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-   -   Algorithm for combining Carmichael numbers (https://www.mersenneforum.org/showthread.php?t=24771)

devarajkandadai 2019-09-14 13:11

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.

devarajkandadai 2019-09-21 05:40

[QUOTE=devarajkandadai;525821]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.[/QUOTE]
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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