Editing Znám's problem (section)

== Number of solutions ==
The number of solutions to Znám's problem for any <math>k</math> is finite, so it makes sense to count the total number of solutions for each <math>k</math>.{{sfn|Janák|Skula|1978}} {{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 similar to that for Sylvester's sequence, but with a different set of initial values.{{sfn|Sun|1983}} The number of solutions for small values of <math>k</math>, starting with <math>k=5</math>, forms the sequence{{sfn|Brenton|Vasiliu|2002}}
:[[2 (number)|2]],&nbsp;[[5 (number)|5]],&thinsp;[[18 (number)|18]],&nbsp;[[96 (number)|96]] {{OEIS|id=A075441}}.
Presently, a few solutions are known for <math>k=9</math> and <math>k=10</math>, but it is unclear how many solutions remain undiscovered for those values of <math>k</math>.
However, there are infinitely many solutions if <math>k</math> is not fixed:
{{harvtxt|Cao|Jing|1998}} showed that there are at least 39 solutions for each <math>k\ge 12</math>, improving earlier results proving the existence of fewer solutions;<ref>{{harvnb|Cao|Liu|Zhang|1987}} {{harvnb|Sun|Cao|1988}}</ref> {{harvtxt|Sun|Cao|1988}} [[conjecture]] that the number of solutions for each value of <math>k</math> grows [[Sequence#Increasing and decreasing|monotonically]] with <math>k</math>.{{sfn|Sun|Cao|1988}} 

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 number|prime]], their product is a [[primary pseudoperfect number]];{{sfn|Butske|Jaje|Mayernik|2000}} it is unknown whether infinitely many solutions of this type exist.