How to solve for solutions to the diophantine 5b^2*c^2 = 4a^2(b^2+c^2)

LittleNewton

How can I judge? Can you be more specific?

I simplified all results using the gcd's.
then I tried randomly picking primitives,
and all seemed legitimate...

LittleNewton

By trial and error, I found that the cube root of limit is enough for the range.
However this might be by chance or might depend on that particular limit.

And that is still a lot of (primitive) starting points...

LittleNewton

I also read a lot of papers on that subject.
Most just state that these generating functions generate infinitely many solutions,
but they are not all the solutions, meaning there might be more.

They sometimes use modular arithmetic to show the non-existence of a solution.

I would like to know when we are sure that we have found all the primitives.

micromass

I'm pretty sure that those are all the solutions. Did you check the link in my post? There they give a proof (or at least an argument) for why these are all the solutions.

SeventhSigma

false

micromass

Please don't post unless you post your solution.

LittleNewton

Yes, I checked it. Somehow only they (this paper and others)
only confidently talk about primitives in case of the basic Pythagorean triples.

I want to know :

1) if I am supposed to extent to negative m,n values
2) what ranges of m & n will give me all the primitives

For example if I want to find all solutions below a certain limit, say 10K,
what is the complete set of primitives I need to start with?

micromass

I'm sorry about this thread. I should have locked it when I had the chance. Now I let it degenerate into a mess. My sincere apologies to all the readers.

micromass

I was always under the impression that we could find all the rational points on a given conic. I'll admit that my knowledge about this is very small. But in the book "Rational points on elliptic curves" by Silverman and Tate, they certainly seem to indicate that all the rational points on a conic can be found with this method.

Can you explain me what makes this situation different from the Pythagorean triple situation?

LittleNewton

I guess I am not very good at this either
(otherwise I would have found the solution by now )

I think they are both the same. I want to know where to stop
finding the primitives and jump to finding the all other non-primitive solutions.
If the answer is that m,n should span 0..limit, then that is a really big range.
I want to know if this is the most efficient way, i.e. we can't avoid the big search.
I tried to trim it by jumping on evens for m, and on odds for n, etc.
but each case missed some solutions. I tried squareroot(limit), which worked.
I tried cuberoot(limit), seemed to work, but I guess it was case-specific.

LittleNewton

It might be that the only problem is efficiency:
How can we efficiently generate all solutions?
i.e. how can we avoid spending time on degenerate solutions,
because it seems like there are lots of them...

ramsey2879

My conjecture is stronger than that since 9*13 is an even power of 3 times a prime of the form 4n +1. 5*9 and 5*113^2 cannot be put into the form 4n^2+m^2 with n coprime to m [even] though 9 and 81 are even powers of 3

ramsey2879

My experience with Mathematica at this problem leads me to conjecture that the number of primitives is 1 if f is a power of 5, 2 if a power of a prime of the form 4n + 1 or 5 times a power of such a prime, for a product of n distinct primes of the form 4n + 1 or 5 times such a product, the number of primitives is 2^n, If f containes a factor of the form 4n+3 there is no primitive root.
My Mathematical program (for odd f )is the following:
t = 1;
tx = t + 401;
While[t < tx,
Print[t, Solve[
4*x*x + y*y == 5*t*t && GCD[x, y] == 1 && x > 0 && y > 0, {x, y},
Integers]]; t += 4]
for investigating a specific product like t = 13*17*37*29 I set tx = 1.

ramsey2879

I have a further observation: If 4x^2+y^2=f has at least one primitive solution and 5 does not divide f, then there are two primitive solutions for f (not divisible by 5) in 4x^2 + y^2 = 5f, (x1,y1) and (x2,y2) all positive values such that x2*y1 +/- x1*y2 = +/- 2f^2). Never mind my post about f being prime, f = 29*17 is a counter example to that conjecture.
Below is a proof;
The pell equation 4a^2+b^2 = 5f^2 has the solution set a = m^2 -2mn -4n^2, b = m^2 +8mn -4n^2, f = m^2 + 4n^2. If (m,n) form one solution set then (m,-n) will form a 2nd solution set (p,q). Also, my experience is that qa + pb = 2*f^2.

We need to show that q*(m^2-2mn-4n^2) + p*(m^2+8mn -4n^2) = 2*(m^2+4n^2)^2.

Substitute q =b'(m,-n) =m^2-8mn-4n^2 and p = a'(m,-n) =m^2+2mn-4n*2
then (m^2-8mn-4n^2)*(m^2-2mn-4n^2) + (m^2+2mn-4n^2)*(m^2+8mn-4n^2) =

m^4 -10m^3*n+(16-8)m^2*n^2+40n^3*m+16n^4 + m^4 +10m^3*n +
(16-8)m^2*n^2 - 40n^3*m + 16n^4 =

2*(m^4 + 8m^2*n^2 + 16n^4) = 2*(m^2+4n^2). Q.E.D.

ramsey2879

This was in response to LittleNewton saying f^3 appears to be an upper limit for searching for primitive solutions.

If you have some information on the upper range for searching for primitive solutions being the cube of f then would you provide it? I am not sure what you have in mind but am interested.

LittleNewton

I did a bit more experiments and found that I can get the range check down to
order of forth root: 2*limit^(0.25) to be precise.

Also using gcd = 1, skips many candidates.
Focusing on signs and parity also cuts it down.

I kept a hash of the results to avoid repeats,
but if I a nice method is outlined there shouldn't be a need
to store the results.

ramsey2879

I don't know what you mean by limit. I don't see why using gcd limits candidates should matter since if (d,e) > 1 then GCD(d,e,f) > 1 and I thought SeventhSigma was only interested in primitive solutions. As for limiting d,e to positive, it seems trival to test both d and -d when testing d will suffice, same for e.; but, then maybe you have a program to compute d and e that will give negative input. If so then go for it.