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 2019-09-14, 13:11 #1 devarajkandadai     May 2004 13C16 Posts Algorithm for combining Carmichael numbers We can combine two Carmichael numbers to form another Carmichael number. An example: 1729 = 7*13*19 294409 = 37*73*109 Both are of type (6m+1)(12m+1)(18m+1); we get the first by putting m = 1 and the second by putting m = 6. First step: Check whether for any given value of m, 6m+1, 13m+1 and 18m+1 are primes.If for two given values of m, we get two Carmichael numbers by applying Korselt's criterion. If so their product will be a Carmichael number subject to satisfaction of Korselt's criterion.
2019-09-21, 05:40   #2

May 2004

4748 Posts

Quote:
 Originally Posted by devarajkandadai We can combine two Carmichael numbers to form another Carmichael number. An example: 1729 = 7*13*19 294409 = 37*73*109 Both are of type (6m+1)(12m+1)(18m+1); we get the first by putting m = 1 and the second by putting m = 6. First step: Check whether for any given value of m, 6m+1, 13m+1 and 18m+1 are primes.If for two given values of m, we get two Carmichael numbers by applying Korselt's criterion. If so their product will be a Carmichael number subject to satisfaction of Korselt's criterion.
Just combined 3 Carmichael numbers to form one Carmichael number; all three are of type (6m+1)(12m+1)(18m+1). 1729*294409*118901521 =60524817082337881.

Conjecture: There can be many Carmichael numbers of this type i.e. k number of Carmichael numbers can be combined to form one Carmichael number where k belongs to N and greater than 2.

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