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Williams numbers. (English) Zbl 1204.11185
Summary: Let $$N$$ be a composite squarefree number; $$N$$ is said to be a Carmichael number if $$p-1$$ divides $$N-1$$ for each prime divisor $$p$$ of $$N$$. H. C. Williams [Can. Math. Bull. 20, 133–143 (1977; Zbl 0368.10011)] has stated an interesting problem of whether there exists a Carmichael number $$N$$ such that $$p+1$$ divides $$N+1$$ for each prime divisor $$p$$ of $$N$$. This is a long standing open question, and it is possible that there is no such number.
For a given nonzero integer $$a$$, we call $$N$$ an $$a$$-Korselt number if $$N$$ is composite, squarefree and $$p-a$$ divides $$N-a$$ for all primes $$p$$ dividing $$N$$. We will say that $$N$$ is an $$a$$-Williams number if $$N$$ is both an $$a$$-Korselt number and a $$(-a)$$-Korselt number.
Extending the problem of Williams, one may ask more generally if for a given nonzero integer $$a$$, there is an $$a$$-Williams number. We give an affirmative answer to the question for $$a = 3p$$, where $$p$$ is a prime number such that $$3p-2$$ and $$3p+2$$ are primes. We also prove that each $$a$$-Williams number has at least three prime factors.

MSC:
 11Y16 Number-theoretic algorithms; complexity 11Y11 Primality 11A51 Factorization; primality