Znám's problem

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. 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 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. This problem does not seem to have been named in the literature, and is referred to here 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.

Origin

Znám's problem is named after the Slovak mathematician Štefan Znám, who suggested it in 1972.

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

2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443 (sequence

in the

Thus, it is a solution to the improper Znám problem, but not a solution to Znám's problem as it is usually defined.

A few calculations will show that

Connection to Egyptian fractions

The solutions to this equation have been applied to the classification of singularities on surfaces, and to the theory of nondeterministic finite automata.

Number of solutions

Sun's solution is based on a recurrence similar to that for Sylvester's sequence, but with a different set of initial values.

2, 5, 18, 96 (sequence A075441 in the OEIS).

It is unknown whether there are any solutions to Znám's problem using only odd numbers. With one exception, all known solutions start with 2. If all numbers in a solution to Znám's problem or the improper Znám problem are prime, their product is a primary pseudoperfect number.