High School Popularizing a property for n-bonacci numbers without publishing it?

[MevsEinstein]

TL;DR

I found a super cool property of n-bonacci numbers but it isn't really worth publishing. Ho do I make it known?

Hi PF!

Everyone knows that: $${\varphi }^2 - \varphi - 1 = 0$$ But guess what? $${\varphi}^3-2{\varphi}^2+1=0$$ Generalizing this for all n-bonacci numbers: $$x^{n+1}+1 = 2x^n$$ where $x$ is the n-bonacci number and $n$ is the degree of the polynomial that the n-bonacci number is a root of, i.e: $x^n-x^{n-1}-...-x-1=0$. To prove this property, what we do is to use the fact that all the terms of lower degree than $n$ are in a geometric series with ratio $x$ and with a beginning term $-1$: $$x^n-\frac{1-x^n}{1-x}=0$$$$x^n=\frac{1-x^n}{1-x}$$$$x^n-x^{n+1}=1-x^n$$$$[{2x^n=x^{n+1}+1}]$$ Now this is a very small property that isn't worth writing a paper on it (I think), but it's cool. Writing in Physics Forums is pretty good already, but how do I spread it quicker?

 

[jedishrfu]

Before you publish it, you might want to talk with a math prof. It may well be that your property is easily derived from a known property or it is a known property to those experts in the field.

 

[malawi_glenn]

I'd say write a short paper about it in LaTeX with everything: introduction, history, your lemma and its proof and bibliography. It is good practice nevertheless.

 

[MevsEinstein]

I'd say write a short paper about it in LaTeX with everything: introduction, history, your lemma and its proof and bibliography. It is good practice nevertheless.

Yeah I thought that since the paper would probably be too small I wouldn't be able to publish it. But maybe, who knows.

 

[TeethWhitener]

It’s quite straightforward to derive:
$$\varphi^2-\varphi-1=0\Rightarrow\varphi^2-\varphi=1$$
$$\therefore\varphi^3-2\varphi^2+1=\varphi^3-2\varphi^2+\varphi^2-\varphi=\varphi^3-\varphi^2-\varphi=0$$
Pull out a factor of $\varphi$ to get the original expression.

 

[MevsEinstein]

$$\therefore\varphi^3-2\varphi^2+1=\varphi^3-2\varphi^2+\varphi^2-\varphi=\varphi^3-\varphi^2-\varphi=0$$

Where did you get $\varphi^3-2\varphi^2+\varphi^2-\varphi = \varphi^3-2\varphi^2+1$? I don't understand this part.

 

[TeethWhitener]

It’s in the first line:
$$\varphi^2-\varphi=1$$

 

[jedishrfu]

Yeah I thought that since the paper would probably be too small I wouldn't be able to publish it. But maybe, who knows.

There are a few short papers in math that got published. One showed a counterexample disproving a theorem.

https://www.openculture.com/2015/04/shortest-known-paper-in-a-serious-math-journal.html

 

[TeethWhitener]

John Forbes Nash’s (of A Beautiful Mind fame) first paper on his eponymous equilibrium is less than a page long and contains no equations:
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063129/pdf/pnas01550-0057.pdf

 

[The Bill]

If you can frame it in the right way, American Mathematical Monthly could be a good place to publish a "small" fact like this. I've come across many little bits and pieces published there as visual proofs or exercises in pedagogy, for example.

 

[Swamp Thing]

Based on your original question & title, this could be a good option:
https://summerofmathexposition.substack.com/p/the-summer-of-math-exposition-is

It was conceived by Grant Sanderson (3Blue 1Brown) and is in its second year. But you have just a week before the submission deadline, so it depends on your readiness as of now.

 

[pbuk]

Summary: I found a super cool property of n-bonacci numbers but it isn't really worth publishing. Ho do I make it known?

Hi PF!

Everyone knows that: $${\varphi }^2 - \varphi - 1 = 0$$ But guess what? $${\varphi}^3-2{\varphi}^2+1=0$$ Generalizing this for all n-bonacci numbers:

Be careful with terminology; we normally use Fibonacci number to refer to a term in the Fibonacci sequence 1, 1, 2, 3, 5, 8... thus the 6th Fibonacci number is 8. We generalise this to refer to the terms of the n-bonacci sequence as n-bonacci numbers, see http://oeis.org/wiki/N-bonacci_numbers.

The numbers $x$ which are roots to the equation $x^{n+1}+1 = 2x^n$ are called n-bonacci constants. This equation is well-known (often written in the form $x + \frac 1 {x^n} = 2$); in fact we define the n-bonacci constant as the single root > 1 of $x + \frac 1 {x^n} = 2$.

Also be very carful not to use the word series when you mean sequence.

 

[KDunc]

Where did you get $\varphi^3-2\varphi^2+\varphi^2-\varphi = \varphi^3-2\varphi^2+1$? I don't understand this part.

They added $\varphi^3-2\varphi^2$ to both sides of the equation. Since they've added these terms to both sides, they cancel each other out, so they haven't changed the equation or relationship. As demonstrated, this can be a very useful tool to make the algebra a bit easier or to demonstrate a property that is not otherwise immediately obvious.