Real-world applications of prime numbers?

Question

I am going through the problems from Project Euler and I notice a strong insistence on Primes and efficient algorithms to compute large primes efficiently.

The problems are interesting per se, but I am still wondering what the real-world applications of primes would be.

What real tasks require the use of prime numbers?

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Edit: A bit more context to the question: I am trying to improve myself as a programmer, and having learned a few good algorithms for calculating primes, I am trying to figure out where I could apply them.

The explanations concerning cryptography are great, but is there nothing else that primes can be used for?

Answer 1

The most popular example I know comes from Cryptography, where many systems rely on problems in number theory, where primes have an important role (since primes are in a sense the "building blocks" of numbers).

Take for example the RSA encryption system: All arithmetic is done modulo $n$, with $n=pq$ and $p,q$ large primes. Decryption in this system relies on computing Euler's phi function, $\varphi(n)$, which is hard to compute (hence the system is hard to break) unless you know the prime factorization of $n$ (which is also hard to compute unless you know it upfront). Hence you need a method to generate primes (the Miller-Rabin primality checking algorithm is usually used here) and then you construct $n$ by multiplying the primes you have found.

Answer 2

Here is a hypothesized real-world application, but it's not by humans...it's by cicadas.

Cicadas are insects which hibernate underground and emerge every 13 or 17 years to mate and die (while the newborn cicadas head underground to repeat the process). Some people have speculated that the 13/17-year hibernation is the result of evolutionary pressures. If cicadas hibernated for X years and had a predator which underwent similar multi-year hibernations, say for Y years, then the cicadas would get eaten if Y divided X. So by "choosing" prime numbers, they made their predators much less likely to wake up at the right time.

(It doesn't matter much anyway, because as I understand it, all of the local bug-eating animals absolutely gorge themselves whenever the cicadas come out!)

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EDIT: I should have refreshed my memory before posting. I just re-read the article, and the cicadas do not hibernate underground. They apparently "suckle on tree roots". The article has a few other mild corrections to my answer, as well.

Answer 3

Yes indeed modern cryptography is a useful branch which requires extensive use of prime numbers. A real world application to them would be how we use large primes in order for us to be able to encode information that is sent wirelessly when making transactions on our debit cards, credit cards, computers,$~\ldots$etc in order to keep our information safe. Now when I say real world I don't mean the physical world. Primes numbers use is only in the computer world, in which we use computers in our physical world; if that makes any sense at all. Primes number had little use until about the 19th century, when mathematicians experimented with them in hopes to uncover some breakthrough with their use. When the times of the war came around, the U.S. defense needed a way of secrecy of all high class confidential information, so files and messages all needed to be encoded, so that enemy lines could not retrieve vital information of plans and routines. Encryption was used, and to make the process of using primes numbers to encode information, computers came into play to create more complex and longer codes that would be much harder to crack by anyone. Primes can also be used in pseudorandom number generators and computer hash tables. There are some biological instances in which primes are used to help in predicting the predator-prey model for a special type of insect to have a higher survival rate which are "Cicada". Something else would be public-key encryption, formally known as RSA.

There are many types of classifications of prime numbers, but the main two are Fermat primes and Mersenne primes.

Have a look at this video here from Terence Tao. Structure and Randomness in Prime Numbers

Articles Here:

Treatment on Primes, They are the very top 9 links by Terry Tao and others.

Powerpoint Link in First Paragraph

Answer 4

There may be some applications (other than to cryptography, already mentioned) in Manfred Schroeder's book, Number Theory in Science and Communication.

Answer 5

Primes are also useful for generating hash codes.

Answer 6

Just to add one more: Primes are also useful when generating Pseudo-Random Numbers with the computer. A few formulas use them to avoid patterns in the output.

Answer 7

You can use prime numbers to plot this fine pattern :)

Intensity of green colour for each pixel was calculated using a function, which can be described with this pseudocode snippet:

g_intensity = ((((y << 32) | x))^((x << 32) | y))) * 15731 + 1376312589) % 256

where x and y are a pixel coordinates in screen space, stored in a 64bit integer variables.

Answer 8

Like yourself, I got into primes since this was a common exercise program to do when learning new programming languages and it was interesting to see which language was faster on the same algorithm/error check plan.

It was only when I was refining my Ada coded program to get the highest number of primes that I could get from a 32-bit machine that I came across the offset logarithmic integral. (I needed to reserve enough - but not too much - memory for my holding array for the primes. The array, of course, had to be declared prior to making any assignments to it. On a 1 GB memory 32-bit machine, I can get primes up to ~ 50 million before stack blows.)

$${\rm Li} (x) = \int_2^x \frac{dt}{\ln t}$$

This function represents the best approximation to the number of primes up to some number, x.

All I'm saying here is that this equation made me wonder about primes in the context of a number of other things that use related functions . . . That led me on to thinking about entropy calculations, particularly about selecting compositions more likely to give rise to metastable crystal forms - possibly even glasses - than other compositions using the same constituent elements.

Answer 9

When I was some 20 years old and living by myself for the first time, I designed a little racetrack with numbered squares on it, along with a handful of coloured tokens that would race along the track at the speed of one square per day. Each token had a household chore and a prime number on it; when a token hit its number, I had to carry out the given task, and it would get reset to zero. So, I washed the dishes every two days, watered the plants every three, vacuumed the carpet every five, ....

It was a good system. It made cleaning fun, it provided variety and structure at the same time, and I was obliged to devote the entire day to chores only once every 1397.73 years.