# Sun's conjecture on sums of primes and triangular numbers

## Primary tabs

Type of Math Object:
Conjecture
Major Section:
Reference

## Mathematics Subject Classification

### Suggestion

You might want to make a manual link to EmpiricalProofThatEverySufficientlyLargeEvenIntegerCanBeExpressedAsTheSumOfAPairOfAbundantNumbers. Just a suggestion.

### conjectures from arxiv GM?

Posting an entry like this seems very strange. A conjecture from arXiv (entry in General Mathematics, i.e. Garbage Machine) seems like the last thing to be posted in the encyclopedia.

1) It is NOT known that this is a really important conjecture
2) It is NOT yet known if it in fact has been proved or disproved previously and just not noticed by the author.
3) It seems highly likely that such a conjecture could have been made long time ago and thus may have been made a long time ago. It shouldn't be given a "name" after whoever conjectured it on arxiv right now without doing lots of research. If it appears in a book or survey by somebody renowned in number theory and given this name, then OK.
4) Given that it just appeared on arxiv, it may also easily be false. Only conjectures with PLENTY of supporting evidence where many experts in the field find them plausible should have any chance of getting here.
5) General Mathematics on arXiv is dubbed Garbage Machine. That's where things generally considered crap are dumped rather than deleted.

I have tens if not hundereds of my own conjectures I could put in a paper, put it on arxiv and name them "Lebl conjecture on foo-bar." I have theorems that appeared in peer-reviewed literature, but I don't think they should appear here.

And most definitely things should NOT BE NAMED AFTER THE AUTHOR until well after the particular theorem or conjecture has been time tested, to be truly original and actually widely useful.

There are plenty of useful things still missing from the encyclopedia. Cluttering it with noise is not a good idea I don't think.

### Re: conjectures from arxiv GM?

There are plenty of useful things still missing from the encyclopedia. Cluttering it with noise is not a good idea I don't think.

Well, stuff the common man can understand is not noise. I hope PlanetMath gets more stuff like this and fewer mathematical masturbations like impractical geometries and fictitious algebras.

### Re: Hooray for Number Theory! Boo to Transfinite Lamarckian ...

Some impractical geometries might have applications in string theory or maybe even physics in general where concerned with black holes and other normalcy-bending phenomena. But fictional algebras make me wonder: What's so deficient with normal algebra that there's a need to invent other algebras? Isn't normal algebra trouble enough?

And who the heck calls ArXiv GM "Garbage Machine?" Nor ArXiv itself. Try this Google search:

"garbage machine" site:arxiv.org

So I'll tell you who. Elitists who j--- off to stuff like transfinite Lamarckian trundles. And _that_ ought to be called garbage.

To brave the rough surf of number theory, where proofs appear to be within grasp and elusive at the same time, now that takes moral fortitude!

### Re: conjectures from arxiv GM?

(My comments are not directed to the entry in question. I haven't read it and I don't know anything about it.)

I agree with jirka. I think the encyclopedia should only contain conjectures, results or definitions that are found to be "useful" or "interesting" and are accepted by the mathematical community.

Of course, this leads to the old debate about what is useful or interesting (or even "accepted") and what it is not.. But one way to avoid incorporating dubious material in the encyclopedia is by trying to:

- Never write an entry about an unpublished or recently published result.
- Never write an entry about something you worked in.
- As for definitions or conjectures, it is for better if they appear in books or at least are cited by many experts in the field.
- Be very careful when putting someone's name in an entry.

There are entries in the encyclopedia that, I think, do not really belong there.. Fortunately I believe they are a very small minority. But still they are a problem, and I don't know what can be done about them besides appealing the authors to be reasonable about their entries.

### Civility

Whether or not the entry belongs in the encyclopedia, this sort
of diatribe does not belong in the forum. While there is nothing
wrong with debating whether it is a good idea to write an entry
inflammatory language --- not only does bickering clutter up the
forum with garbage, it lowers the reputation of the site as a
whole and repels people who might be interested in discussing math
here. As our community guidelines say:

"If you do have disagreements and want to discuss it in public, do it
in a constructive manner. Try not to make judgment or use sarcasm in
any negative way. Do not make unfounded accusatory comments. Avoid
starting a flame war."

### Re: conjectures from arxiv GM?

Please post your conjectures at Open Problem Garden, a site specifically intended to put math conjectures there:

http://garden.irmacs.sfu.ca/
--
Victor Porton - http://www.mathematics21.org
* Algebraic General Topology and Math Synthesis
* Category Theory - new concepts

### Re: conjectures from arxiv GM?

I can make plenty of "conjectures" that anybody can understand. There are infinitely many simple "facts" about the natural numbers. That does not make all of them useful, insightful, or viable for an encyclopedia. Most statements one can make about natural numbers are just that: noise, albeit true.

arXiv is where cutting edge new research appears. A large amount of it is also crap that will never get published. I am not saying that this necessarily is crap. If in 10 or 20 years this appears in several places in the literature, then that's when it has maybe earned the right to be in an encyclopedia like this, IMO.

About "garbage machine": The moderators for mathematics do not in general remove articles, no matter how stupid they are. If an article appears like crap (much lower standards than any real peer reviewed journal) then it is reclassified to "General Mathematics." Hence it has earned the nickname "Garbage Machine."

My big complaint is putting things in Planetmath or Wikipedia or wherever just to be able to put your (or someone you know) name on it. Which is what this seems like to me. Even worse this is an uproven conjecture.

I'm not sure what "impractical geometries" and "fictitious algebras" you are talking about. There are other entries like the infamous Smarandache nonsense which have no merit.

IMO, no concept should be in the encyclopedia unless it appears in at least one book from a major publisher. Or at least a survey article in a well respected peer-reviewed journal.

If you go through all "introduction to ..." books in your library, we most likely have but a small percent of all major concepts / theorems from those books in the encyclopedia.

### Re: conjectures from arxiv GM?

It is with great trepidation that I put Zhi-Wei Sun in the same paragraph with Florentin Smarandache. One is a respected mathematician with a kind of prime number named after him (an assignment accepted by both the OEIS and MathWorld), who edits the International Journal of Modern Mathematics and has Erdos number 2. The other teaches at a community college and his Erdos number is at least 4, and many don't even want to give him that.

Also, even "impractical geometries" and "fictitious algebras" can have their uses, such as the string theory in physics which someone already mentioned. But if the only use of an impractical geometry or fictitious algebra is to serve as a status symbol, then I think it would be better off in a desk's bottom drawer.

### Very interesting conjecture

PrimeFan, thank you very much for informing us about this very interesting conjecture. Sun's paper may not be on the level of the landmark Tao-Green paper on primes in arithmetic progression (hailed by Neil Sloane when it was an ArXiv preprint) but it certainly is very erudite and illuminating. Sun's papers on mixed-kind sums are already generating buzz.

Having read Sun's paper on primes and triangulars a couple of times, I think this is the most fascinating conjecture of the lot. The other ones are still interesting, but this one is more so for the fact alone that its exception set has a single element {216}. At times I too feel like the proof of it is within my grasp and then it slips away. Of course, if it has eluded Sun, I doubt I could catch it.

Lisa

P.S. to jirka: a proven conjecture is no longer a conjecture. ;-)

### Re: conjectures from arxiv GM?

Even if Erdos was still alive and posted an article on arxiv with some sort of conjecture, I do not think it should go into the encyclopedia as "Erdos conjecture on foo-bar" the moment it appeared there. This is especially true if said article was relegated to GM by the arXiv moderators. Even if said conjecture appeared in a research article in a peer reviewed journal would I be against putting it immediately on planetmath. There are plenty of wonderful things that appeared in the Annals that were either wrong, incomplete, or proved unimportant with the passage of time.

BTW, Smarandache also has stuff on OEIS and MathWorld, neither of the two does too much editorial scrutiny. That is not to say I am considering Sun a crank, or a hack or anything of the sort. I understand he is a well known mathematician within his field with many articles published in respectable journals (as a quick search of mathscinet reveals). That is not the point.

### Re: The Zumkeller-Lebl conjecture is just an expansion of Ch...

Given that Zumkeller thought this up by himself back in 2004, long before Ji\V{r}i Lebl did with a large amount of venomously derisive dismissiveness, I don't think Lebl merits to have his name associated with such an interesting open problem.

### Re: The Zumkeller-Lebl conjecture is just an expansion of Ch...

That would be viciously petty, and this whole business clearly calls for turning the other cheek. Besides, it would be passing up an opportunity to use haceks.

### Re: The Zumkeller-Lebl conjecture is just an expansion of Ch...

You're right.

So, I've been thinking about this Zumkeller-Lebl conjecture, and how it could possibly be proven or disproven. It really unifies Chen's theorem with various conjectures, such as Goldbach's conjecture and Levy's conjecture. Proving Zumkeller-Lebl could automatically prove the other ones, but proving the other ones would not necessarily prove Zumkeller-Lebl.

### Re: The Zumkeller-Lebl conjecture is just an expansion of Ch...

Let's not forget that Zumkeller only stated it in regards to distinct primes. Ji\V{r}\'i Lebl's contribution was also to consider nondistinct primes, which greatly increases the probability of a number having a representation as stated.

### Re: The Zumkeller-Lebl conjecture is just an expansion of Ch...

Good point. Lebl said that the exception sets (obtained with non-optimized code), which he stated explicitly, were much smaller when nondistinct primes were allowed.

### Re: The Zumkeller-Lebl conjecture is just an expansion of Ch...

Though it wouldn't hurt to doublecheck the smaller exception sets. They give way too many results when looked up in the OEIS, whereas the bigger exception specifically gives the relevant sequence.

### Re: The Zumkeller-Lebl conjecture is just an expansion of Ch...

I see that. 1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,24,30,36,42,60 brings up just A100952. 1,2,3,4,5 brings up over three thousand results, which is way too many to go over. 1,2,3,4,5,6,7,14 brings up just six results, four of which pertain to base 7, one to base 8 and one to base 10.