Janus numbers—Classification and infinite families

We study the equations relating the product of two integers of equal length to the concatenation of those integers or of their digit reversals. We show that four of them are empty and that the remaining four reduce, by exchanging the roles of the factors, to two distinct problems, called symmetric and asymmetric. For the symmetric equation we exhibit a family of length-five solutions indexed by the divisors of 𝑀^2 lying between 𝑀 and 2𝑀 , and reduce the search for unbounded-length families of a certain shape to a finite system, whose only solution below an explicit bound occurs in base eight. For the asymmetric equation we exhibit a family of unbounded length in every composite base, indexed by the divisors of 𝑀 𝑛 between 2 and 𝑀 − 1.