Finding a counter-example for Polya's conjecture.

Before we start, let us define two functions, let $n$ a positive integer and $m$ is the number of total prime divisors of $n$ (for example, if $n=12$ then $m=|\{2,2,3\}|=3$, and we define $\lambda(n)=(-1)^m$. And also : \[L(n)=\sum_{k=1}^{n} \lambda(k)\] Polya conjectured that $L(n)\leq 0$ for any $n\geq 2$, but this was disproved fifty years ago and the counter example was a very large number (almost a billion).

Our task : let us assume that we do not know that there is a counter-example and we try to find one, what do you suggest as the best programming language for this ? and how can we speed up the search for a counter example ?

Please share your codes and running time and the machine specifications and remember that they made it work on a 1958 computer, it should be doable on a 2014 computer.

Wiki page :Polya's conjecture.

#NumberTheory #ComputerScience #Python #Cprogramming #IntegerProgramming

Easy Math Editor

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Vous n'êtes pas mauvais en maths, êtes-vous? Je voudrais avoir le temps de travailler sur ce plus mais je n'ai pas. Il est très intéressant!

Finn Hulse - 6 years, 7 months ago

I didn't know that you speak French! Ce problème est très intéressant car les solutions classique utilise beaucoup de mémoire RAM (par exemple l'utilisation des cribles). Tant que les RAMs des années cinquante étaient au plus 500 mega, je crois qu'il y a des optimisation majeures qui peut être mise ici.

Anyway, thanks for replying. And for your question : I'm not good too.

Haroun Meghaichi - 6 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$