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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?

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?

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    $\begingroup$ Modern cryptography uses large primes. See this previous question and answer $\endgroup$
    – Arturo Magidin
    Jun 4, 2011 at 3:44
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    $\begingroup$ As far as "real tasks" (if you don't consider mathematical research to be a real task) cryptography is the main use, though no doubt they make appearances in many other algorithms used all over the place, they don't have the "leading role", as it were, that they have in cryptography. $\endgroup$
    – Arturo Magidin
    Jun 4, 2011 at 4:17
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    $\begingroup$ No offense taken. When my dad's advisor was teaching a course in automata theory in the sixties, a student asked "Is there any practical application of automata theory?" After thinking about it for about 10 seconds, he replied "I know that at least me and thirty odd other people in the country make a living by doing automata theory. If you can come up with something more practical than that, let me know." $\endgroup$
    – Arturo Magidin
    Jun 4, 2011 at 21:08
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    $\begingroup$ Beside cryptography is coding theory. Random number generators, error correcting codes, and hashes often involve primes: either directly or indirectly. Another not so obvious (indirect) application: many libraries which perform arithmetic on large integers, or polynomials involve reductions modulo primes (see Hensel's lemma) for computational complexity reason. $\endgroup$
    – user2468
    Jul 16, 2012 at 15:56
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    $\begingroup$ I use them for animation. This allows different parts of a website to animate always at a different time. $\endgroup$
    – Petah
    Nov 11, 2015 at 6:54

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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.

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    $\begingroup$ Note that this encryption system will be utterly useless as soon as quantum computers are reasonably usable. $\endgroup$
    – akkkk
    Nov 30, 2012 at 22:48
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    $\begingroup$ Indeed. However, it is still not clear whether quantum computers will even be reasonably useful at a level that allowed them to break real-world RSA ciphers, and meanwhile RSA is used practically everywhere, so RSA is a good example of practical use of primes even if someday it will be obsolete. $\endgroup$
    – Gadi A
    Dec 3, 2012 at 9:45
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    $\begingroup$ I've heard that unicorns are also adept at breaking RSA, so watch out. $\endgroup$
    – R R
    Aug 16, 2014 at 23:56
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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!)

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.

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    $\begingroup$ I somehow don't think 13 and 17 are "large primes" that need computing, though, even if you are a cicada... $\endgroup$
    – Arturo Magidin
    Jun 4, 2011 at 4:54
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    $\begingroup$ Cicada's don't have the computing power that we do, so they stuck with smaller primes. Anyway, I realize my answer is not quite was the OP was looking for, but I still thought it was neat. $\endgroup$
    – Jeff
    Jun 4, 2011 at 4:58
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    $\begingroup$ Still, it's a very nice "real world application of primes". $\endgroup$
    – Gadi A
    Jun 4, 2011 at 4:58
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    $\begingroup$ And the computation is not done by the cicadas anyway, but by the predators who ate all the 15- and 16-year cicadas. $\endgroup$
    – MJD
    Jul 16, 2012 at 16:14
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    $\begingroup$ @Jeff: to expand on Mark's answer, it's not a matter of a computational power, because the burden of proof is on the predators. It's more likely because 13 and 17 were the smallest primes that allowed them to avoid most predators. A hypothetical group of 89-year period cicadas would grow much more slowly while not avoiding many more predators, so it would not be favored by evolution. $\endgroup$
    – Generic Human
    Jul 25, 2012 at 13:46
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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.

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    $\begingroup$ Interesting, lol, maybe I will do something like this some day. $\endgroup$
    – GarouDan
    Jun 5, 2013 at 0:25
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You can use prime numbers to plot this fine pattern :)

enter image description here

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.

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    $\begingroup$ Nice picture! FWIW this is equivalent to ((x^y)*115 + 13) % 256 and it has nothing to do with prime numbers, but rather with the fact that 115 is odd and has a binary representation that is "random enough". $\endgroup$
    – Generic Human
    Jul 25, 2012 at 14:05
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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.

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From the world of real things...

Prime number are used in developing machine tools. Utilizing primes you can avoid setting up harmonics which "eat" your very expensive tools. Tools chatter, (bounce up and down), as they are being used. Allowing those vibrations to propagate intensifying the chatter and the wear.

You ever wonder how the metal racks in a microwave get designed? Again they use primes to assure that there are no harmonic possibilities, and you don't get the light show you would on an older microwave.

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    $\begingroup$ Can you be more specific? How are primes used here? (Links would do - I'm interested...) $\endgroup$
    – user1729
    Aug 14, 2013 at 14:31
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    $\begingroup$ To overly simplify it...Machine tools are made of "composites", many parts. Taking the first part and multiplying it by a prime, your can reduce the harmionics caused by vibration. This continues through the whole tool set up. A Microwave metal grill is made the same way. If you look at the grill in your microwave you will note all of the cross beams are set at strange distances to each other. This is to eliminate arcing problems often seen in early microwaves. $\endgroup$
    – Real World Guy
    Aug 27, 2013 at 1:07
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Primes are also useful for generating hash codes.

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  • $\begingroup$ How would they be used for that purpose? Is it different from the cryptographic use? $\endgroup$
    – Sylverdrag
    Jun 4, 2011 at 6:56
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    $\begingroup$ The requirements for a hash are a little different: you want to minimize collisions and you don't really care whether the "encoding" is easy to undo or not. Though both randomizing functions and encryption functions can be used to generate hashes. $\endgroup$
    – trutheality
    Jun 4, 2011 at 7:05
  • $\begingroup$ Another reason prime numbers are used is that when the size of a hash table is prime, collisions are less likely. $\endgroup$
    – trutheality
    Jun 4, 2011 at 7:07
  • $\begingroup$ Maybe you want to expand your answer - explain what hash codes are and how primes are used to generate them. $\endgroup$
    – Gadi A
    Jun 4, 2011 at 8:49
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    $\begingroup$ No I don't. I'm severely underqualified for that. Those interested are better off searching for further details on Wikipedia or Google. $\endgroup$
    – trutheality
    Jun 4, 2011 at 8:56
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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.

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    $\begingroup$ Not sure this answers the question... $\endgroup$
    – J. M. ain't a mathematician
    Jul 16, 2012 at 16:07
  • $\begingroup$ A metastable phase of an existing substance is effectively a whole new material. It has its own individual properties, some (e.g. magnetic properties of metallic glasses, mechanical properties of diamond-like carbon, abrasive properties of cubic boron nitride, . . ) potentially very useful to mankind. The mathematical approach to predicting such compositions likely to obviate the usual kinetics of crystallisation has to be cheaper and simpler than existing approaches, like rapid solidification, huge external magnetic fields, phase prediction based on existing thermochemical data, etc. $\endgroup$
    – Deek
    Jul 16, 2012 at 19:12
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Uniform usage of gears with relatively prime number of teeth

Straight from Wikipedia:

For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of mathematics[b] other than the use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance.

enter image description here

The small gear has 13 teeth the middle gear has 21, relatively prime to 13. Image source.

Arecibo message image dimensions

The Arecibo message consisted of a rectangle with prime width and height.

This guarantees that aliens can only interpret the image in 2 ways (modulo rotations and excluding a flat 1 x N image).

For example, for a 2 x 3 x 5 image with 30 bits can be read as either of:

 2 x 15
 3 x 10
 5 x  6
 6 x  5
10 x  3
25 x  2

But a similarly sized 3 x 11 image with 33 bits can only be read as:

 3 x 11
11 x  3

If that counts as "real world", I leave up to you.

A good visualization of why this is for kids is that you cannot make a rectangle out of a prime number, so there is only one possibility for each edge.

enter image description here

Image source.

Indicator of intelligent life in interstellar communications

Another application in the alien communication vein: in the 1980's documentary series Cosmos, Carl Sagan proposes that we use a signal like:

X XX XXX XXXXX XXXXXXX XXXXXXXXXXX

with prime numbers length strings at the start of messages we send to aliens.

The rationale is that this is likely to catch their attention, since there are no (?) natural processes that generate such a sequence, even though it is a sequence we expect alien mathematicians to immediately understand, and thus recognize as a sign of intelligence.

So in this case, the lack of physical application of primes leads to their usefulness!

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A simple answer is finding GCF and LCD for whole numbers which allows us to efficiently manipulate fractions, both arithmetic and algebraic. Another is rationalizing and simplifying radical expressions. Prime number manipulation is a basic and not-so-basic tool of mathematics.

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Quadratic Reciprocity is stated in terms of the residues modulo primes. This "Golden Theorem" as called by Gauss, is one of the main threads leading up to Langlands program and eventually to the geometric Langlands Program. This later area of research has been shown to have ties with S-duality in string theory. String theory is just now being proven useful in understanding phenomena in condensed matter physics. Also it is the techniques that are used to prove results about prime numbers that have applications rather than a particular theorem about specific families of primes. Prime numbers are often test beds for more general results used in other areas of mathematics.

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Thought I'd mention an application (or more like an explicit effect, rather than a direct application) that prime numbers have on computing fast Fourier transforms (FFTs), which are of fundamental use to many fields (e.g. signal processing, electrical engineering, computer vision).

It turns out that most algorithms for computing FFTs go fastest on inputs of power-of-two size and slowest on those of prime size. This effect is not small; in fact, it is often recommended, when memory is not an issue compared to time, to pad one's input to a power of 2 (increasing the input size to earn a speedup).

Papers on this have been written: e.g. see Discrete Fourier transforms when the number of data samples is prime by Rader. And github issues like this suggest it is still an issue.

Very specific algorithms (e.g. see this one using the Chinese remainder theorem for cases where the size is a product of relative primes) have been developed that, in my opinion, constitute some relevancy of primality to these applications.

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

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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

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You know the Queen song, We Will Rock You? How it has that stamp-stamp-clap-pause beat? According to Wikipedia,

"The stamping effects were created by the band overdubbing the sounds of themselves stomping and clapping many times and adding delay effects to create a sound like many people were participating. The durations of the delays were in the ratios of prime numbers, a technique now known as non-harmonic reverberation."

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My prime pattern

Primes are really strange... I created this simple pattern out of bordom. I haven't seen any of similarity online. As you see, the picture has lines of absence depending on the scale you choose, this is ranging from values 1 to 1000000

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I used prime numbers to help group entities using two factors using Excel. I needed to calculate a. how large an entity was (in terms of turnover), and b. how tardy it was (by number of months) in filing annual returns. Each band had a consecutive number, and each size (e.g. turnover between $125,000 and $2 million) was assigned a prime number higher than the total number of time bands. Using prime numbers ensured that multiplying "size" by "lateness" values resulted in a unique "lateness score" that could then be used used to group entities based on size and degree of lateness.

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Prime numbers are often used to make puzzles. Specially series puzzles. For instance, I made this one few months ago:

Finding the next term in the sequence: $10,37,521,8177,33550457$

It is OEIS-proof.

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    $\begingroup$ In what sense a puzzle is a "real-world application"? $\endgroup$
    – mlc
    May 25, 2017 at 20:12
  • $\begingroup$ I was expecting this question. Competitive tests have such questions. $\endgroup$
    – Nikhil Bhavar
    May 25, 2017 at 20:14
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    $\begingroup$ Your answer reminds me the Tibetan saying about "giving a green answer to a blue question". $\endgroup$
    – mlc
    May 25, 2017 at 20:18
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Prime numbers are used to generate Pseudo-Random numbers---which are used for coding-decoding exam.papers and digital signals . Also they are useful for testing new designs of computers . For example--

1/7=0.142857_142857_142857_14...(the decimal numbers repeat after six digits)

1/7^2=0.020408163265306122448979591836734693877551_020408163265306122448979591836734693877551_020408163265306122448979591836734693877551_02041...(Decimal digits repeat after 42 =6*7 digits)

1/7^3.=0.002915451895043731778425655976676384839650145772594752186588921282798833819241982507288629737609329446064139941690962099125364431486880466472303206997084548104956268221574344023323615160349854227405247813411078717201166180758017492711370262390670553935860058309037900874635568513119533527696793_002915451895043731778425655976676384839650145772594752186588921282798833819241982507288629737609329446064139941690962099125364431486880466472303206997084548104956268221574344023323615160349854227405247813411078717201166180758017492711370262390670553935860058309037900874635568513119533527696793_00291545189504...( Decimal digits repeat after294=42*7 =6*7^2digits)

period of repetition of decimal digits 1/7^n = 6*7^(n-1)

You can use multiples of 6 digits or 7digits as a code.

If the signal you are sending has 200 words ,you can use 6digit code .

6*7^(n-1)>2000*6 ......7^(n-1)>2000 ....so n>5

Also you can calculate p/7^101 ( p is any prime number) and check if the numbers repeat after .............6*7^100=19406859057748547948067886614601300865143219193427752405603371988350148757821568360006 digits --if your computer can handle that many digits.

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Prime numbers are used in public key cryptography. It is used because you generally don't think of the really big prime numbers, so it is great for codes and to keep things safe.

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