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Some comments on Baillie-PSW pseudoprimes. (English) Zbl 1052.11006
A 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 $$f(W)\equiv z\pmod {mn}$$.

##### MSC:
 11A51 Factorization; primality 11N25 Distribution of integers with specified multiplicative constraints 11A07 Congruences; primitive roots; residue systems 11B39 Fibonacci and Lucas numbers and polynomials and generalizations