# Prime Numbers

Can anyone tell me, what is so important about prime numbers in science?

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matt grime
Homework Helper
Do you count mathematics as science? From the phrasing of the question it would seem not.

In science (as in the experimental branches such as physics) then I'd point you in the direction of, say, the links between random matrices and number theory. This provides (at some level) an analogy between prime numbers and the spectrum of atoms.

Hi Eire2003! There are a lot of interesting things dealing with prime numbers. I believe the most important application of prime numbers however is in dealing with encryption.

If you buy something, from say Amazon.com, with a credit card at some point in that transaction your credit card information is encrypted. The algorithm to encrypt important information such as credit cards is based on the theory of prime numbers.

This is why corporations will pay money to people who find large prime numbers, because they can be used in even more secure ways of encryption.

Theoretically speaking prime numbers are a foundation for number theory as well.

Can anyone tell me, what is so important about prime numbers in science?
Well, Eire, naturally any response to your question must have something of a personal nature about it. In other words, what is important to one person, might not be important to another.

So here is my personal belief.

It seems to me that God put order into the universe when, or soon after, He created it.

My quest as a mathematician is to find and appreciate that order. It's part of my way of thanking God for creating the universe and everything in it, including me.

With that as an introduction, now let me address the question of prime numbers.

First of all, it seems to most mathematicians, including me, that the concept of prime numbers is a lot more like something that is inherent in the nature of numbers as opposed to an artifact created by man.

Naturally, for those mathematicians (like me) who are on a quest to discover the order that they believe God has put into the universe, this makes the prime numbers a particularly interesting object of study.

And, (for my peculair point of view) we have not been disappointed!

I will give but one example of the beautiful mysteries that are hidden with in prime numbers.

Namely, If you take an integer "x" and divide it by the number of primes less than "x," the ratio approaches the log to the base "e" of "x" as "x" gets bigger and bigger. (This fact is called "the prime number theorem.")

Now, it's not supriising that the ratio approaches some well defined limit. It's not even suprising that it approaches the logarithm (to some base) of "x," but it is definitely suprising that it the base "e" that appears here.

"e" is Euler's constant. e = 2.718281828 ... .

OK, maybe that might not be suprising to you if you never heard of "e," before, after all it's not suprising that it goes to the log of "x" for some base, so, why not call the base "e." So, to explain why it is suprising, here is one definition of the constant, "e" (there are many others).

If you sum the series of the reciprocals of the factorials from 0 to x, the sum approaches "e" as "x" gets bigger and bigger.

In other words,

e = 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + 1/720 + ...

where

0! = 1,
1! = 1,
2! = 2,
3! = 2x3 = 6,
4! = 2x3x4 = 24,
5! = 2x3x4x5 = 120,
6! = 2x3x4x5x6 = 720, etc.

Now you've got to admit that's incredible. Why should the distribution of the prime numbers have anything to do with an infinite sum of factorials?

As far as I know, that is a mystery that has not been completely explained by what we know about mathematics so far. It's is only relatively recent (say 100 years ago) that mathematicians were able to prove that the factorials and the primes are related as described above. So, it's not suprising that there is still some mystery surrounding "the real reason why."

Perhaps future generations will find more intuitive proofs of the prime number theorem than are available to us today and which will illucidate the reason that "e" is the base of the logarithm that makes the prime number theorem work out correctly.

The most intuitive proof of the prime number theorem that we have today is built on the study of the Riemann zeta function, an "analytic" function defined with "complex" numbers. Complex numbers are numbers that have the square root of -1 in them. That is a mystery in itself, why do mathematicians have to use complex numbers to get a "somewhat understandable" (and some would even say "intuitive") proof that the prime number theorem is true. Who knows, maybe future generations will find and "intuitive" elementary proof. (Erdos and somebody else did find an elementary proof about 50 years ago, but it is not at all "intuitive.")

Actually the Riemannian proof of the prime number theorem itself involves a lot of mystery. For example, the proof gives an exact formula for the number of primes less than each integer x. But the formula involves an infinite series and is not "effectively computable" (whatever that means). And the derivation of the formulas seems more like magic tha science (to me at least).

In fact obody knows if an effectively computable forumla that gives the number of primes less than an integer x even exists! There is an incredibly complicated polynomial whose positive values give precisely the prime numbers, but there is no "effective way" to tell when that polynomial is going to give a positive value.

So, perhaps that gives you some idea why some people (like me, for example) think that the prime numbers are improtant. And, even more than important, definitely worthy of our attention.

DJ

P.S. Fellow PhysicsForumItes: Please post corrections if you see any mistakes.

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Do you count mathematics as science? From the phrasing of the question it would seem not.
Five years in college studying physics, yes of course I would regard mathematics as a science. Mathematics describing the trajectories of particles/projectiles ie. trigonometry, dynamics of systems, rates of change in calculas, vector and scaler fields such as electromagnetic waves and temperature, spirals in nature such as petals of a flower, hurricanes and whirlpools, etc.
Mathematics and physics being intertwined as the fundamental science underpinning our entire universe.
However, I have to admit, I do not have much interest in mathematics of number theory because I never really sat down to read it. I found it rivetting for the reason that I could not see it put into use in reality. In saying this, I would love to read about number theory and random matrices if I could see where it was applied to in nature, such as symmetry in flowers and snow flakes- Fractal patterns?

random matrices and number theory. This provides (at some level) an analogy between prime numbers and the spectrum of atoms.
Can you explain to me the analogy between prime numbers and the spectra of atoms?

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Hi Eire2003! There are a lot of interesting things dealing with prime numbers. I believe the most important application of prime numbers however is in dealing with encryption.

If you buy something, from say Amazon.com, with a credit card at some point in that transaction your credit card information is encrypted. The algorithm to encrypt important information such as credit cards is based on the theory of prime numbers.
Interesting, thank you

God put order into the universe when, or soon after he created it.

The concept of prime numbers is a lot more like something that is inherent in the nature of numbers.

Naturally, for mathematicians who are on a quest to discover the order that they believe God has put into the universe, this makes the prime numbers a particularly interesting object of study.

And, (for my peculair point of view) we have not been disappointed!

I will give but one example of the beautiful mysteries that are hidden within prime numbers.
I wish I could feel the beauty in the mathematics of order in nature also.

If you take an integer "x" and divide it by the number of primes less than "x," the ratio approaches the log to the base "e" of "x" as "x" gets bigger and bigger. (This fact is called "the prime number theorem.")

Now, it's not supriising that the ratio approaches some well defined limit. It's not even suprising that it approaches the logarithm (to some base) of "x," but it is definitely suprising that its the base "e" that appears here.

"e" is Euler's constant. e = 2.718281828 ... .

So, to explain why it is suprising, here is one definition of the constant, "e" (there are many others).

If you sum the series of the reciprocals of the factorials from 0 to x, the sum approaches "e" as "x" gets bigger and bigger.

In other words,

e = 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + 1/720 + ...

where

0! = 1,
1! = 1,
2! = 2,
3! = 2x3 = 6,
4! = 2x3x4 = 24,
5! = 2x3x4x5 = 120,
6! = 2x3x4x5x6 = 720, etc.

Now you've got to admit that's incredible. Why should the distribution of the prime numbers have anything to do with an infinite sum of factorials?

As far as I know, that is a mystery that has not been completely explained by what we know about mathematics so far. It's is only relatively recent (say 100 years ago) that mathematicians were able to prove that the factorials and the primes are related as described above. So, it's not suprising that there is still some mystery surrounding "the real reason why."

Perhaps future generations will find more intuitive proofs of the prime number theorem than are available to us today and which will illucidate the reason that "e" is the base of the logarithm that makes the prime number theorem work out correctly.

The most intuitive proof of the prime number theorem that we have today is built on the study of the Riemann zeta function, an "analytic" function defined with "complex" numbers. Complex numbers are numbers that have the square root of -1 in them. That is a mystery in itself, why do mathematicians have to use complex numbers to get a "somewhat understandable" (and some would even say "intuitive") proof that the prime number theorem is true. Who knows, maybe future generations will find and "intuitive" elementary proof. (Erdos and somebody else did find an elementary proof about 50 years ago, but it is not at all "intuitive.")

Actually the Riemannian proof of the prime number theorem itself involves a lot of mystery. For example, the proof gives an exact formula for the number of primes less than each integer x. But the formula involves an infinite series and is not "effectively computable" (whatever that means). And the derivation of the formulas seems more like magic tha science (to me at least).

In fact obody knows if an effectively computable forumla that gives the number of primes less than an integer x even exists! There is an incredibly complicated polynomial whose positive values give precisely the prime numbers, but there is no "effective way" to tell when that polynomial is going to give a positive value.

So, perhaps that gives you some idea why some people (like me, for example) think that the prime numbers are improtant. And, even more than important, definitely worthy of our attention.
Profound! Thank you for sharing the time to express your thoughts on Gods creation. I will certainly go read about number theory, it will be a long struggle haha, but worth it I know!

There are insects, http://www.abc.net.au/science/k2/moments/s421251.htm" [Broken], that spend their lives underground, before emerging, every 7th, 13th, or 17th year, to mate. There is a theory that nature applied evolutionary selection as Eratosthenes' sieve: certain predators of cicadas also come out periodically, such as every second, or third year, therefore those cicadas species have the highest chance of survival which emerge after a number of years that is not divisible by a smaller number (apart from the number one - but that is unavoidable).
Suppose a variant of cicadas emerge every 12th year - then they will meet all those predators that come out every 2nd, 3rd, 4th, or 6th year, and so that variant will not exist for long.

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Hurkyl
Staff Emeritus
Gold Member
This is why corporations will pay money to people who find large prime numbers, because they can be used in even more secure ways of encryption.
Huh? That doesn't sound right at all.

CRGreathouse
Homework Helper
This is why corporations will pay money to people who find large prime numbers, because they can be used in even more secure ways of encryption.
Huh? That doesn't sound right at all.
I *think* the intended reference was to the RSA challenge numbers: paying people to factor numbers into primes, so that everyone else could rest secure in the knowledge that factoring larger numbers into primes is 'hard' since the prizes are still unclaimed.

the prizes for the RSA factor challenge will continue to be unclaimed, as the challenge is no longer active.

CRGreathouse
Homework Helper
the prizes for the RSA factor challenge will continue to be unclaimed, as the challenge is no longer active.
Yes, I'm aware of that. But while they were still offered, they gave a good idea of what was state of the art -- many people tried to factor them, and several teams succeeded. Had they been able to factor more they likely would have tried: the fame of factoring one of the numbers would have made it worthwhile.

Thank you for sharing the time to express your thoughts on Gods creation. I will certainly go read about number theory, it will be a long struggle haha, but worth it I know!
You are welcome! It gives me great pleasure that I struck a responsive chord! DJ

P.S. Talking about reading about prime numbers, I would suggest browsing this forum for starters. Maybe a serach on "the prime number thoerem" or maybe just "primes" or "primes."

DIVIDE X by (SQUARE ROOT X)!
! BEING A FUNCTION ON YOUR CALCULATOR THAT SUMS 1 x 2 x 3 etc.

IF PRIME THEN X WILL NOT BE DIVISIBLE BY ANY OF THE NUMBERS MULTIPLIED BY EACH OTHER BELOW THE SQUARE ROOT OF X AND SO WILL REMAIN UNCHANGED BY THE DIVISION.

IF NOT PRIME THEN ONE OF IT's FACTORS CAN BE FOUND BELOW THE SQUARE ROOT OF IT AND THE TOP HALF OF THE EQUATION NAMELY X WILL BE DIVIDED BY IT AND REDUCED, OTHERWISE IT'S PRIME!

DIVIDE X by (SQUARE ROOT X)!
! BEING A FUNCTION ON YOUR CALCULATOR THAT SUMS 1 x 2 x 3 etc.

IF PRIME THEN X WILL NOT BE DIVISIBLE BY ANY OF THE NUMBERS MULTIPLIED BY EACH OTHER BELOW THE SQUARE ROOT OF X AND SO WILL REMAIN UNCHANGED BY THE DIVISION.

IF NOT PRIME THEN ONE OF IT's FACTORS CAN BE FOUND BELOW THE SQUARE ROOT OF IT AND THE TOP HALF OF THE EQUATION NAMELY X WILL BE DIVIDED BY IT AND REDUCED, OTHERWISE IT'S PRIME!

I think you have your caps lock turned on. You can turn it off, just press it... Anyway, what you say isn't really correct. Consider 2. Then you need to calculate $$\frac{2}{\sqrt{2}!}$$. Even if you could make sense of $$\sqrt{2}!$$ (through gamma functions), this calculation will not remain unchanged!
The same thing will happen if you take composite numbers...

I *think* the intended reference was to the RSA challenge numbers: paying people to factor numbers into primes, so that everyone else could rest secure in the knowledge that factoring larger numbers into primes is 'hard' since the prizes are still unclaimed.
I thought he meant that corporations want people to find large primes so that the modulus in the public key of the RSA system is even harder to crack.

If you square root X you must round down.

So take 2 for example Square root rounded down is 1. Then 2/1! remains as 2/1 so unchanged and is prime. Because there are no factors under the square root of 2 bigger than 1.

Take 6 (which is 3 x 2), square root rounded down is 2. 2! is 1 x 2 = 2. 6/2 = 3/1 which is changed and therefore not prime. Because both halfs of the fraction are divisible by 2.

It works by your calculator dividing the top half of the fraction and the bottom half of the fraction by the biggest denomination under the square root of X. Ultimately finding the multiple of X bigger than 1 and smaller than the square root of X if there is one to be found.

OK, take 11, square root rounded down is 3. But 11/3! is not 11, so it doesn't remain unchanged...

In this case the number as a whole changes from 11 to 11/3! but 11 itself doesn't change at all. If it were 25 and I divided it by 120 it would change to 5 and not remain the same. Basically I can write a fraction of X/(Square root of X rounded down)! on a piece of paper or calculator and change it to smaller numbers on both sides of the division line by either myself or the calculator if it is not prime.

I believe PrimeNumbers wants to say that if $$GCD(N,[\sqrt{N}]!) = 1$$ then N is prime.

gb7nash
Homework Helper
I believe PrimeNumbers wants to say that if $$GCD(N,[\sqrt{N}]!) = 1$$ then N is prime.
This is true, and can be proved using prime decomposition. Is it practical though? I'm not sure. If you want to determine if a humungous number is prime, calculating factorials and then the gcd can be a very excrutiating process.

Clearly it's not computational feasible for large numbers, however there are some interesting results for example if $$gcd(N, [\sqrt{N}]!) = P, P > 1$$ then P is a factor of N.

It could become computational feasible if someone finds good algorithms for adding, subtracting and finding modulus that work in the factorial base (Cantor discovered that we can write any number in factorial base, for example 15 = 1! + 1*2! + 2*3!). Because we can easily find the representation of N in factorial base all we would need is fast computational algorithms for this base.

P.S. EIRE2003 see how many interesting questions are prime numbers raising? This kind of questions and their answers have made great improvements allover mathematics! Those numbers look uninteresting until you ask a question about them, try it.

Char. Limit
Gold Member
DIVIDE X by (SQUARE ROOT X)!
! BEING A FUNCTION ON YOUR CALCULATOR THAT SUMS 1 x 2 x 3 etc.

IF PRIME THEN X WILL NOT BE DIVISIBLE BY ANY OF THE NUMBERS MULTIPLIED BY EACH OTHER BELOW THE SQUARE ROOT OF X AND SO WILL REMAIN UNCHANGED BY THE DIVISION.

IF NOT PRIME THEN ONE OF IT's FACTORS CAN BE FOUND BELOW THE SQUARE ROOT OF IT AND THE TOP HALF OF THE EQUATION NAMELY X WILL BE DIVIDED BY IT AND REDUCED, OTHERWISE IT'S PRIME!
Huh? Take off the caps lock and size changes, please. And explain better. Have a comma: ","

epenguin
Homework Helper
Gold Member
It was surely not for any of the applications that have been mentioned. It was cultivated for centuries before they were dreamt of. And also prime numbers specifically are not really essentially connected with cryptography. It is just that factorisation into prime numbers is one, just one, example of a hard (computationally very long) problem whose inverse (multiplying the factors) is not hard, if I understand. There are other such hard problems ready to take over for cryptography if ever anyone cracks the factorisation problem.

I think of it as having a pile of pebbles, can I arrange them in a regularly spaced rectangle? If not I have a prime number of pebbles. Could be tempted to wonder if it is worthy of a grown man's attention. Tempted to believe that it would be if it were simple - could be explained, followed, carried in the head it would be revealing of a structure. But if it is so difficult and complicated that no one understands the solution when it is found, will it be revealing in the same way? I believe this question is discussed about some of today's very difficult proofs. Unless it throws light on other problems whose significance is more apparent. We are told this would be so, but I suggest we do need to be told.

Now you've got to admit that's incredible. Why should the distribution of the prime numbers have anything to do with an infinite sum of factorials?

As far as I know, that is a mystery that has not been completely explained by what we know about mathematics so far. It's is only relatively recent (say 100 years ago) that mathematicians were able to prove that the factorials and the primes are related as described above. So, it's not suprising that there is still some mystery surrounding "the real reason why."
didn't ken ono recently establish something like 'factorials of primes follow a fractal pattern'?
can someone post the proof please?