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EIRE2003
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Can anyone tell me, what is so important about prime numbers in science?
EIRE2003 said:Can anyone tell me, what is so important about prime numbers in science?
matt grime said:Do you count mathematics as science? From the phrasing of the question it would seem not.
random matrices and number theory. This provides (at some level) an analogy between prime numbers and the spectrum of atoms.
Feldoh said: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.
DeaconJohn said: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.
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.
Huh? That doesn't sound right at all.Feldoh said: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.
Hurkyl said:Feldoh said: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.
matticus said:the prizes for the RSA factor challenge will continue to be unclaimed, as the challenge is no longer active.
EIRE2003 said: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!
PrimeNumbers said: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!
CRGreathouse said: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.
atomthick said:I believe PrimeNumbers wants to say that if [tex]GCD(N,[\sqrt{N}]!) = 1[/tex] then N is prime.
PrimeNumbers said: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!
DeaconJohn said: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."
Prime numbers are positive integers (whole numbers) that have exactly two distinct divisors, 1 and itself. This means that prime numbers can only be divided by 1 and itself, and are not divisible by any other numbers.
Prime numbers are important in science because they are the building blocks of all other numbers. They are also closely related to many important concepts in mathematics and physics, such as the distribution of prime numbers and the Riemann Hypothesis.
The Riemann Hypothesis is a famous unsolved problem in mathematics that deals with the distribution of prime numbers. It states that all non-trivial zeros of the Riemann zeta function lie on the critical line of 1/2. The Riemann zeta function is a mathematical function that is closely related to the distribution of prime numbers.
Prime numbers have many practical applications in fields such as cryptography, computer science, and data encryption. They are also used in the development of algorithms and mathematical models for predicting and analyzing complex systems.
Scientists study the mysterious nature of prime numbers through various mathematical and computational methods. They also use advanced tools and techniques such as prime number sieves, prime number factorization algorithms, and mathematical models to analyze and understand the complex patterns and properties of prime numbers.