Carmichael number Determine is in general true or false Reca

Carmichael number Determine is, in general, true or false. Recall that a universal statement is true if it is true for all possible cases while it is false if there is even one counterexample. Be prepared to prove that your answer is correct by supplying a proof or counterexample, whichever is appropriate

Suppose that m> 0 and that 4m +1, 8m + 1, and 12m+ 1 are prime. Then n= (4m+ 1) (8m + 1) (12m+ 1) is a Carmichael number.

Solution:

- In order to prove that n is a Carmichael number we need to verify that n -1 is divisible by LCM of 6m, 12m and 18m that is 26m|n-1

Consider n= (4m+ 1) (8m + 1) (12m+ 1) n =1296m^3 +396m^2 +36m+1

n= 36m(36m^2 +11m+1)+1

n-1=36 m k n -1 is divisible by 36m

Hence, n is a Carmichael number

Could you check it for me please is it correct or not?

Solution

There is a mistake in question first of all. IT should be

Suppose that m> 0 and that 6m +1, 12m + 1, and 18m+ 1 are prime. Then n= (6m+ 1) (12m + 1) (18m+ 1) is a Carmichael number.

Same correction in proof

n= (6m+ 1) (12m + 1) (18m+ 1)

Everything else is correct.