# What kind of special numbers are not suitable as RSA keys?

I have read that some integers are not appropriate to be chosen as the modulus in an RSA cryptosystem. Some of these numbers are those that, given a modulus $$n=pq$$, then $$p-1$$ or $$q-1$$ do not have large factors. This is due to the fact that there are factorization algorithms that allow this type of modulus to be factored efficiently.

My question is, what other types of integers are not suitable to be used as a modulus and why?

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Edit:

Other kind of numbers not suitable as RSA modulus:

• Probably the Q duplicates this Is it reasonable to assure that p-1 and q-1 aren't smooth? and this Williams' p+1 in tandem with Pollard's p−1? and this Finding strong primes and finally this Generation of strong primes Commented Oct 20, 2022 at 11:32
• It is edited in 2000, and it is old. Check the links that I've provided to you to see. Commented Oct 20, 2022 at 11:36
• Thanks @kelalaka, it seems that due to Williams p+1 we also have to worry about large factors at p+1 and q+1, which is very interesting. But I think my question remains open. What other kinds of numbers do we have to worry about? Commented Oct 20, 2022 at 11:39
• close primes due to Fermat's factoring Commented Oct 20, 2022 at 11:43
• Thans again kelalaka, i've edited the question. Let's see if anyone knows of other cases. Commented Oct 20, 2022 at 11:53