Timeline for Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $
Current License: CC BY-SA 3.0
13 events
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| Feb 19, 2018 at 19:41 | history | edited | Will Jagy | CC BY-SA 3.0 |
added 7 characters in body
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| Mar 9, 2014 at 19:26 | history | edited | Will Jagy | CC BY-SA 3.0 |
added 818 characters in body
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| Mar 29, 2011 at 18:15 | comment | added | Kevin Buzzard | Will [or anyone else reading]: the reason this question got bumped was that I edited my answer. Initially I wrote "here's a strategy to prove iff and it should work" but I spent some time recently trying to finish the job, and failed. I edited my answer (which is now rather a mess :-/ ) to reflect this. The answer still answers the question you raise, but it does not do the other implication (it doesn't show that if C isn't a solution to the Pell then C is represented). See my edit after the statement of Conj 1 in my answer. | |
| Jan 29, 2010 at 18:57 | vote | accept | Will Jagy | ||
| Jan 29, 2010 at 11:59 | answer | added | Kevin Buzzard | timeline score: 34 | |
| Jan 28, 2010 at 3:47 | history | edited | Will Jagy | CC BY-SA 2.5 |
new partial check
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| Jan 27, 2010 at 22:22 | history | edited | Will Jagy |
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| Jan 27, 2010 at 21:26 | history | edited | Will Jagy |
yet another tag
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| Jan 27, 2010 at 21:15 | history | edited | Will Jagy |
added tag
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| Jan 21, 2010 at 17:28 | comment | added | Will Jagy |
Personally, I believe it is iff in your succinct version. But research about indefinite inhomogeneous polynomials has been rare. Usually there are positivity restrictions, see R.C.Vaughan book The Hardy-Littlewood Method.'' So I do not know of any theorem that says an indefinite homogeneous polynomial represents any number unless there is a good reason for failure'' but that is something I believe to be true. And I believe the +-N symmetry, which fails in the related problem for 3 x^2 + 2 xy + 4 y^2 + z^3 - z^2 - z because of the z^2 term. See website. or email me. Thanks for appreciation
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| Jan 21, 2010 at 7:43 | comment | added | Kevin Buzzard | Much easier question: can someone show that N is of the form 2x^2+xy+3y^2+z^3-z iff -N is? Will: do you believe this? | |
| Jan 21, 2010 at 7:43 | comment | added | Kevin Buzzard | More succinctly, you're asking if it's true that a positive integer c is not of the form +-(2x^2+xy+3y^2+z^3-z) if (c,6)=1 and 27c^2-4 is 23 times a square? Is it even iff? What a great question! Looping with |x|,|y|,|z|<=100 I do indeed pick up every integer between -1000 and 1000 other than +-1 and +-599, and 1, 599 are precisely the two values of c less than 1000 and coprime to 6 satisfying the Pell equation. Of course I understand that you're asserting that you've computed much much further, I'm just (a) independently verifying a part of the calculations and (b) giving the problem a plug. | |
| Jan 21, 2010 at 2:40 | history | asked | Will Jagy | CC BY-SA 2.5 |