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There Are Infinitely Many Prime Twins
fustflum writes "R. F. Arenstorf from Vanderbilt University has presented a 38-page possible proof of the twin-prime conjecture using methods from classical analytic number theory. The paper is on arxiv.org and is freely available to the public. Twin primes are pairs of primes where both p and p + 2 are prime. "It is conjectured that there are an infinite number of twin primes ... but proving this remains one of the most elusive open problems in number theory." More information about twin primes can be found on Mathworld."
Something I read in Science the other day: There's a new proof in review that there are infintely many sequences such as 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 -- primes that differ by a constant offset. See also Mathworld.
The sum of any two consecutive odd numbers is divisible by 4. I misread a question in first year and proved it.
I thought that was fairly neat, it makes me the life of the party when I tell it to people. (Well, not really. Depressing.) Does anyone know any other little tricks like that?
It's easy (for a mathematican) to prove that PI is infinite.
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I started trying to write out a proof, but it looks too messy in slashdot
Have a look at something like:
http://www.lrz-muenchen.de/~hr/numb/pi-irr.html
You can't assume that a certain positive percentage of *all* primes differ by two as stated in number two, because that's an analogous statement to what you're trying to prove.
;)
Say I have an infinite number of socks. All are white, except 3, which are grey. I have a positive percentage of grey socks, but that doesn't mean anything since that percentage is infinitessimal. It will be infinitessimal for any number of grey socks, so you can't say that you are assumed have a positive percentage of grey socks *unless* you have an infinite number of grey socks, and that's a tautological argument.
Chocolate chip, please
Karma: Contrapositive
If it turns out to be true, this will be super-duper-extraordinary - the man is probably in his 70s. G. H. Hardy wrote: "No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game". Wiles proved FLT at 40, Perelman of the purported Poincare proof is in his 30s... this is similar-level stuff. The only thing I can think of that even comes close is Fred Galvin in his 50s (?) proving the Dinitz conjecture.
You can follow discussions on sci.math and fr.sci.maths. Or read about how similar asymptotic proofs about properties of primes failed. Remember, this is arxiv - in the age of electronic preprints, you get many good proofs like Perelman's along with almost-proofs like Castro-Mahecha's and Dunwoody's.
Exactly, which is why that definition is no good either - there is an infinite amount of numbers which are a power of 2, so saying their percentage is 0% makes no sense, or conveys no interesting information. By that definition, an empty, a finite and even an infinite set could be 0% of all natural numbers.
No, you do not understand his proof. His proof makes the assumption that there is a finite number of primes. Then he disproves that assumption. Again:
..., Pn. If that set were finite, then the number (P1*P2*...*Pn)+1. would not have any prime factors. Therefore, that number would also be prime. Hence, there cannot be a finite set of primes.
Say there were a finite set of primes. Call the elements of that set P1, P2,
You are correct in saying that in the real world, multiplying the first N prime numbers together and adding 1 won't necessarily produce a prime.
For example, 2*3*5*7*11*13 + 1 = 30031 = 59*509.
However, the fact that such a number might have a rogue factor does not deny the proof of its validity, because the existence of a rogue prime factor would also discount the finite set of prime numbers. However, the above proof is valid without this.
Interestingly, it can be proven that there is a series of n consecutive composite (nonprime) numbers for ANY number n! This means there is some sequence of 10 trillion nonprime numbers. It seems almost contradictory to the infinicy of primes (though it is not).
s t2b.html -
..... . n! + n shows this conjecture must be true. The number n!, where n = 5, for example, has a value of 1 x 2 x 3 x 4 x 5, or 120. In the general case, n! + 2 is evenly divisible by 2, n! + 3 is evenly divisible by 3, and so on. Finally, n! + n is evenly divisible by n. Therefore, all the numbers in the sequence are composite. The sequence can be made arbitrarily long by picking a sufficiently large number n.
From http://www.fortunecity.com/emachines/e11/86/touri
At the same time, it's relatively easy to prove that consecutive primes can be as far apart as anyone would want. The sequence of numbers n! + 2, n! +3, n!
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To find an infinite number of prime siblings, you first need to find an appropriate set of numbers. To cut down on processing time, you should note that these numbers are seperated by 6, or a multiple thereof.
My son figured this out - with the help of some Lego - the answer is 332 (except for the Cherry ones that take a few less):
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http://www.sjbaker.org/gallery/lickomatic/index.h
www.sjbaker.org