How do people come up with a list of large prime numbers?
84537
Let's say that we want 1024 bit key. We want to generate two primes that are 512 bit long.\n It is known that the number of primes less than n is about n/ln(n) when n is sufficiently big. So the probability of a number near n being prime is about 1/ln(n). n is 512 bit long, so the probability is about 1/ln(2^512) = 1/(512*ln(2)) = 1/350.\n Since we know that even numbers are not primes(except for 2), we only need to check for odd numbers. This doubles the probability to about 1/180. Therefore, we only need few hundred trials to generate RSA key that is 1024 bit long.\n\nAs you can see, generating a RSA key is actually simple. On the other hand, factoring takes forever. This is why RSA is very secure(for now).
652436
Dear Visit and search in the left side(Curios search):\nhttp://primes.utm.edu/curios/page.php...
850647
Too long to print but here's the website\n\n(2^3021376)(2^3021377-1)\n\nThere are 1,819,050 digits\n\nhttp://www.calendarhome.com/prime/perfect4.html\n\nPlease note that the highest perfect prime number currently known, will continue to change when a higher one is found.
1045069
There are currently 43 known Mersenne Primes (of the form 2^p-1, where p is prime).  All numbers up to the 39th prime have been exhaustively searched (see the mersenne.org status page) but there are gaps beyond that so it is possible that there is/are more Mersenne Primes between some of the later numbers.  I've denoted those with ? in the sequence numbering.  \n\nThe numbers get pretty large, so after awhile all I can give you is the number in the form 2^p-1 and the number of digits in the number.  You can go look up the long form if you really need it. :) \n\n1) 2^2-1 = 3\n2) 2^3-1 = 7\n3) 2^5-1 = 31\n4) 2^7-1 = 127\n5) 2^13-1 = 8191\n6) 2^17-1 = 131071\n7) 2^19-1 = 524287\n8) 2^31-1 = 2147483647\n9) 2^61-1 = 2 305 843 009 213 693 951\n10) 2^89-1 = 618 970 019 642 690 137 449 562 111\n11) 2^107-1 = 162 259 276 829 213 363 391 578 010 288 127\n12) 2^127-1 = 170 141 183 460 469 231 731 687 303 715 884 105 727\n13) 2^521-1 = 157 digits\n14) 2^607-1 = 183 digits\n15) 2^1,279-1 = 386 digits\n16) 2^2,203-1 = 664 digits\n17) 2^2,281-1 = 687 digits\n18) 2^3,217-1 = 969 digits\n19) 2^4,253-1 = 1,281 digits\n20) 2^4,423-1 = 1,332 digits\n21) 2^9,689-1 = 2,917 digits\n22) 2^9,941-1 = 2,993 digits\n23) 2^11,213-1 = 3,376 digits\n24) 2^19,937-1 = 6,002 digits\n25) 2^21,701-1 = 6,533 digits\n26) 2^23,209-1 = 6,987 digits\n27) 2^44,497-1 = 13,395 digits\n28) 2^86,243-1 = 25,962 digits\n29) 2^110,503-1 = 33,265 digits\n30) 2^132,049-1 = 39,751 digits\n31) 2^216,091-1 = 65,050 digits\n32) 2^756,839-1 = 227,832 digits\n33) 2^859,433-1 = 258,716 digits\n34) 2^1,257,787-1 = 378,632 digits\n35) 2^1,398,269-1 = 420,921 digits\n36) 2^2,976,221-1 = 895,932 digits\n37) 2^3,021,377-1 = 909,526 digits\n38) 2^6,972,593-1 = 2,098,960 digits\n39) 2^13,466,917-1 = 4,053,946 digits\n40?) 2^20,996,011-1 = 6,320,430 digits\n41?) 2^24,036,583-1 = 7,235,733 digits\n42?) 2^25,964,951-1 = 7,816,230 digits\n43?) 2^30,402,457-1 = 9,152,052 digits\n\nAnd if you want to know the English name of that 43rd known Mersenne 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.)
890978
To get prime factors, keep dividing by numbers and divide those until you can't divide them no more!  For example,\n\n34 \n17 * 2\n\n(Can't divide 17 any more and can't divide 2 -- both of these are prime numbers)\n\nAnother example,\n\n120\n12 * 10\n2 * 6 * 2 * 5\n2 * 2 * 3 * 2 * 5 (<-- all prime numbers here)\n\nTry to get the rest yourself and post the answers, we'll tell you if you got them right!  ;-)
571951
Start at 2, and eliminate every second number after that.\nThen start at 3, and eliminate every third number after that.\n4 will already have been eliminated in step 1.\nStart at 5, and eliminate every fifth number after that.\nContinue this way.  The numbers left uneliminated are prime.
1098310
There are lots of them.  The easy way is to multiply 5 times 6 times 7.  This number is divisible by 5, 6, and 7.  So -- just add two to it.\n\nThis is 212.\n\nYou want another?  How about 2102?  210000000002?\n422? 420002?\n\nI got a million of 'em.
22351
Many have tried, yet none have found a formula. As noted on MathWorld,\n\n"Euler commented "Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate" (Havil 2003, p. 163). In a 1975 lecture, D. Zagier commented "There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision" (Havil 2003, p. 171)."
982295
A prime number is a positive integer such that the only integers which evenly divide it are 1 and itself.\n\nFor example, 7 is prime because only 1 and 7 divide it evenly.\n\nThe sequence of prime numbers begins thus:\n2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 ...\nAnd there are infinitely many of them.\n\nhttp://en.wikipedia.org/wiki/Prime_number
588013
I don't think the average man on the street thinks about prime numbers at all. He really doesn't have any reason to.\n\nHowever, mathematics is the foundation of almost all science and the study of prime numbers has led to many breakthroughs in mathematics, and useful theorems in number theory which have proved crucial in developing different branches of mathematics, and these in turn have led to direct applications in the sciences and technologies which affect us all. If it wasn't for prime numbers many of the electronic transactions that we make in various ways would not be secure. Prime numbers and number theory are extremely useful.
1090956
25, if you're talking about whole, positive numbers.\n\nThey are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.\n\nMany people debate whether or not 0 and 1 should also be considered prime.  0 is just weird and 1 is part of the definition of prime, so I don't think they really are prime, personally.
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.)
357003
if you take the number N and add one it must be prime... \n\ntake any number it can be writen as a product of primes, so in this case let us just examin N+1/p_j\n\nclearly N/p_j is an integer, and 1/p_j is clearly not an integer. \n\nSo if N were not prime then there would exist p_j such that N+1/p_j was an integer.   But that can not be true, therefore the assumption that there are only finitly many primes is incorrect, and the opposite is true, there are infinitly many primes QED
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.
236479
Any number can be broken down into it's prime factors.  So find all the prime factors of a number, and see what there are 3 of.\n\n8 is an easy example - the first prime number that goes evenly into 8 is 2 so\n\n2*4 = 8; but 4 is not a prime number.  The prime number 2 also goes into 4 so\n\n2*(2*2) = 8; 2*2*2 = 8; 2^3 = 8, so 2 is the cube root of 8.\n\nA slightly trickier one - 3,375.  Five is a prime number that clearly goes into this, so\n\n5*675 = 3,375; now 5 will also go into 675\n\n5*5*135 = 3,375; again, you can factor out another 5\n5*5*5*27 = 3,375; no more fives, but the prime number 3 will go into 27\n\n5*5*5*3*9 = 3,375; and 9 is still not a prime number, but 3 goes again - \n\n5*5*5*3*3*3 = 3,375\n\nOk, now it's broken down into prime numbers, so see what there are 3 of - there are 3 5's and 3 3's so you could regroup like this:\n\n5*3*5*3*5*3 = 3,375 or\n15*15*15 = 15^3 = 3,375\n\nSo the cube root of 3,375 is 15\n\n~~Now~~  If there had been leftover numbers that there weren't three of, say the number had been 48 - first pull out the prime factors -\n\n2*24 = 48\n2*2*12 = 48\n2*2*2*6 = 48\n2*2*2*2*3 = 48\n\nYou would group 3 of the 2's and then group the leftovers\n\n2^3 * 2*3 = 48\n2^3 * 6 = 48\n\nThen the cube root of 48 = cuberoot(2^3) * cuberoot(6), or cuberoot48 = 2*(cuberoot6)
207196
There is an infinite number of primes.  They'll never be done.\n\nHow Many Primes are There? \n\nWe will now consider the question of how many prime numbers there are.  The following investigation of this question is over two thousand years old, and is attributed to Euclid. \n\nTwo possible answers to the question are that there are either finitely many primes, or there are infinitely many of them.  We will show that the first answer is not possible, and therefore there are infinitely many primes.  (This approach is referred to as "proof by contradiction."  In general, proof by contradiction works like this: we know that there are only two possible answers to a question; we assume that the first answer is the correct one, and then discover that this leads to an inconsistency; and so we conclude that it is actually the second of the two answers that is correct.) \n\nEuclid's Proof \n\nSo let's suppose that there are only finitely many primes, say n of them.  We could then make a list of these n primes, say p1, p2, ..., pn.  Then consider the integer \n\n\nobtained by multiplying all of the primes, and then adding 1.  This new number m is either prime or it is not.  If it is not, then it is divisible by some prime.  But we are supposing that the only primes are p1 through pn--and none of these primes divides m, for when m is divided by one of these primes, the remainder is 1.  This contradicts the known fact that every positive integer greater than 1 is divisible (evenly) by some prime.  So the conclusion must be that m is prime.  But clearly m is greater than all of the primes p1, p2, ..., pn , so this contradicts the supposition that p1, p2, ..., pn is a complete list of all primes.  Our conclusion is that there are not finitely many primes; there are infinitely many.
293052
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97\n\n101 is the next prime number ... \nso 24 prime numbers between 1 - 100