# Optical methods for number theory?

I found a paper: 'A New Method of Finding the Distribution of Prime Number', saying

We stack discs and annuluses with certain rules then turn on the light to illuminate. The projection of annuluses corresponds prime number,and the projection of discs corresponds composite number.

Are there other physics methods for number theory?

Edit: Answers and comments here and to the corresponding meta question have shown that the answer is yes, and much broader and more recent than the sieve of Eratosthenes. Also, a more informative link to the above paper is http://wenku.baidu.com/view/1d602350be23482fb4da4cc6.html?re=view

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The paper you refer to doesn't seem to be very accessible -- when I click the PDF link, I see a page asking in Chinese for a login password. – Stefan Kohl Jun 18 '14 at 12:51
@StefanKohl,the paper has 4 pages,main content is in Chinese ,here you can see the first 2 pages, wenku.baidu.com/view/1d602350be23482fb4da4cc6.html?re=view, the last 2 pages are more examples. – Mike Jun 18 '14 at 13:22
Here is a physical implementation of the more efficient version: en.wikipedia.org/wiki/Lehmer_sieve – Zack Wolske Jun 18 '14 at 17:04
@ZackWolske ,yes it is sieve of Eratosthenes. – Mike Jun 19 '14 at 5:59
I would like to query the closure - have posted on meta: meta.mathoverflow.net/questions/1749 – Carl Jun 19 '14 at 8:01

## 5 Answers

There's a way to use physics to calculate the digits of $\pi$. I quote from http://math.stackexchange.com/questions/138289/intuitive-reasoning-behind-pis-appearance-in-bouncing-balls

Let the mass of two balls be $M$ and $m$ respectively. Assume that $M=16\times100^nm$. Now, we will roll the ball with mass $M$ towards the lighter ball which is near a wall. How many times do the balls touch each other before the larger ball changes direction? (The large ball hits the small ball which bounces off the wall)

The solution is the first $n+1$ digits of $\pi$.

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That's cute, but already for $n=1$ real-world collisions surely cannot be so perfectly elastic and aligned (nor $M/m$ controlled so precisely) that one could come close to guaranteeing a count of $31$; even $n=0$ may be too ambitious. – Noam D. Elkies Jun 25 '14 at 16:38
@Noam I agree with this. Remarkably elastic and aligned quantum systems have been demonstrated, though, for example the quantum Newton's cradle: nature.com/nature/journal/v440/n7086/abs/nature04693.html – Carl Jun 26 '14 at 7:33

Probably, the best-known application of ideas from physics to computational number theory is Shor's algorithm.

You may also be interested in other examples from “unusual and physical methods for finding prime numbers”.

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The physics of the Riemann Hypothesis D. Schumayer and D. A. W. Hutchinson, Rev. Mod. Phys. 83 307-330 (2011). abstract here

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Actually such optical devices were at one point in the 20th century the state of the art for doing certain number-theoretical calculations, notably factoring. See http://en.wikipedia.org/wiki/Lehmer_sieve (specifically the 1932 device, pictured in pages 18-20 of http://people.ucalgary.ca/~hwilliam/Sieve_Pictures.pdf).

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This is the best pdf. – Mike Jul 4 '14 at 1:42

I agree with Noam (great pictures!) Some of the links from comments are quite informative. Here are two specific cases

1. D. N. Lehmer first used a device made of bicycle chains and rods and later devices made of gears with holes in them. This article relates in rather breathless prose that one of the gear mechanisms (combined with theory) proved in a few minutes that

the great unconquered nineteen digit number $3,011,347,479,614,249,131$ , known to be a factor of $2^{95} + 1$ and suspected to be prime, is in fact prime.

2. A few years back (maybe 2000?) there was great excitement over rumors that Adi Shamir had a breakthrough which would speed up (the sieving step of) the number theory sieve by "several orders of magnitude". The then record factoring of an RSA key was for a 465 bit integer and the breakthrough was rumored to make 512 bit keys "very vulnerable." When the details came out of the punnily named TWINKLE device it turned out to be an electro optical device using LEDs and filters. It is agreed to be quite clever, it has never been built. Enhanced (theoretical) versions might threaten 768 bit keys in under 9 months (That estimate was in 2000, for an organization willing to invest in 80,000 pentium 2 PCS and 5000 TWINKLE devices ). 1024 bit keys would probably have been well beyond that. I think that the state of the art in unbuilt (or so they say...) special purpose devices is no longer optical.

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