Problems & Puzzles: Puzzles

Puzzle 739. Carmichael numbers of special form Emmanuel Vantieghem sent the following puzzle: It is well known that, when  6m+1, 12m+1  and  18m+1  are prime, their product is a Carmichael number.   However, there are many cases in which only two of the three numbers 6m+1, 12m+1, 18m+1 are prime while their product is a Carmichael number.  Examples : 31*61*91, 67*133*199, 91*181*271  (and there are  44  such numbers for  m < 10000).    There are  11  cases (for  m < 10^6) in which  only one of the three numbers 6m+1, 12m+1, 18m+1 is prime while their product is a Carmichael number: m = 22, 105, 225, 609, 12259, 15125, 37961, 85410, 108585, 449905, 477044.    But, for  m <3*10^9, there is no Carmichael number of the form (6m+1)(12m+1)(18m+1) in which none of the three numbers 6m+1, 12m+1, 18m+1 is prime.   So, we can ask:   Q. Find an  m  such that  6m+1, 12m+1  and 18m+1  are composite while  (6m+1)(12m+1)(18m+1)  is a Carmichael number or prove that no such  m  exist.

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Emmanuel added: "I continued my search till  m = 10^10.  No such  m  was found."

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