For which positive integers $\ n\ $ is $\ N:=n^2+1\ $ a Carmichael-number ?
The smallest examples are summarized in this table (the program is currently still in progress) :
gp > forstep(m=2,10^9,2,n=m^2+1;if(Mod(2,n)^(n-1)==1,if(carm(n)==1,print(m," ",n," ",factor(n)))))
216 46657 [13, 1; 37, 1; 97, 1]
1560 2433601 [17, 1; 37, 1; 53, 1; 73, 1]
8208 67371265 [5, 1; 13, 1; 37, 1; 109, 1; 257, 1]
592956 351596817937 [17, 1; 37, 1; 197, 1; 433, 1; 6553, 1]
649800 422240040001 [13, 1; 37, 1; 41, 1; 73, 1; 241, 1; 1217, 1]
4321800 18677955240001 [29, 1; 61, 1; 101, 1; 197, 1; 701, 1; 757, 1]
21415680 458631349862401 [37, 1; 53, 1; 89, 1; 19009, 1; 138241, 1]
Is it known whether infinite many such Carmichael numbers exist ? And how can I enumerate them more efficiently ?