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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
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