squares or not squares

[m_f_h]

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...

[R.D. Silverman]

Originally Posted by m_f_h

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._

[m_f_h]

Originally Posted by R.D. Silverman

(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."

[maxal]

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.

[mfgoode]

Originally Posted by m_f_h

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

[m_f_h]

Originally Posted by mfgoode

"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]

Originally Posted by maxal

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)

[maxal]

Originally Posted by m_f_h

(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.

[mfgoode]

Originally Posted by m_f_h

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

[m_f_h]

Originally Posted by mfgoode

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.

Lagrange had an established base to build upon.
Pythagoras had none. That's why I admire the ancient one.

I agree completely !!

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]

Originally Posted by maxal

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.