the conditions of Znám's problem, except that the largest value equals one plus the product of the other terms, rather than being a proper divisor. Thus, it is a solution to the improper Znám problem, but not a solution to Znám's problem as it is usually defined. One solution to the proper Znám problem, for , is . A few calculations will show that Connection to Egyptian fractions Any solution to the improper Znám problem is equivalent (via division by the product of the values ) to a solution to the equation where as well as each must be an integer, and conversely any such solution corresponds to a solution to the improper Znám problem. However, all known solutions have , so they satisfy the equation That is, they lead to an Egyptian fraction representation of the number one as a sum of unit fractions. Several of the cited papers on Znám's problem study also the solutions to this equation. describe an application of the equation in topology, to the classification of singularities on surfaces, and describe an application to the theory of nondeterministic finite automata. Number of solutions The number of solutions to Znám's problem for any is finite, so it makes sense to count the total number of solutions for each . 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 number of solutions for small values of , starting with , forms the sequence 2, 5, 18, 96 . Presently, a few solutions are known for and , but it is unclear how many solutions remain undiscovered for those values of . However, there are infinitely many solutions if is not fixed: showed that there are at least 39 solutions for each , improving earlier results proving the existence of fewer solutions; conjecture that the number of solutions for each value of grows monotonically with . 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; it is unknown whether infinitely many solutions of this type exist. References Notes Sources .