Talk:Polite number

"Powers of two are impolite numbers" Should this not be "multiples of two?"

Rationale: if n is an integer, two consecutive integers could be expressed as n, n+1.

n + n + 1 = 2n+1 = any odd integer.

But I didn't want to jump to correcting right away.

64.231.183.253 13:23, 16 August 2007 (UTC)

You are incorrect, because you only proved that the odd numbers are polite, not that even numbers are impolite. For example, 6 = 1 + 2 + 3 is even and polite. Samohyl Jan 17:19, 16 August 2007 (UTC)

Oops. Didn't see the 'or more' consecutive integers. Thanks.

137.207.238.106 13:59, 17 August 2007 (UTC)

So many references

I question the value of some of these additions. I have no intentions of deleting here, but I ask if the bloat justifies the content. Tell me if I am really being too picky.

Two of my main beefs are 1) Terminology creation which serves no real purpose. 2) Citations of articles which really add very little besides making the Wikipedia article look highly footnoted.

It is a common problem for beginning number theory or even pre-college math: How many ways can n be expressed as a sum of consecutive integers?

Sometimes people present this nice problem in a journal for teachers or students. Sometimes someone solves the problem, perhaps only partially, and their solution is published in such a journal.

The name "polite number" (in one or another presentation of the problem) is not touted as an important or even worthy new class of numbers, just a shortcut in that paper until one realizes it means - not a power of 2.

Do we need to cite a 1 page paper in The Mathematical Gazette to see that a triangular number with unique odd divisor is the product of an odd prime p with (half of) either (p-1) or (p+1) where that other factor has no odd factors, i.e. is a power of two. Hence a prime of the form 2^k-1 or 2^k+1 which is a Mersenne or Fermat prime. P.S. 1 is an odd divisor so really it should say unique odd divisor besides 1 or something.

Sylvestor's result is that the number of partitions of n into distinct parts which fall into k maximal sequences of consecutive integers [i.e. take a trapezoidal number T color the rows in 2k-1 stripes black white black white..black where stripes can have various widths but all at least 1 erase the white stripes, obtain a partition of n] IS the same as the number of partitions into odd numbers with repeats allowed but only k distinct odds showing up.

Is that SO related to polite numbers? Note that T above could be a single integer or a triangular number. Or do we want to note that a temporary term in that paper (which does have a great title) can be interpreted to include trapezoidal numbers (along with single numbers and triangular numbers)

It sounds good to have an article from the Journal of Integer Sequences but is it really? It is an online journal so why not give a link? http://www.emis.de/journals/JIS/trapzoid.html

One looks and finds: I was playing with my 3 year old who is so smart she can already count. I know that primes are the numbers that can't be made into nontrivial rectangles. I will define T.n to be the number of trapezoid arrangements including a single row and a triangular arrangement. I found a short sequence with numbers that have only 3 and 5 as factors. Does that continue? I can't find it in the old print version of OEIS. OK I found it in the OEIS. T.n is number of odd divisors but here is a program to find it.

There is just nothing new there, why mention it?

OK I'll stop. --Gentlemath (talk) 21:30, 26 March 2009 (UTC)

There are several different questions here:

1. Is the bijection between odd divisors and consecutive sums sufficiently noteworthy to write an encyclopedia article about? Clearly yes, because so many published papers have noted it. It has been noted, ergo it is notable.
2. What is the point of including so many citations to papers that all have the same content? To hammer home the fact (that I find quite impressive, although it is less about mathematics and more about academia in general) that the same idea has been discovered repeatedly by so many mathematicians working independently of each other. If I could point to a survey article and say "many other mathematicians have rediscovered the same result; see [so and so] for a listing" I would instead, but I don't know of such a survey; Apostol at least has this idea of trying to find past work on the subject, but his bibliography is short.
3. Why call it "polite numbers"? Because we have to call it something and that name has been used in the literature. I have no great attachment to it but I think using that name is greatly preferable to titling this article "Equivalence between odd divisors and representations as sums of consecutive positive numbers" or something equally cumbersome.
4. Since the number sequence is just the complement of the powers of two, why not include this material in our article about powers of two rather than making it a separate article? I took that attitude some months ago and suggested a merge, but this was declined after some discussion and I now agree with that decision. Because to me, the article isn't about powers of two at all, it's about the relation between divisibility by odd numbers and consecutive sum representations, neither of which applies to powers of two. We want to define concepts by what they are, rather than by what they are not.
5. Is Sylvester relevant? Yes. Almost everything in this article was proven by him and then proven again later by other people. Sylvester does explicitly talk about the "first class" case, not just the more general result from which this case follows.
6. Why don't we count 1 as an odd divisor, and correspondingly why don't we count n = n as a consecutive sum? I agree, it would be more logical to do it that way. And that would obviate the "isn't this the same as powers of two" objection. But most of the literature doesn't allow n = n and makes a big deal about the exclusion of the powers of two, and rather than committing original research we should follow the literature to the extent that it's reasonable.

—David Eppstein (talk) 21:56, 26 March 2009 (UTC)

Polite number is the term used in presenting the problem to 7th graders (see http://www.wyevalley.bucks.sch.uk/Curriculum/KS3/Mathematics%20VLE/Level%204%20Pack%201/L4Pack1_26.pdf )

This is a popular problem to give kids at that level and many of the references are of that type.

Do you at least peruse articles before adding them to the references? In 1999 CHRIS JONES 80 Eton Avenue, East Barnet, Herts EN4 8TY NICK LORD Tonbridge School, Kent TN9 IJP were a student and a school teacher

In 2005 Chris was noted for his success at fives (so he can't have been that old in 1999) Soon he will be cloned all over the internet as a trail-blazer for trapezoidal numbers.

I suppose that it is petty of me to be tracking down that info, but it just seems clear that this is not real notable.

It says something about academia that the solution to a middle school problem can get published over and over. But these journals are not really for breaking new ground. They are for encouraging people to engage.

I will stop watching this article and go on to other things but as final remarks:

Mason (1912) gives that the number of expressions as a sum of consequtive integers where we include just one integer (itself) and negative integers is twice the number of odd divisors. Not that different.

Leveque is considering the average number of representations over a range.

You say above:

I agree, it would be more logical to do it that way. And that would obviate the "isn't this the same as powers of two" objection. But most of the literature doesn't allow n = n and makes a big deal about the exclusion of the powers of two, and rather than committing original research we should follow the literature to the extent that it's reasonable.

That is just not so. Half your references do allow n = n. The big deal about excluding powers of two is maybe for middle schoolers (or high schoolers) to have a fun problem and get started on discovering the more general result.

Sylvestor is great stuff. He does not belong in Polite numbers. If you don't want another title then you need to say:

The only polite numbers that may be non-trapezoidal are the triangular numbers with only one odd divisor **aside from the divisor 1**

To see the connection between odd divisors ** besides 1** and polite representations

When you say:A generalization of this result states that, for any n, the number of partitions of n into odd numbers having k distinct values equals the number of partitions of n into distinct numbers having k runs of consecutive numbers.[13][26][27] The bijection between polite representations and odd divisors is the case k = 1 of this result.

You should say:

1)The bijection between polite representations and odd divisors **greater than 1** is the case k = 1 of this result with the divisor 1 and the singleton partition removed.

2) His result allows runs of length 1 but requires a missing number between runs

Sylvestor has a nice result, the trivial case is here. He uses generating function techniques and may not give an actual bijection.

Stockhofe, Dieter Bijektive Abbildungen auf der Menge der Partitionen einer natürlichen Zahl. (German) [Bijective mappings on the set of partitions of a natural number] Bayreuth. Math. Schr. No. 10 (1982), 1--59 may be the first actual bijective proof of Sylvester's very general result (or maybe I am wrong).

OK Peace, --Gentlemath (talk) 05:38, 27 March 2009 (UTC)

Re: Sylvestor is great stuff. He does not belong in Polite numbers.: this is pure snobbery and an inappropriately neutral sentiment to be using to guide our editing here. Sylvester did serious mathematics. This article is not serious mathematics. But it does not follow that Sylvester does not belong here. Did you read his paper? He handles the "first class" case combinatorially before moving on to generating functions. Your idea that the work of a schoolkid (when reliably published) is too declassé to use here is, again, inappropriate snobbery, as is your bizarre implication that notability has anything to do with mathematical depth. But I'll have to look at the Stockhofe paper (assuming Google translate can help enough with the German for it to make sense to me) — thanks for the reference. —David Eppstein (talk) 05:48, 27 March 2009 (UTC)

Etymology and notability of the term "Polite number"

To the casual reader, this article seems to be the ramblings of disturbed individuals who obsessively over-analyze the obvious and create new terminology for integers that aren't a power of two. I suggest detailing the origin and history of the term "polite" and the noteworthiness of its use to describe such numbers in mathematical circles. Krushia (talk) 14:39, 31 December 2012 (UTC)

Most of your concerns are addressed in the long discussion above your comment. I suggest you read it. In particular, the name "polite" isn't what's significant about this subject. —David Eppstein (talk) 18:07, 31 December 2012 (UTC)

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