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

Conjecture. (Zhi-Wei Sun) Any positive integer $n\neq 216$ is either a prime number $p_{i}$, a triangular number $T_{j}$ or the sum of a prime number and a triangular number $p_{i}+T_{j}$.

For example, 47 = 37 + 10; 37 is a prime number and 10 is a triangular number. (There are other solutions).

Although there are infinitely many triangular numbers, they are spaced farther apart as their indexes get larger. But no matter how large $T_{j}$ might be, the next higher triangular number is always $j+1$ away. To put it algebraically, $T_{{j+1}}=T_{j}+(j+1)$, with $T_{1}=1$ (or $T_{0}=0$ if one likes). It is also true that there are infinitely many prime numbers, but the distance between one prime and the next is not as neatly predictable as it is for the triangular numbers; there is no known simple formula that takes $n$ as its sole argument and gives the $n$th prime as its output. There could very well be a record-setting prime gap that is immediately followed by a prime quadruplet, but that’s another conjecture altogether. The point is that the larger $n$ is the less likely it is to be a prime or a triangular number. From this line of reasoning, it would be easy to jump to the conclusion that it is also less likely to be the sum of a prime and a triangular number.

That would be hasty. In contrast to the decreasing likelyhood of primality or triangularity, for larger $n$ the value of $\pi(n)$ (the prime counting function) increases, as does $\tau(n)$ (the triangular number counting function, here represented by that much overloaded Greek letter). Thus there is a greater likelihood that one of the values of $n-p_{i}$ (for $0<i<\pi(n)$) will be a triangular number, or that one of the values of $n-T_{j}$ (for $0<j<\tau(n)$) will be a prime number. For most numbers less than 512, there is more than one representation as the sum of a prime and a triangular number.

There are many theorems and conjectures in additive number theory in the mold of “Every sufficiently large number is the sum of two or three numbers of a specific kind,” with the word “sufficiently large” indicating a greater than inequality. For example, every even $n>46$ is the sum of two abundant numbers. So, besides involving numbers of two different kinds rather than two or three of the same kind, this conjecture is also unusual in that it gives a negated equality rather than a greater than inequality.

Sun has checked it for $n$ below 17000000. He admits 0 as a triangular number and almost tacitly 0 as a prime number. The conjecture has been stated the differently here so as to not identify 0 as a prime number. In the range 1 to 512, only in the cases of 2 and 61 is it necessary to accept 0 as a triangular number. The differences $216-T_{j}$ are: 6, 26, 45, 63, 80, 96, 111, 125, 138, 150, 161, 171, 180, 188, 195, 201, 206, 210, 213, 215; a list that includes some semiprimes.

# References

- 1 Zhi-Wei Sun, “On Sums of Primes and Triangular Numbers” ArXiv preprint, 10 April (2008): 2, Conjecture 1.2

## Mathematics Subject Classification

11A25*no label found*

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## Comments

## 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

about this conjecture, there is a lot wrong with using nasty,

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.

## transfinite Lamarckian trundle theory

Let $U$ be an universe with a variable number of dimensions $\mathcal{W}$. The number of dimensions $\mathcal{W}$ is a number satisfying $\mathcal{W}$ \not\exists \mathbb{C}$ if $$\bigcup_{i = {\mathcal{O}\rho}^{aleph_{\mathcal{W}} U(\mathbb{M})$$, and certainly never $\mathcal{W} \exists \mathbb{Z}$, that being absolutely forbidden with utmost prejudice. When the frame of movement $\mathcal{J}$ matches $$\int_{\mathcal{O} - \rho}^{aleph_{\mathcal{W} + \rho} U(\mathbb{M}\mathcal{W}),$$ the excess dimensions of $U$ are shifted into the category $\mathcal{C}$ having a pointed topology $\rho$. If the frame moves both slower and faster than $$\int_{\mathcal{O} + \rho}^{aleph_{\mathcal{W} - \rho} U(\mathbb{M}\mathcal{W}),$$ it becomes a trundle which both can and cannot revert to a bundle except if $U$ ceases to exist prior to its creation. But if a parallel dimension $\rho\epsilon$ with Klerbian geometry intersects the principal dimension of $U$ at regular intervals, bundle reversion can occur if viscosity $\Sigma\star$ holds true. Otherwise the trundle is maintained but is no longer transfinite.

Our own universe, owing to its many deficiencies, is incapable of maintaining any sort of Lamarckian trundle at any time.

## Re: conjectures from arxiv GM?

Maybe you should've suggested PrimeFan remove Sun's name from the title. That might've played a lot better with number theory aficionadi than referring to number theory as "garbage."

Somehow I doubt John "PrimeFan" Smith is related to Zhi-Wei Sun, or even knows him personally.

And yes, Smarandache does have some stuff in the OEIS, Mathworld and Wikipedia. It's very little compared to Sun. And what little there is at Wikipedia, has been fought over so much you can be sure the crap has been long flushed. I myself have defended Smarandache-Wellin numbers, been lukeward about the Smarandache function and argued against generalized Smarandache palindromes.

## Re: conjectures from arxiv GM?

(For those that are confused: I have nothing against number theory and I do not think that Sun is a bad mathematician)

I did not refer to number theory as garbage, I have no clue where you got that. I only mentioned that "General Mathematics" on arxiv is colloquially known as "Garbage Machine." Of course it is not officially known as that. Hence you won't find that terminology on arxiv itself. All mathematicians I know would never ever read any paper in GM. Try looking through the GM section to see why.

When a paper is relegated to GM on arxiv it does mean that one of the moderators thought the paper was probably not good enough to put into the NT section. Given the already low standards of arxiv, it is not saying much to get a paper not reclassified.

But being part of arxiv GM was only a small part of my issue. I think the conjecture and similar conjectures should not appear on planetmath at all. At least until they become widely accepted as useful or important. And that's especially true of conjectures.

If PrimeFan just thought the conjecture is interesting and wanted to point it out, it is better to just post on the forums about it. The planetmath encyclopedia should not become arxiv, we already have one arxiv. This should be an encyclopedia of well established concepts. Something just appearing on arxiv is NOT well established. Arxiv only does very cursory moderation, it does not do peer review at all. Even if the paper is correct, it is not known that this a fruitful or a useful avenue leading anywhere other than giving yet another true statement about natural numbers.

It also does not matter that PrimeFan does not know Sun. It is still wrong to name things like this. Important theorems/conjectures/definitions are given someone's name after a long time in print, and only by respected experts in the field. Even if no ego stroking was involved, giving something a name on planetmath will only confuse matters. Only give facts, unless a name is already established. Citing a paper is fine, but not to make up new terminology.

So to sum up:

1) I think the entry shouldn't appear here at all.

2) I think it should not have anyone's name attached to it if it does.

PS: I have not seen a single Smarandache entry which is justified. Yes some of the things he proves are true. So what. There is no motivation behind them and they have not found general use.

## Re: Very interesting conjecture

I also agree. That could very well be the most interesting thing about 216, even more interesting than the fact that it is the sum of three consecutive cubes. Zhi-Wei Sun, befriending the integers, shows himself to be in the noble spirit of Ramanujan.

## Re: conjectures from arxiv GM?

I completely agree.

I think most people have been debating the wrong issues here. This is nothing against number theory, nor against Sun. Also, this not a question of how easy or difficult a subject may be or how many applications it has (what the hell is a ficticious algebra, by the way?). It's about adding fairly recent and unpublished material to the encyclopedia (with someone's name on it).

The encyclopedia should contain solely time-tested concepts!

I could write an entry about Hilbert's approach to Operator Theory and derive some consequences in other areas. But it turned out that Hilbert's work in this subject (although a major breakthrough) was not the most insightful way to the this. And no one does that anymore for more than 50 years! Any entry I could write using Hilbert's approach would be considered silly by any expert in Functional Analysis or Operator Theory!

Of course, this doesn't mean that Hilbert was not a great mathematician nor that Operator Theory is not important!

The PM encyclopedia should be treated as an encyclopedia, not as a journal of mathematics!

P.S. - There are other problematic entries besides the Smarandachian ones.

## 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.

Yes, there are plenty of things missing from the encyclopedia. Zero was missing for a long time, with things like the transfinite Laplacian trundles getting added much earlier (and we're still waiting for history of zero!). Not that those don't belong. But when tons of broad, general terms get linked to extremely narrow and rare applications, that IS noise. To complain about a conjecture which could very well have come from the 18th Century that is titled with a suitably long title that won't get autolinked irrelevantly.

PlanetMath should have something about singular Levi-flat hypersurfaces in complex projective space. But the person adding that entry should have the _decency_ not to put "flat" in the Defines list! Nor "space" nor "sides" nor "both"!

## Re: conjectures from arxiv GM?

> But the person adding that entry should have the _decency_ not

> to put "flat" in the Defines list! Nor "space" nor "sides" nor "both"!

No, that is what linking policies are for. It would be alright to

put words like "flat", "space", and "sides" on the list of defined

terms of your hypothetical entry provided that one would also make

a linking policy for that entry which would confine linking of those

terms within the same subject area, thus preventing them from

appearing in entries, such as entries on elementary geometry, where

they should not be linked.

## Re: conjectures from arxiv GM?

When were the prime numbers created? By the ancient Greeks? Or maybe by God at the Big Bang? And how about triangular numbers? Maybe God would let the Pythagoreans claim those. Or who knows, maybe one of Sun's ancestors from thousand of years back came up with the same result.

1) I think the entry shouldn't appear here at all.

So let's delete the entries on prime numbers and triangular numbers while we're at it!

## Re: conjectures from arxiv GM?

Let us illustrate what I'm complaining about. Let me take a stab at this conjecturing game. This took me the last 15 minutes to come up with and test on a computer:

Conjecture: Every natural number except those in the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 30, 36, 42, 60} can be written as the sum

p + q r

Where p, q and r are DISTINCT primes.

Furthermore, if you do not require p,q,r to be distinct, the list of exceptions is just the list {1,2,3,4,5}. If you only require that q

and r are distinct, the list of exceptions is {1,2,3,4,5,6,7,14}.

I have quickly verified the conjecture up to 100000 by very unoptimized computer code.

Is it interesting? I argue, with high probability it is not. Is it original? I have no clue, I doubt it. Does it have an easy proof? Don't know, haven't tried. Should it be in the encyclopedia? When you show me at least a survey paper that mentions it (NOT wikipeida, mathworld, OAIS, usenet, etc..), then and only then yes.

Just because the statement is simple and elementary to understand, does not make it worthy of an encyclopedia.

## Re: conjectures from arxiv GM?

Huh? Would you please read what I wrote before replying rather than reading a single sentence of what I wrote?

## Re: conjectures from arxiv GM?

Contrived as it is from such a high level of animosity against number theory, as opposed to arising in a paper you submit for publication, I doubt anyone would care to investigate that conjecture you've just given right now. Sun's conjectures from his papers, on the other hand, have inspired others to explore them.

## Re: conjectures from arxiv GM?

I really don't think anyone has expressed any animosity towards number theory; moreover, keep in mind, just for the record, that not all number theorists are concerned with twin primes and representing a positive integer as the sum of a prime and a triangular number...that's just one small area of number theory. Regarding the object recently added by PrimeFan, while I agree that it's place in the encyclopaedia may be questionable, at least it is a well-written article conforming to PlanetMath stylistic standards. If we're truly concerned about the quality of entries, we should be looking at the various entries with "failure function" in the title, which are written in a manner unsuitable for the encyclopaedia (some still have type-oh's), and which are related to a concept which I have yet to find satisfactorily defined here. Moreover, if all the Smarandache nonsense and topics like Mangammal and Shantha primes are included on PM, I don't see how we can argue against PrimeFan's entry.

## Re: conjectures from arxiv GM?

> ... at least it [PrimeFan's entry] is a well-written article conforming to PlanetMath stylistic standards

I must disagree. The entry is written in a colloquial and vague style that has nothing to do with the way mathematics should be written. Expressions such as "thin out on the approach to infinity, but never completely disappear", "but this thinning out is fitful and erratic", "a healthy proportion", etc are just too vague for an encyclopedia. These are musings by the author which belong in the forum not in the collection. Let me be more specific:

1) "there is no simple recurrence relation that gives the next prime number from just the knowledge of that prime's index and value." This is not true, i.e. we do not know if this is true or not. None have been found but nobody has proved that there are no such formulas. What does the author mean by "simple" anyway? There are formulas that spit out the nth prime solely from the value of n, albeit PrimeFan would not find them to be "simple" I imagine.

2) "That would be hasty. In contrast to the decreasing likelyhood of primality or triangularity, for larger $n$ the value of $\pi(n)$ (the prime counting function) increases, as does $\tau(n)$ (the triangular number counting function, here represented by that much overloaded Greek letter). Thus there is a greater likelihood that $n - p_i$ (for $0 < i < \pi(n)$) will be a triangular number, or that $n - T_j$ (for $0 < j < \tau(n)$) will be a prime number."

That is not a proof by any means that the probability that n-p_i is a triangular number increases. It is a heuristic without much explanation or validity. Again, these are the author's musings and intuition which may be right or wrong, but they do not belong in an encyclopedia which others may read and believe to be facts.

3) "In fact, generally, for most numbers there is more than one representation as the sum of a prime and a triangular number." The 'generally' invalidates the whole sentence. If that sentence was true, the conjecture would follow, so I see no point to this remark unless the author is willing to provide a reasonable heuristic to the number of representations for every integer.

4) "I have stated the conjecture differently (I would rather accept 1 as a prime before accepting 0 as such) but with the same meaning. I've only checked it up to 512,..." who is "I" in an encyclopedia entry?? A entry should not refer to the author and references to the writer should be avoided at all costs. Instead, it should be rephrased in the passive voice: "The conjecture has been stated differently here..."

Etc.

T

## Re: conjectures from arxiv GM?

Alright, I agree with you on these, but if you are going to point them out, then you should really take issue with objects like Failure Function Exponential Function or Shantha Prime...these give virtually no information at all and the latter includes vague assertions like those you're citing in PrimeFan's entry. At the very least PrimeFan's topic is mathematical in nature. Anyone could make up a name, like Kidwell Prime, and say that it's any prime of the form 2^j-1, where j is the class number of a number field of degree a power of 2...these types of things really do not belong in the encyclopaedia.

## Re: conjectures from arxiv GM?

> Alright, I agree with you on these, but if you are going to

> point them out, then you should really take issue with

> objects like Failure Function Exponential Function or

> Shantha Prime...

Completely agree, there are far worse entries. Perhaps the newly appointed board members will finally get a Content Committee rolling and

those entries will be taken care of properly.

T

## Re: conjectures from arxiv GM?

The Content Committee will review all of these entries and determine their appropriateness.

In the meantime, if you see an entry that you think is not appropriate, please post a correction notice to it. Hopefully it will get resolved there. If not, and if you still feel that the entry is inappropriate, you can inform any one of the Content Committee members:

Ray (rspuzio),

Roger (rm50),

Ratboy,

Warren (Wkbj79), or

Chi (CWoo)

## Re: conjectures from arxiv GM?

Although I agree with Torquemada, that the entry is rather vague, I think you touched a very important aspect: there are entries whose content is completely unsuitable and lowers the quality of the encyclopedia. Much more than PimeFan's entry!

Still, the fact that there are much worst entries isn't a reason for me to accept (eventually) a well written entry coming directly from the arxiv.. but if I had to complain, yeah, PrimeFan's entry wouldn't be my top priority!

## Re: Linking policies

I had stopped paying attention to that. Now when I see links in entries I've just created, I just assume they will be irrelevant. If any entry of mine is showing up as an irrelevant link, I would appreciate being told.

## The Zumkeller-Lebl conjecture is just an expansion of Chen's...

Conjecture. (Reinhard Zumkeller \& Ji\V{r}\'i Lebl) Every natural number $n \exists \mathbb{N}$ except those in the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 30, 36, 42, 60} can be written as the sum $p + qr$, where $p$, $q$ and $r$ are {\em distinct} primes.

Furthermore, if you do not require $p$, $q$ and $r$ to be distinct, the list of exceptions is just the list {1, 2, 3, 4, 5}. If you only require that $q$ and $r$ are distinct, the list of exceptions is {1, 2, 3, 4, 5, 6, 7, 14}.

----

According to Jonathan vos Post, the Zumkeller-Lebl conjecture "effectively extends" Chen's theorem from just even integers to both odd and even integers.

Chen, J.-R. "On the Representation of a Large Even Number as the Sum of a Prime and the Product of at Most Two Primes, II." {\it Sci. Sinica} {\bf 21}, 421-430, 1978.

Hardy, G. H. \& Wright, W. M. "Unsolved Problems Concerning Primes." Appendix Â§3 in {\it An Introduction to the Theory of Numbers}, 5th ed. Oxford, England: Oxford University Press, pp. 415-416, 1979.

Ross, P. M. "On Chen's Theorem that Each Large Even Number has the Form $p_1 + p_2$ or $p_1 + p_2p_3$." {\it J. London Math. Soc.} {\bf 10}, 500-506, 1975.

## 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.