A question about twin primes?
482620
All primes (ignoring 2) are odd.\n\nDivide odd numbers into 3 groups: those of the form 6n - 1, those of the form 6n + 1, and those of the form 6n + 3.\n\nIgnoring 3 itself, all primes are of the form 6n + 1 or 6n - 1, as all numbers of the form 6n + 3 or 6n - 3 are divisible by 3 and not prime. \n\nYour proposition follows from that fact.
955219
ok so prime factorization is basically factoring a number into it's multiple primes. for example, to prime factorize 12, u'd figure out what it's multiples are- 4 and 3 (or 6&2..1&12 u get the point), ok now break down 4 to prime numbers- 2 and 2. and since 3 is already prime u don't have to do anything to it. so the prime factorization of 12 is 2^2 and 3
1046445
the first one is 2, the last one is.. doesn't exist.\n\nLet Pn be the n-th prime number. For all interger N, (P1xP2xP3x....xPN)+1 is not divisible by Pn for all n from 1 to N, and hence is another prime number. So, for every discovered prime number, there is at least one other prime ((P1xP2xP3x..xPN)+1) larger than it.\n\n\nregarding your additional comment..\n\nprime number is defined to be is a positive integer that has no positive integer p>1 divisors other than 1 and p itself. So, from definition alone, it has excluded 0 and 1.\n\nActually, 0/0 is not 0. It is undefined. However, u can simply define the discontinuity at 0/0 to be 0 in this case. But even so, this has nothing to do with prime number, from the definition of prime.\n\nI know u find 0 fascinating. In fact, like infinity, it is. It always has a special place in mathematics. A lot of time, when defining mathematical objects, 0 is defined explicitly because it has a special place. It is the point of inflection of +ve and negative, it is a additive identity, A+0=A, and for any number is its multiplicative identity, A0=0...with the exception of infinity(infinity is not a number anyway). For example, the set {0} with its normal + - and x operations is a Ring. And if u define 0/0=0, and adding / operation, it is a Field. But, this is trivial and of not much use. The point is, 0 is indeed special although it is not a prime.
165290
Yes - again I agree with Aseem (why is he so much faster than me?) :-)\n\nFirst you have to ask is 7 a prime number ; the answer is yes (see my answer to your other prime number question) so you go to the second test. Can you divide 60 by 7 with no remainder? Answer is no; so 7 is not a prime factor of 60. The prime factors of 60 are 2(2), 3, 5;\n\n60/2 = 30\n30/2 = 15\n15/3 = 5\n5 is a prime so you go no further; \nSo you end up with 2, 2, 3, 5
203743
SpawnOfSpock, the fact that there are an infinite amount of numbers does not imply there are also an infinite amount of primes. It is a serious enough question to have been proven in myriad ways by some of the greatest mathematicians.\nThe following is a common one. Suppose there were a finite amount of primes. Then the product p of all these primes is divisible by each prime. Consider the number p+1. Since p+1 was not in the set, it must be divisible by a prime number. Let that prime be s. But (p+1)/s = q + 1/s for some integer q and thus p+1 must also be prime. Thus our assumption that the set of all primes is finite must be wrong.
1023056
There is no largest prime, since there are an infinite number of prime numbers, but there is a largest *known* prime number.  It's a Mersenne prime (of the form 2^p-1) where p is another prime.  Specifically it is 2 ^ 30,402,457 - 1.  The number is too big to represent here because it is 9,152,052 digits long.\n\nThis number and most of the recent Mersenne primes were calculated using a collective effort known as the 'Greater Internet Mersenne Prime Search' (or GIMPS) for short.  Several tens of thousands of volunteers around the world are using background processing time on their PCs to check these high-digit Mersenne primes.\n\nThe number 30,402,457 was tested as the exponent in 2^p-1 and the result was found to be prime as part of this GIMPS project.  The credit goes to both the project (all contributors to the GIMPS project) and to the people who were running that specific test.  In this case it was Dr. Curtis Cooper and Dr. Steven Boone, professors at Central Missouri State University who discovered it December 15, 2005.  The official credit for this new discovery is listed as "Cooper, Boone, Woltman, Kurowski, et. al.".  By the way, I'm one of the et. al. having racked up several years of computational time running tests.\n\nThe search continues and there is a prize for the person that finds the next prime number of over 10 million digits... join now if you want to share part of the $100,000 pot.\n\nIf you want to know the English name of the current largest known prime, it starts, "three hundred fifteen tremillia millia quin quagin millia sescendo octo gintillion, four hundred sixteen tremillia millia quin quagin millia sescenun octo gintillion..." (See the last link for the full name.)
653675
Yes.  If you assume there are a finite number prime numbers, then a list of them would look like a1, a2, ..., an.  If you can make this number...\n\n(a1)(a2)(a3)...(an) + 1\n\n...then dividing this number by any of the known primes, you would have a remainder of 1, so therefore this number is a prime.  This is a contradiction that the primes are finite, so the primes must be infinite.
216703
Yes, 1000003 happens to be prime but unless I'm missing something your reasoning is not convincing. How do we know that r + r' is not divisible by n ?
118857
Ahkens provided the correct answer and Aseem provided the correct answer and the formulaic method of determining the relevant prime numbers. If you are confused, an easier way of obtaining the prime numbers for low numbers like 255 is simply to start with the lowest primes and divide them into 255 until you exhaust the possibilities. Firstly, the definition of a prime number is a number that can be divided, without a remainder, only by itself and by 1. (The number 1 is itself, however, NOT considered to be a prime - notice that Aseem correctly excluded it from his answer despite it appearing in the results of his formulae). \n\nUsing this definition, the prime numbers between 0 and 100 are:\n\n2,3,5,7,11,13,17,19,23,29,\n31,37,41,43,47,53,59,61,\n67,71,73,79,83,89,97\n\nSo 2 is the first prime that fits the above definition; but 2 doesnt divide into 255 so it doesn't fit into this answer.\n\nNext is 3; 255 / 3 = 85 so 3 is your first number.\nNext prime is 5; 85 / 5 = 17 so 5 is your next number.\n17 is a prime because it too fits the definition so 17 is your next number and as there are no remainders you have solved the problem. 3,5,17
27160
If you are referring to twin prime numbers (primes separated by only 2), then no, it has not yet been proven that they are infinite in number. It has been conjectured, but the proof has been elusive.
594189
34,\n\nYou are having the series as 2 times <prime numbers>\n2*2, 2*3, 2*5, 2*7, 2*11, 2*13, 2*17, 2*19, 2*23...\n = \n4, 6, 10, 14, 22, 26, 34, 38, 46 ...
7923
There isn't one. Mathematicians are working on it, but they have yet to find a pattern that covers all prime numbers (except for the fact they are divisible only by themselves and one.) There are patterns that cover some prime numbers, such as paired primes (where the difference between the two prime numbers is 2, such as 5 and 7), but there is no currently known way to quickly produce or check primes, which has important implications for the RSA algorithm (used in PGP for example), which relies on the complexity of primes for its security.
652436
Dear Visit and search in the left side(Curios search):\nhttp://primes.utm.edu/curios/page.php...
698319
You will have to be realistic.  A character file containing all the primes up to only 10^12 would occupy about 400 gigabytes.  Compare that with your broadband download ration, and see how many more powers of 10 you could manage even if the files existed.\n\nThere are world-class researchers exploring the density of primes.  They make use of refined approximation formulae, instead of an actual list of primes, once they get up towards 10^20.
263111
I don't particularly have a favorite number, but I have some interesting observations from other's numbers.\n\nOut of 44 whole numbers provided so far, 29 of them are prime numbers. \n\nA full 66% of all numbers were prime. Interesting, because there are only 12.6% prime numbers between 1 and 7919. I would suggest that there is a significantly high proportion of primes to non-primes.\n\nThe highest prime as a favorite was 347. I wonder if this was a deliberate choice, or a subconcious preference. \n\nI could hypothesize that people are naturally drawn towards primes because they seem 'unique' and 'different' - Everyone wants to stand out. \n\nAnyway, Interesting. I wonder if anyone has ever done a study that would relate primes to people's unique number choices.
1067732
If the N numbers:\n\np, p + d, p + 2d, ..., p + (N-1)d\n\nare all prime, and if q is a prime, then either:\n\nq divides d and all elements of the sequence have the same residue modulo q (which would be nonzero unless N=1 and p=q)\n\nor\n\nany q consecutive elements of the sequence make up a complete residue set modulo q, i.e., their remainders modulo q will be the set {0, 1, 2, ..., 1-1} not necessarily in that order.\n\nthen note that unless 3 divides d that the first three taken modulo 3 form a complete residue set (i.e., the remainders modulo 3 will be the set {0, 1, 2} in some order).  Likewise the first five taken modulo five will have remainders modulo five of {0, 1, 2, 3, 4} modulo five.  So searches for N consecutive primes in arithmetic progression usually take N to be a multiple of all of the primes less than or equal to N.\n\nA search for\n\nprime number records consecutive arithmetic progression\n\nin Google,\n\nhttp://www.google.com/search?q=prime+number+records+consecutive+arithmetic+progression\n\nyields as its first link\n\nPrime Arithmetic Progression -- from Wolfram MathWorld\nhttp://mathworld.wolfram.com/PrimeArithmeticProgression.html\n\nwhich contains some sample sequences (smaller p and larger N, smaller N and larger p) and says it was last updated Friday, May 12, 2006 -- sample quote from that page:\n\n"\nFor example, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 is a 10-term arithmetic progression of primes with difference 210.\n"\n\n(note that 210 = 2 * 3 * 5 * 7 is the product of all primes less than or equal to ten).\n\nDan D
842088
111 is divisible by 3 (plus others) so is not a prime\n63  is divisible by 3 (plus others) so is not a prime\n15 is divisible by 3 (plus others) so is not a prime\n41 and 43 are twin primes