Some comments on Baillie-PSW pseudoprimes.

*(English)*Zbl 1052.11006A Baillie-PSW pseudoprime is defined as a composite number which is both a base 2-pseudoprime and a Lucas pseudoprime. In the present paper, some calculations are proposed, related to the construction of Baillie-PSW pseudoprimes. The authors warn us that they have no example of such a number, although they are certain they could construct one if only they “could search through a rather large space in which such an example will live.”

Finally, the following claim is proposed, to be proved or disproved: Let \(m\) and \(n\) be relatively prime integers. Let \(A\) and \(B\) be disjoint sets of primes, with no prime dividing \(mn\). Suppose that for each reduced residue class \(x\) of \(m\) and \(y\) of \(n\) there are nonempty subsets \(S\), \(T\) of \(A\) and \(U, V\) of \(B\) such that \(f(S)\equiv x\pmod m\) and \(f(U)\equiv x\pmod m\), \(f(T)\equiv y\pmod n\) and \(f(V)\equiv y\pmod n\). Then for each reduced residue class \(z\) of \(mn\), there is a subset \(W\) of \(A\cup B\) such that \(f(W)\equiv z\pmod {mn}\).

Finally, the following claim is proposed, to be proved or disproved: Let \(m\) and \(n\) be relatively prime integers. Let \(A\) and \(B\) be disjoint sets of primes, with no prime dividing \(mn\). Suppose that for each reduced residue class \(x\) of \(m\) and \(y\) of \(n\) there are nonempty subsets \(S\), \(T\) of \(A\) and \(U, V\) of \(B\) such that \(f(S)\equiv x\pmod m\) and \(f(U)\equiv x\pmod m\), \(f(T)\equiv y\pmod n\) and \(f(V)\equiv y\pmod n\). Then for each reduced residue class \(z\) of \(mn\), there is a subset \(W\) of \(A\cup B\) such that \(f(W)\equiv z\pmod {mn}\).

Reviewer: Solomon Marcus (Bucureşti)