Question

a) Use Exercise 48 to show that every integer of the form $(6m+1)(12m+1)(18m+1)$, where m is a positive integer and 6m+1, 12m+1, and 18m+1 are all primes, is a Carmichael number.

b) Use part (a) to show that 172, 947, 529 is a Carmichael number.

Short answer

a) If we multiply out this expression, we get $n=1296m^3+396m^2+36m+1$

Clearly $6m|n-1,12m|n-1\text{, and}18m\mid n-1$Therefore, the conditions of Exercise 48 are met, and we conclude that n is a Carmichael number.

b) Letting m=51 gives n=172,947,529

Step-by-step solution

Step: 1

c) If we multiply out this expression, we get $n=1296m^3+396m^2+36m+1$

Clearly $6m|n-1,12m|n-1\text{, and}18m\mid n-1$Therefore, the conditions of Exercise 48 are met, and we conclude that n is a Carmichael number.

d) Letting m=51 gives n=172,947,529