(m^2-1)=y^2) into maple and fiddle a bit to make sure to get the smallest solution.)

PS: (not more powerful than D.Alpern's applet, but since he seems to be another "bioinformatics number theorist" (well...)): http://www.bioinfo.rpi.edu/~zukerm/cgi-bin/dq.html)

Last fiddled with by m_f_h on 2007-05-25 at 19:37
m_f_h is offline   Reply With Quote
Old 2007-05-25, 19:18   #8
maxal
 
maxal's Avatar
 
Feb 2005

22·32·7 Posts
Default

Quote:
Originally Posted by m_f_h View Post
(it's sufficient to type isolve(719*(m^2-1)=y^2) into maple and fiddle a bit to make sure to get the smallest solution.)
As all solutions are given by recurrent sequences, it is easy to prove that these sequences are monotone and find a smallest positive element in each.
maxal is offline   Reply With Quote
Old 2007-05-26, 05:20   #9
mfgoode
Bronze Medalist
 
mfgoode's Avatar
 
Jan 2004
Mumbai,India

40048 Posts
Thumbs up

Quote:
Originally Posted by m_f_h View Post
well,
"the finest thing acheived before Lagrange"
is a stronger compliment than
"the finest thing acheived before Pythagoras".
(?)


Well! Well! m_f_h.

It all depends how fundamental it is.

Lagrange had an established base to build upon.

Pythagoras had none. That's why I admire the ancient one.

He was not only a mathematician, but a musician and mystic combined and half of his revolutionary theories have been lost in antiquity.

Who do you reckon as the greater ? the artisan who cut the stone for the great pyramid or the man who designed it? And with such proportions that we still do not know!

Mally
mfgoode is offline   Reply With Quote
Old 2007-05-26, 12:22   #10
m_f_h
 
m_f_h's Avatar
 
Feb 2007

1B016 Posts
Default

Quote:
Originally Posted by mfgoode View Post
Well! Well! m_f_h.
It all depends how fundamental it is.
no, I did not mean to judge the importance of these mathematicians, but:
max { f(y) | -oo < y < 0 } <= max { f(y} | -oo < y < 1800 }
where f(y) is a measure of the greatest acheivment in the year y.

greatest up to J.C. = l.h.s.
greatest up to 18th century = r.h.s.

Thus, it is a stronger statement to say : C is equal to the second max
than to say: C is equal to the first max.
Quote:
Lagrange had an established base to build upon.
Pythagoras had none. That's why I admire the ancient one.
I agree completely !!
Quote:
He was not only a mathematician, but a musician and mystic combined and half of his revolutionary theories have been lost in antiquity.
Of course... (I suppose the same is true for several other great spirits we don't know of: chinese, south american, ... who may have found several things centuries before European science even started to emerge.)
m_f_h is offline   Reply With Quote
Old 2007-05-26, 12:48   #11
m_f_h
 
m_f_h's Avatar
 
Feb 2007

24×33 Posts
Default

Quote:
Originally Posted by maxal View Post
As all solutions are given by recurrent sequences, it is easy to prove that these sequences are monotone and find a smallest positive element in each.
I agree, that's what I mean by "fiddle a bit", since maple just spits out 8 (or 8n) sets
{ x=..., y=...} , { x=..., y=...} , { x=..., y=...} , { x=..., y=...} ...
in no apparent order (of course groups of 4 of them always correspond to different combinations of signs of x,y), where the r.h.s. depends on arbitrary integer _Z1, so the easiest way (I see at first glance) is to substitute at least 2 different values, so that you can find the smallest possible solution by minimizing.
I conjecture that Maple's algorithm is such that the smallest possible solution is always obtained for either _Z1=1 or _Z1=0 (e.g. for 634), but I'm not 100% sure of that.
m_f_h is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Find the Squares a1call Puzzles 18 2018-03-02 16:47
Regarding Squares a1call Miscellaneous Math 42 2017-02-03 01:29
Perfect squares in NFS paul0 Computer Science & Computational Number Theory 2 2015-01-02 14:21
A Sum of Squares Problem MattcAnderson Puzzles 7 2014-08-17 07:20
parsimonious squares fivemack Miscellaneous Math 11 2011-06-29 02:09

All times are UTC. The time now is 15:31.

Sat Apr 17 15:31:47 UTC 2021 up 9 days, 10:12, 0 users, load averages: 1.16, 1.56, 1.49

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2021, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.