Editing Znám's problem (section)

{{Short description|On divisibility among sets of integers}}
[[Image:Znam-2-3-11-23-31.svg|thumb|301px|Graphical demonstration that 1 = 1/2 + 1/3 + 1/11 + 1/23 + 1/31 + 1/(2×3×11×23×31). Each row of {{mvar|k}} squares of side length&thinsp;1/{{mvar|k}} has total [[area]]&thinsp;1/{{mvar|k}}, and all the squares together exactly cover a larger square with area 1. The bottom row of 47058 squares with side length 1/47058 is too small to see in the figure and is not shown.]]
In [[number theory]], '''Znám's problem''' asks which [[set (mathematics)|sets]] of [[integer]]s 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. {{harvtxt|Sun|1983}} showed that there is at least one solution to the (proper) Znám problem for each <math>k\ge 5</math>. Sun's solution is based on a [[recurrence relation|recurrence]] similar to that for Sylvester's sequence, but with a different set of initial values.

The Znám problem is closely related to [[Egyptian fraction]]s. It is known that there are only finitely many solutions for any fixed <math>k</math>. It is unknown whether there are any solutions to Znám's problem using only [[odd number]]s, and there remain several other [[open problem|open questions]].