20070524, 17:50  #1 
Feb 2007
660_{8} Posts 
squares or not squares
Find integers m>1 such that n(m^21)+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 squarefree (d) in the general case. PS: partial answers are accepted... 
20070524, 17:56  #2  
Nov 2003
16444_{8} Posts 
Quote:
(2) Factor (1n) over the ring of integers of Q(sqrt(n)). Solutions are the norms of the (factors times powers of the fundamental units._ 

20070524, 19:34  #3  
Feb 2007
2^{4}·3^{3} Posts 
Quote:
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 20070524 at 19:57 Reason: +WP link 

20070525, 00:12  #4 
Feb 2005
2^{2}·3^{2}·7 Posts 
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 20070525 at 00:19 
20070525, 09:41  #5  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}×3^{3}×19 Posts 
citation!
Quote:
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 

20070525, 19:06  #6 
Feb 2007
432_{10} Posts 

20070525, 19:10  #7  
Feb 2007
2^{4}·3^{3} Posts 
Quote:
Well, for computers its not difficult to find. (it's sufficient to type isolve(719*(m^21)=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/cgibin/dq.html) Last fiddled with by m_f_h on 20070525 at 19:37 

20070525, 19:18  #8 
Feb 2005
2^{2}·3^{2}·7 Posts 
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.

20070526, 05:20  #9  
Bronze Medalist
Jan 2004
Mumbai,India
4004_{8} Posts 
Quote:
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 

20070526, 12:22  #10  
Feb 2007
1B0_{16} Posts 
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:
Quote:


20070526, 12:48  #11  
Feb 2007
2^{4}×3^{3} Posts 
Quote:
{ 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. 

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