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 2019-08-11, 05:01 #1 devarajkandadai     May 2004 22·79 Posts Continued product Carmichael numbers Let me give an example of a set of continued product Carmichael numbers: a)2465 = 5*17*29 b)278545 = 5*17*29*113 c)93969665=5*17*29*113*337 d)63174284545 = 5*17*29*113*337*673 and e)169875651141505 = 5*17*29*113*337*673*2689 Algorithm for this type of c.p.Carmichael numbers is simple and I will illustrate how to derive b) above starting from a). Largest prime factor of a) is 29. Check the first prime generated by 28*k + 1; when k = 4 we get 113.
2019-08-13, 04:43   #2

May 2004

22·79 Posts

Quote:
 Originally Posted by devarajkandadai Let me give an example of a set of continued product Carmichael numbers: a)2465 = 5*17*29 b)278545 = 5*17*29*113 c)93969665=5*17*29*113*337 d)63174284545 = 5*17*29*113*337*673 and e)169875651141505 = 5*17*29*113*337*673*2689 Algorithm for this type of c.p.Carmichael numbers is simple and I will illustrate how to derive b) above starting from a). Largest prime factor of a) is 29. Check the first prime generated by 28*k + 1; when k = 4 we get 113.
Another set of continued product Carmichael numbers ( prefer to call them "spiral Carmichael numbers"): a)2821 = 7*13* 31
b)172081= 7*13*31*61
c)31146661 = 7*13*31*61*181
d)16850343601= 7*13*31*61*181*541
Important point: possibility of constructing such spiral Carmichael numbers strengthens my conjecture that, r, the number of prime factors of a Carmichael number is not bounded.

Last fiddled with by devarajkandadai on 2019-08-13 at 04:50

2019-09-24, 03:14   #3

May 2004

22×79 Posts

Quote:
 Originally Posted by devarajkandadai Another set of continued product Carmichael numbers ( prefer to call them "spiral Carmichael numbers"): a)2821 = 7*13* 31 b)172081= 7*13*31*61 c)31146661 = 7*13*31*61*181 d)16850343601= 7*13*31*61*181*541 Important point: possibility of constructing such spiral Carmichael numbers strengthens my conjecture that, r, the number of prime factors of a Carmichael number is not bounded.
Another set of spiral Carmichael numbers: 252601 = 41*61*101
151813201 = 41*61*101*601
182327654401=41*61*101*601*1201
875355068779201 = 41*61*101*601*1201*4801*
12605988345489273601 = 41*61*101*601*1201*4801*14401
726117534688527648691201 = 41*61*101*601*1201*4801*14401*57601

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