In number theory, Znám's problem asks which sets of integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. Znám's problem is named after the Slovak mathematician Štefan Znám, who suggested it in 1972, although other mathematicians had considered similar problems around the same time. The initial terms of Sylvester's sequence almost solve this problem, except that the last chosen term equals one plus the product of the others, rather than being a proper divisor. showed that there is at least one solution to the (proper) Znám problem for each . Sun's solution is based on a recurrence similar to that for Sylvester's sequence, but with a different set of initial values. The Znám problem is closely related to Egyptian fractions. It is known that there are only finitely many solutions for any fixed . It is unknown whether there are any solutions to Znám's problem using only odd numbers, and there remain several other open questions. The problem Znám's problem asks which sets of integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. That is, given , what sets of integers are there such that, for each , divides but is not equal to A closely related problem concerns sets of integers in which each integer in the set is a divisor, but not necessarily a proper divisor, of one plus the product of the other integers in the set. This problem does not seem to have been named in the literature, and will be referred to as the improper Znám problem. Any solution to Znám's problem is also a solution to the improper Znám problem, but not necessarily vice versa. History Znám's problem is named after the Slovak mathematician Štefan Znám, who suggested it in 1972. had posed the improper Znám problem for , and , independently of Znám, found all solutions to the improper problem for . showed that Znám's problem is unsolvable for , and credited J. Janák with finding the solution for . Examples Sylvester's sequence is an integer sequence in which each term is one plus the product of the previous terms. The first few terms of the sequence are Stopping the sequence early produces a set like that almost meets