--- In

primenumbers@yahoogroups.com, miltbrown@... wrote:

>

> A Carmichael Number, C, has a form such that

>

> C-1 has as a factor (2^n*3^m) where (n+m)>=3

>

> Now, 10^5+3 as a PRP is prime because 10^5+2=2*3*166667,

> not a Carmichael Number.

>

> Also,

> 10^11+3 is prime because 10^11+2 = 2*3*7*1543*1543067

> 10^17+3 is prime because 10^17+2 = 2*3*7*61*65701*594085421

> 10^18+3 is prime because 10^18+2 = 2*3*17*131*1427*52445056723

> 10^39+3 is prime because 10^39+2 = 2*23*41*107*1053497*32333333

> 10^101+3 is prime because 10^101+2 = (2*3)^1*(primes >3 0

>

> In fact, any PRP of the form 10^x+3 is Prime,

> because it can not have a factor (2^n*3^m) where n>1 and m>1

>

?

for(n=1,7,N=10^n+3;if(!isprime(N),for(a=2,N-2,if(Mod(a,N)^(N-1)==1,print("Not

prime "N" is "a"-PRP and factors of N-1 are "factor(N-1));break))))

Not prime 1003 is 237-PRP and factors of N-1 are [2, 1; 3, 1; 167, 1]

Not prime 10003 is 664-PRP and factors of N-1 are [2, 1; 3, 1; 1667, 1]

Not prime 10000003 is 104503-PRP and factors of N-1 are [2, 1; 3, 1;

47, 1; 35461, 1]

Move along; there is nothing to see here. Really!

Paul