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-   -   Is this solvable in general. (https://www.mersenneforum.org/showthread.php?t=22578)

 jwaltos 2017-09-14 22:26

[QUOTE=retina;467759]You can replace x and y with any value you like. Simplifying and coalescing all the constants gives this [b]formula[/b]:

f(x,y) = a*x^3 + (b+c*y)*x^2 + (d+e*y)*x + f*y^2 + g*y + h[/QUOTE]

This approach, transforming the structure of the polynomial expression is necessary to better understand the context of what I am asking.

[QUOTE=R. Gerbicz;467798] Why would be your longer and higher/ degree polynom is easier than mine?[/QUOTE]
Because it is an optimal and very specific entry point to a difficult set of questions for which the conceptual tools may/may not exist to resolve them.. one way or the other. Every variable and operation has a very specific interpretation much like equations within general physics where dimensional analysis is applicable.
For example, the uniformization theorem and the geometrization conjecture are contexts that I have explored. As a non-specialist such topics are heavy going but worth it.

[QUOTE=CRGreathouse;467797]I think you've reduced factoring to a problem harder than factoring. :unsure:[/QUOTE]
I don't think so. I've come to appreciate that factoring is a very challenging problem requiring a special set of wits.
Sometimes making something more complicated initially will simplify things drastically..or create a huge intractable swamp. As Laplace once stated, "Read Euler.."

As will be noted through my posts I do have my weaknesses, however, what I have asked should be clear.

 jwaltos 2017-09-14 23:43

I should have included this earlier as this is an aspect of testing and implementing certain results obtained relative to the question asked above;
Regarding computation, in addition to mainstream CAS's has anyone worked with ACL2 and/or Vermaseren's FORM?

 science_man_88 2017-09-15 00:18

[QUOTE=jwaltos;467756]Soliciting advice on how to solve this equation:

-2*x^3+(19/2-60*a-60*b*y)*x^2+(21/2+314*a+30*d*y+30*c+314*b*y)*x+2+205*a+11*c+900*b*y*c+420*a*b*y+210*a^2+210*b^2*y^2+11*d*y+900*a*d*y+900*b*y^2*d+205*b*y+900*a*c

Assuming a, b, c and d are known, what is the best method to resolve the remaining variables: x, y, such that the resulting number is a specific integer.[/QUOTE]

well the only parts that can change it from even to odd etc. ( assuming all variable are integer) are 19/2 * x^2 , 21/2 * x, 205 * a, 11 * c, 11 * d * y, 205 * b * y, the rest would all set it as even in the integers. so if for example a,b,c,d are all even then it will come down to the ones that contain only x in that list to determine parity. other considerations can be made of course.

 CRGreathouse 2017-09-15 13:39

[QUOTE=science_man_88;467811]well the only parts that can change it from even to odd etc. ( assuming all variable are integer) are 19/2 * x^2 , 21/2 * x, 205 * a, 11 * c, 11 * d * y, 205 * b * y, the rest would all set it as even in the integers. so if for example a,b,c,d are all even then it will come down to the ones that contain only x in that list to determine parity. other considerations can be made of course.[/QUOTE]

If x is odd, you can multiply everything (including the hidden constant A that everything equals) by 2 to normalize the terms to integers. If x is even, then everything is already an integer.

 CRGreathouse 2017-09-15 13:59

[QUOTE=jwaltos;467810]I should have included this earlier as this is an aspect of testing and implementing certain results obtained relative to the question asked above;
Regarding computation, in addition to mainstream CAS's has anyone worked with ACL2 and/or Vermaseren's FORM?[/QUOTE]

These really aren't the right tools for the sort of investigation you're doing.

You want Sage or MAGMA for state-of-the-art integral or S-integral computations on elliptic curves, or [url=http://johncremona.github.io/ecdata/]John Cremona's tables[/url] if you know the conductor and it's small enough (less than 400,000 at present). Look for integral_points/S_integral_points in Sage and IntegralPoints/SIntegralPoints in MAGMA.

For subexponential algorithms on Diophantine quadratics -- generalized Pell's equation and the like -- PARI/GP is state-of-the-art as far as I know. See the help for the functions quadunit, bnfisintnorm, and qfsolve.

 Dr Sardonicus 2017-09-15 14:11

F(x,y,a,b,c,d) = -2*x^3+(19/2-60*a-60*b*y)*x^2+(21/2+314*a+30*d*y+30*c+314*b*y)*x+2+205*a+11*c+900*b*y*c+420*a*b*y+210*a^2+210*b^2*y^2+11*d*y+900*a*d*y+900*b*y^2*d+205*b*y+900*a*c

1) Does solving F(x,y,a,b,c,d) = N bear directly on factoring N?

2) If not, how does the value of N relate to the number to be factored?

3) Where does this function come from?

Without an answer to his last question -- which has already been asked twice, but not really answered -- it seems pointless to try to proceed. Phrases like "part of an integer factorization toolkit," "derived in my attempts to develop a simple method of factoring integers," "an optimal and very specific entry point to a difficult set of questions," or "Every variable and operation has a very specific interpretation much like equations within general physics where dimensional analysis is applicable" don't really address the question.

As examples of the sort of questions whose answers might actually be informative: Is F(x,y,a,b,c,d) obtained from a function in more variables by assigning values to some of them? If so, what was that function? Was F obtained from a homogeneous function?

 jwaltos 2017-09-15 16:22

[QUOTE=CRGreathouse;467829]These really aren't the right tools for the sort of investigation you're doing.

You want Sage or MAGMA for state-of-the-art integral or S-integral computations on elliptic curves, or [URL="http://johncremona.github.io/ecdata/"]John Cremona's tables[/URL] if you know the conductor and it's small enough (less than 400,000 at present). Look for integral_points/S_integral_points in Sage and IntegralPoints/SIntegralPoints in MAGMA.

For subexponential algorithms on Diophantine quadratics -- generalized Pell's equation and the like -- PARI/GP is state-of-the-art as far as I know. See the help for the functions quadunit, bnfisintnorm, and qfsolve.[/QUOTE]

Thank you CR. I have used Sage but not MAGMA (yet). MAGMA has an implementation from Denis Simon which pertains to my question and I have looked at the originating paper. Years ago, John Cremona was gracious enough to answer a question of mine, again allied to the question here, providing guidance for a successful resolution to that query. I also use Pari and agree with your assessment regarding certain state-of-the-art functionality. I am usually thorough researching a topic but I always try to account for blind spots, hence my posting. Again, thank you for your insights and bringing them to my attention.

[QUOTE=Dr Sardonicus;467831]

1) Does solving F(x,y,a,b,c,d) = N bear directly on factoring N?

2) If not, how does the value of N relate to the number to be factored?

3) Where does this function come from?

[/QUOTE]

1. Yes.
2. See above.
3. I invented it.

Sardonicus, I sincerely appreciate the time taken to respond but I would be repeating myself if I were to address your questions fully. Sorry.

Thank you to those who responded to my question as it has helped me to better understand how others view what I am looking at.

Moderators, this thread can be closed anytime now. Thank you.

 CRGreathouse 2017-09-15 17:21

[QUOTE=Dr Sardonicus;467831]3) Where does this function come from?

Without an answer to his last question -- which has already been asked twice, but not really answered -- it seems pointless to try to proceed. Phrases like "part of an integer factorization toolkit," "derived in my attempts to develop a simple method of factoring integers," "an optimal and very specific entry point to a difficult set of questions," or "Every variable and operation has a very specific interpretation much like equations within general physics where dimensional analysis is applicable" don't really address the question.

As examples of the sort of questions whose answers might actually be informative: Is F(x,y,a,b,c,d) obtained from a function in more variables by assigning values to some of them? If so, what was that function? Was F obtained from a homogeneous function?[/QUOTE]

[QUOTE=jwaltos;467847]3. I invented it.

Sardonicus, I sincerely appreciate the time taken to respond but I would be repeating myself if I were to address your questions fully.[/QUOTE]

I don't think you've made a [i]bona fide[/i] attempt to answer Dr Sardonicus' question, which is a pity -- it would probably help us answer your question. :down:

 CRGreathouse 2017-09-15 17:34

[QUOTE=jwaltos;467843]I also use Pari and agree with your assessment regarding certain state-of-the-art functionality.[/QUOTE]

Note that qfsolve is relatively recent -- added in 2.8.1 -- though Denis SIMON's script from which it was adapted has been around for ages.

 R. Gerbicz 2017-09-15 17:53

[QUOTE=CRGreathouse;467855]Note that qfsolve is relatively recent -- added in 2.8.1 -- though Denis SIMON's script from which it was adapted has been around for ages.[/QUOTE]

qfsolve solves a quadratic equation, so not playing here.

 CRGreathouse 2017-09-15 18:20

[QUOTE=R. Gerbicz;467856]qfsolve solves a quadratic equation, so not playing here.[/QUOTE]

I mentioned that in my post:

[QUOTE=CRGreathouse;467829]These really aren't the right tools for the sort of investigation you're doing.

You want Sage or MAGMA for state-of-the-art integral or S-integral computations on elliptic curves, or [url=http://johncremona.github.io/ecdata/]John Cremona's tables[/url] if you know the conductor and it's small enough (less than 400,000 at present). Look for integral_points/S_integral_points in Sage and IntegralPoints/SIntegralPoints in MAGMA.

For subexponential algorithms on Diophantine quadratics -- generalized Pell's equation and the like -- PARI/GP is state-of-the-art as far as I know. See the help for the functions quadunit, bnfisintnorm, and qfsolve.[/QUOTE]

But since we're missing all but the final step in jwaltos' process, it seems likely (given the origin of the problem) that there will be quadratics at some point, so I thought it would be useful to mention tools for working with them. And indeed, he says he works with Pari:

[QUOTE=jwaltos;467843]I also use Pari and agree with your assessment regarding certain state-of-the-art functionality.[/QUOTE]

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