Prime numbers and primality testing Yahoo Group Creating primes =============================================== cashogor Message 1 of 9 May 12, 2004 ----------------------------------------------- Hi all! (Please forgive my english) I´ve noticed that given p1, p2,...,pn correlative prime numbers: p1*p2*p3*....*pi + p(i+1)*p(i+2)*....*pn usually gives a prime number. Example: 2*3*5*7*11 + 13*17*19*23*29 = 2803043 I expected to have a result divisible by a 'near' prime such as 31 or 37, but I obtained a prime number. Any observations? N. =============================================== Payam Samidoost Message 2 of 9 May 12, 2004 ----------------------------------------------- Hi N. > 2*3*5*7*11 + 13*17*19*23*29 = 2803043 = 311 * 9013 P. =============================================== cashogor Message 3 of 9 May 12, 2004 ----------------------------------------------- Ups!!! 2803043 is not prime!!! :P I was looking for the smallest factor of the result to see how 'near' it is from the others used in the calculus. 2 + 3 = 5 ---------> 5 2*3 + 5*7 = 41 -----------> 41 2*3*5 + 7*11*13 = 1031 -----------> 1031 2*3*5*7 + 11*13*17*19 = 46399 -----------> 46399 2*3*5*7*11 + 13*17*19*23*29 = 2803043 ---------> 311*9013 N. =============================================== Payam Samidoost Message 4 of 9 May 12, 2004 ----------------------------------------------- for each n<=17 I have written n followed by the factorization of (p[1]*...*p[n])+(p[n+1]*...*p[2n]) (1) 5 (2) 41 (3) 1031 (4) 46399 (5) 311 * 9013 (6) 127 * 1945991 (7) 137 * 187060999 (8) 179 * 37897 * 495289 (9) 181 * 2904630622811 (10) 98299 * 358373 * 2448049 (11) 501923753 * 31963547227 (12) 10499 * 305078184752140339 (13) 619 * 827 * 1495106677900766629 (14) 18016051 * 10888330227638071729 (15) 571252434769 * 89991171579989521 (16) 8353 * 84982897 * 181369373 * 125339844677 (17) 631 * 1317703 * 6264170627636194115699149 It is prime for n=1,2,3,4,24,25,45,59 and no mor for n<100 further observations: (a) the numbers have no small factor.(small means less than p[2n]). (b) they are pairwise coprime (c) they are squarefree Proof is needed. Payam =============================================== David Cleaver Message 5 of 9 May 12, 2004 ----------------------------------------------- How about when you just use + and * operations on the first k primes, using one plus operator and the rest as multiplications? Like this: 2 + 3 = 5 2*3 + 5 = 11 2 + 3*5 = 17 2*3*5 + 7 = 37 2*3 + 5*7 = 41 2 + 3*5*7 = 107 2*3*5 + 7*11 = 107 (the rest with 5 primes are all composite) 2*3*5*7 + 11*13 = 353 2*3*5 + 7*11*13 = 1031 2*3 + 5*7*11*13 = 5011 2 + 3*5*7*11*13 = 15017 2*3*5*7*11*13 + 17 = 30047 2*3*5*7*11 + 13*17 = 2531 2*3*5 + 7*11*13*17 = 17047 2*3 + 5*7*11*13*17 = 85091 ... I guess we could check with all combinations of addition and multiplication. [obviously we coulnt' have ALL multiplications cuz that would negate what this group is all about ;) ] 2 + 3 = 5 2 + 3 + 5 = 10 (no good) 2 * 3 + 5 = 11 2 + 3 * 5 = 17 2 + 3 + 5 + 7 = 17 2 + 3 + 5 * 7 = 40 (no good) 2 + 3 * 5 + 7 = 24 (no good) 2 + 3 * 5 * 7 = 107 2 * 3 + 5 + 7 = 18 (no good) 2 * 3 + 5 * 7 = 41 2 * 3 * 5 + 7 = 37 ... So, to name these, we could call them N_k [ N for the creator, k for the number of primes] so that the above list could be written as: N_2(+) = 5 N_3(+,+) = 10 (no good) N_3(*,+) = 11 N_3(+,*) = 17 N_4(+,+,+) = 17 ... (or maybe use 1 for addition and 0 for multiplication) ... N_2(1) = 5 N_3(11) = 10 (no good) N_3(01) = 11 N_3(10) = 17 N_4(111) = 17 ... (or maybe now, convert the 1's and 0's from binary to decimal #'s) ... N_2(1) = 5 N_3(3) = 10 (no good) N_3(1) = 11 N_3(2) = 17 N_4(7) = 17 ... This way, if anyone was interested in searching these types of numbers, we could all have an easy system to reference these numbers. Just an idea I had. Its late and I'm about to fall asleep, so if this is all just jibba-jabba, then feel free to ignore me. -David C. cashogor wrote: > > Ups!!! > > 2803043 is not prime!!! > > :P > > I was looking for the smallest factor of the result to see how 'near' > it is from the others used in the calculus. > > 2 + 3 = 5 ---------> 5 > 2*3 + 5*7 = 41 -----------> 41 > 2*3*5 + 7*11*13 = 1031 -----------> 1031 > 2*3*5*7 + 11*13*17*19 = 46399 -----------> 46399 > 2*3*5*7*11 + 13*17*19*23*29 = 2803043 ---------> 311*9013 > > N. =============================================== Payam Samidoost Message 6 of 9 May 12, 2004 ----------------------------------------------- > (a) the numbers have no small factor.(small means less than p[2n]). > Proof is needed. It is trivial! If p<p[2n] divides p[1]*...*p[n]+p[n+1]*...*p[2n], since it divides one side of the +, it must divide the other side which is impossible. QED Payam =============================================== Payam Samidoost Message 7 of 9 May 12, 2004 ----------------------------------------------- The same proof works as well for the numbers proposed by David Cleaver. Payam =============================================== Payam Samidoost Message 8 of 9 May 12, 2004 ----------------------------------------------- > (p[1]*...*p[n])+(p[n+1]*...*p[2n]) > (b) they are pairwise coprime > Proof is needed. It is NOT true: Let f(n)=(p[1]*...*p[n])+(p[n+1]*...*p[2n]) Then 1283 divides both f(57) and f(80) 3221 divides both f(81) and f(109) 14939 divides both f(241) and f(242) 5569 divides both f(164) and f(262) Payam =============================================== Jens Kruse Andersen Message 9 of 9 May 12, 2004 ----------------------------------------------- Payam Samidoost wrote: > (p[1]*...*p[n])+(p[n+1]*...*p[2n]) > It is prime for n=1,2,3,4,24,25,45,59 > and no mor for n<100 Primeform/GW can search much deeper with a simple input file: ABC2 p($a)#+p(2*$a)#/p($a)# a: from 1 to 1600 pfgw agrees: It is 3-prp for n=1,2,3,4,24,25,45,59 ... and for n=1238. That number is 10091# + 22091#/10091# (5193 digits). Not much of a prime creator. There is nothing unusual about n=1,2,3,4 when they are small numbers without the smallest prime factors. -- Jens Kruse Andersen =============================================== Cached by Georg Fischer at Nov 14 2019 12:47 with clean_yahoo.pl V1.4