don't really know the "usefulness" of prime numbers...
However, they are segregated because these numbers have NO factors at all!
(factors meaning, e.g. 21=(7)x(3) where 7 and 3 are factors of 21)
Yup, none whatsoever. THis is what makes these numbers special and we call them P R I M E.
Also, make clear that although 1 has no possible factor, it is NOT considered a prime number either.
One more point, that the prime numbers don't have any INTEGRAL factors. (I mentioned this irrelevant one, 'cause I didn't want any arguments that decimal fractions are available for 37!) Hope I helped, Pin~Jinx /anarchist ***************************************************** 09-27-02, 08:53 AM Pin~Jinx K,
don't really know the "usefulness" of prime numbers...
However, they are segregated because these numbers have NO factors at all!
(factors meaning, e.g. 21=(7)x(3) where 7 and 3 are factors of 21)
Yup, none whatsoever. THis is what makes these numbers special and we call them P R I M E.
Also, make clear that although 1 has no possible factor, it is NOT considered a prime number either.
One more point, that the prime numbers don't have any INTEGRAL factors. (I mentioned this irrelevant one, 'cause I didn't want any arguments that decimal fractions are available for 37!) Hope I helped, Pin~Jinx /anarchist
09-27-02, 11:29 AM WiteoutKing I think I heard somwhere a useful tip. Every integer >2 is a sum of diferent primes and 1.
2 = 2 3 = 3 or 1+2 4 = 1+3 5 = 2+3 6 = 1+2+3 or 1+5 7 = 2+5 etc.
09-27-02, 01:17 PM FlyingHellfish WoK,
It sounds like you're referring to the Goldbach Conjecture, which states something along the lines of, "Any integer [greater than 5] can be expressed as the sum of 3 prime numbers." Although no one has proven or disproven it, a counterexample has yet to be found.
With regard to the original question, prime numbers are the so-called building blocks of our number system. According to the Fundamental Theorem of Arithmetic, no two distinct numbers can be expressed as the product of prime factors in the same way. In more advanced mathematics, it can be shown that the distribution of prime numbers is closely related to natural logarithms, and hence the number e. Prime numbers are also intertwined with the value of pi; there are numerous ways to express pi as some function or arrangement of prime numbers. For instance, the probability that any two random numbers are relatively prime to each other can be expressed as a complicated infinite product involving primes which happens to equal 6/pi^2.
The Riemann Zeta Function, one of the most important functions and the basis of one the most famous unsolved math problems today, is closely related to the distribution of primes, and can even be expressed in terms of functions containing infinite products of prime numbers for certain values in its domain. The RZF can relate prime numbers to physics and complicated integration.
09-28-02, 06:36 AM Pin~Jinx FHf,
interesting info there~
I'm getting curious, could you please elaborate on the link between Prime Numbers and pi / e
Thankyou for sharing, Pin~Jinx /anarchist
09-29-02, 04:34 AM FlyingHellfish But be forewarned-although the mathematics behind most of this is not that complicated, there are some leaps in logic that are mindboggling. Since it's late right now, I'll post my answer in parts over the next day or two.
First what I will prove is that the infinite sum of the reciprocals of squares is equal to the reciprocal of the infinite product of 1-1/p2, or equivalently:
Sinf1 1/x2 = [P (1-1/p2)]-1
where p ranges through all prime numbers.
The reason for this is that when written in expanded, each of the factors of the right side of the equation is equal to the sum of a geometric series with the first term equal to 1 and a common ratio of 1/p2, or equivalently:
1 + 1/p2 + 1/p4 + 1/p6 + .....
Here is where it gets tricky. There are an infinite number of these geometric series being multiplied by each other, one for each prime number. Keeping in mind what I mentioned in an earlier post about the Fundamental Theorem of Arithmetic and how it states that every integer has a distinct factorization of prime numbers, think of what happens when all of these geometric series get multiplied by each other. The numerators will all be 1, but the denominators will be in the form p1a*p2b*p3c*..., with every denominator being made up of its own unique set of prime factors. This last statement is key, since it implies that since all prime numbers were represented as a geometric series, then every single natural number is represented in the final sum, or in other words:
1/12 + 1/22 + 1/32 + 1/42 + ....
Does that make sense? As I said, the math itself is not difficult-it's the logic behind it.
I'll finish up the rest of this later. Hopefully the UBB Code came out alright. If not, I'll fix it later unless someone wants to have a go at it.
Addendum: I should note here that although I only proved it for one case, the original equation stands true for all integral powers of x and p.
Sinf1 1/xn = [P (1-1/pn)]-1
09-29-02, 06:13 PM FlyingHellfish The next part of the proof goes on to show that:
Sinf1 1/x2 = p2/6
And this is where I become too lazy to type out a proof, so I'll make reference to an excellent website that provides 14 different ways of showing this:
Click here. Most of these proofs are way over my head, but I'm at least somewhat familiar with #1, #7, and #14. You may notice the term z(2) thrown around a bit, but for our purposes, it has the same value as Sinf1 1/x2
From the above web page, we see that:
[P (1-1/pn)]-1 = Sinf1 1/x2 = p2/6
Hence, p can be expressed in terms of prime numbers.
To be continued....
09-29-02, 06:33 PM FlyingHellfish As simple [and boring] as it sounds, a very important theorem in number theory is called the Prime Number Theorem. Essentially, it says that:
p(n) ~ Integral2 to n dx/ln x
where p(n) is the prime counting function. It's value is equal to the number of primes less than or equal n. For instance, p(22) = 8 because there are 8 prime numbers (2, 3, 5, 7, 11, 13, 17, and 19) less than or equal to 22.
The importance of this is that it tells us the approximate distribution of prime numbers and what those numbers might be.
09-30-02, 06:30 AM Pin~Jinx [b].........but Quite a Head Ache for me![b]
Firstly: let me THANK YOU for replying to me so promptly and explaining the topic to me so comprehensively. Thnx a Bunch!
Ah Well, you see,
I am just a student of grade11, who happens to like the subject of Math... and got plain curious by the connection between pi and prime numbers.
Most of the things you wrote are way past my head; further, (I am sure you mustve linked the page correctly) but my browser refuses to open the link!!!
Geez......... (to embarrassed to even sign-off!) red face
09-30-02, 08:55 AM FlyingHellfish Don't worry, those proofs are far from easy for almost anybody! I'll just hit the high points...
The first 2 proofs are meant to show that p can be expressed in terms of prime numbers by showing that:
p2/6 = Sinf1 1/x2 = [P (1-1/p2)]-1
where P is the symbol representing the infinite product function. For example, P n = 1*2*3*4*5*6....*infinity
The last is an explanation of how natural logarithms (logarithms with base e, denoted ln x) play a major role in determining the distribution of prime numbers. For hundreds of years, mathematicians have been trying to come up with methods to determine the frequency of prime numbers in our number system. It's easy to find small prime numbers and their density among all integers, but it's quite difficult to find large primes because they don't follow any set pattern. The Prime Number Theorem gives us a way of finding a good approximation of just how many primes there are in a certain range of integers. If you were confused about the expression:
Integral2Inf dx/ln x, then let me explain. Picture the graph of a function (above the x-axis, preferably). The Integral of a function is basically the area bounded on top by the function, on the bottom by the x-axis, and on the sides by 2 vertical lines. In this specific case, the area is bounded on top by the function 1/ln x (The dx is there, but is not part of the function. For computational purposes, you can pretend it's not there), on the bottom by the x-axis, by the vertical line x=2 and by the vertical line x=infinity.
Does that help to clarify anything?
Also, you need Adobe Acrobat Reader to open up the link I gave. If you don't have it, it's free to download. It's not that important, though. It would take me quite a while to come up with the simplest of these 14 proofs, and about 80% of what is in there is well over my head.
09-30-02, 12:00 PM Pin~Jinx Oh!
NOw maybe something is forming shape in my head. Your explanation is good enough!
Thankyou so much for your incessant help!
F, I do have the AdobeFoto Shop installed,
however, my browser was giving some errors then. Perhaps the server of Tht site must've been down. Dunno.
Thnx once again,
PiN~Jinx
09-30-02, 02:50 PM Professor First, I have to agree with Pin~Jinx that FlyingHellfish is an excellent teacher!
Regarding Kendor's question about usefulness of primes:
The study of prime numbers belongs to a branch of mathematics called Number Theory, which deals with the integers. Number Theory was long-ago dubbed the "Queen of Mathematics" and regarded as the purest, most up-in-the-clouds area of math. It was the pure mathematician's last refuge from the unrelenting onslaught of applied mathematics over the past few centuries.
When mainframe computers became common in the 1960's, the search was on for larger & larger primes, but finding them was like collecting rare butterflies -- intellectual curiosity, aesthetics, and the thrill of discovery were sufficient reasons.
Then in the late 1970's prime numbers suddenly took center stage in an area of technology known as cryptography (encryption of data).
Two books to look at are The Code Book by Simon Singh, and Crypto: How The Code Rebels Beat the Government by Steven Levy.
The basic idea is to use computers to search for very large prime numbers, let's say one hundred digits. Then, a computer can very easily multiply them together to form an even larger product of about 200 digits.
The trick is: If I give you just the product of the two primes and ask you to find the orignal primes, this factorization problem cannot be solved by a computer easily. In fact, using present-day algorithms on present-day supercomputers it might take centuries (or even many times the lifetime of the universe) before the answer is found.
This is an example of a one-way function, where (in this case) forming a product by multiplying together two primes is trivial, but the reverse process of factoring (pulling the primes back out of the product) is effectively impossible. This forms the basis of RSA public-key encryption.
Not to say that some future mathematician might not devise some devilishly clever factorization algorithm that greatly speeds the time required to factor a large number. I don't think it's been proven computationally impossible. Anyone know?
Anyway, this is the great thing about math. You never know which areas of inquiry -- old or new -- might turn out to have important applications in the future. George Boole, of Boolean algebra fame, doubtless never anticipated that he was laying the foundation of digital electronics.
Another example: Fractals, chaos, and various "string theories" of modern physics all draw upon topology in ways never originally anticipated for that area of math.
On Conjectures about Primes:
I think there'another conjecture that every even number can be expressed as the sum of two (odd) primes (this was parodied by Douglas Hofstadter as "Every odd number...sum of even primes!" in Godel, Escher, & Bach).
I recall one about primes occuring in pairs, n and n+2. There (is?/isn't?) a bound N beyond which (n>N) no such pairs exist.
Great question, Kendor!
10-01-02, 08:47 AM FlyingHellfish Thank you both for the wonderful compliment!
Professor, the 2 conjectures you stated about primes at the end of your post are well-known due to the fact that to this day they remain conjectures!
...every even number can be expressed as the sum of two (odd) primes... is another version of Goldbach's Conjecture. There are several different versions-the original, the weak, the strong, etc-all of which are very similar to each other. By proving one, you essentially can prove them all.
...primes occuring in pairs, n and n+2... is also another famous concept. It is believed that there are an infinite number of prime pairs, such as (3,5), (17, 19), and (41, 43), but once again, no one has been able to prove it true or false. Doing a quick internet search, the largest known prime pair is (242206083 * 238880 + 1,242206083 * 238880 - 1), both with 11713 digits!
10-05-02, 06:11 PM WiteoutKing The best actual usefulness I've found for primes and factorization is that you can quadrisect the amount of numbers to manually divide when finding prime factorizations. The range being {2<=n<=sqrtx|n is prime}, x being the number to factor, n being the factor itself.
"Use the primes to find the primes. 'Tis a prime method of speed and time."
10-06-02, 08:55 PM Kendor Very good information; thank you! [way above head]
Does anyone need a beer?
This message has been edited. Last edited by: DorianGreyed,
it is NOT considered a prime number either.
Hope I helped,
