561 is the smallest Carmichael number in the ring of integers. However it is only a pseudoprime in the ring of Gaussian integers. We can use pari to find the bases to which 561 is a pseudoprime. I found it is pseudoprime to bases 12 + i, 22 +i, 33 + i and 44 + i. Needless to say it is a pseudoprime to other bases too. More on this later.

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## Comments

## Fallout of amateur research

We can use pari to find the smallest divisor of phi(n) for which a congruence holds good. Example: consider the pseudoprime 341. This is pseudo to the base 2. To find d, the smallest divisor we run the program p(n) = (2^n -1)/341. It was found that when n = 10, the function is exactly divisible by 341. Application: When the program p(n) = (2^n+97)/341 was run for n = 1,10, no divisibility was found. Hence 2^n + 97 is not divisible for any value of n. However (2^n + 1007) is divisible by 341 before the program reaches 10. Similarly 40 is the smallest divisor of phi(561) ( 561 is a Carmichael number). Hence any relevant program pertaining to an exponential expression has to be run only till n reaches 40.