Editing Znám's problem (section)

== Connection to Egyptian fractions ==
Any solution to the improper Znám problem is equivalent (via division by the product of the values <math>x_i</math>) to a solution to the equation
<math display=block>\sum\frac1{x_i} + \prod\frac1{x_i}=y,</math>
where <math>y</math> as well as each <math>x_i</math> must be an integer, and [[converse (logic)|conversely]] any such solution corresponds to a solution to the improper Znám problem. However, all known solutions have <math>y=1</math>, so they satisfy the equation
<math display=block>\sum\frac1{x_i} + \prod\frac1{x_i}=1.</math>
That is, they lead to an [[Egyptian fraction]] representation of the number one as a sum of [[unit fraction]]s. Several of the cited papers on Znám's problem study also the solutions to this equation. {{harvtxt|Brenton|Hill|1988}} describe an application of the equation in [[topology]], to the classification of [[singularity (mathematics)|singularities]] on surfaces,{{sfn|Brenton|Hill|1988}} and {{harvtxt|Domaratzki|Ellul|Shallit|Wang|2005}} describe an application to the theory of [[nondeterministic finite automata]].{{sfn|Domaratzki|Ellul|Shallit|Wang|2005}}