Editing Znám's problem (section)

== 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
{{bi|left=1.6|2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443 {{OEIS|id=A000058}}.}}
Stopping the sequence early produces a set like <math>\{2, 3, 7, 43\}</math> that almost meets 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.{{sfn|Brenton|Hill|1988}} 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 <math>k=5</math>, is <math>\{2, 3, 7, 47, 395\}</math>. A few calculations will show that

{| style="margin-left:1.6em"
|align="right" | 3 × 7 × 47 × 395 || + 1 = || 389866, || &nbsp; || which is divisible by but unequal to 2,
|-
|align="right" | 2 × 7 × 47 × 395 || + 1 = || 259911, || &nbsp; || which is divisible by but unequal to 3,
|-
|align="right" | 2 × 3 × 47 × 395 || + 1 = || 111391, || &nbsp; || which is divisible by but unequal to 7,
|-
|align="right" | 2 × 3 × 7 × 395 || + 1 = || 16591, || &nbsp; || which is divisible by but unequal to 47, and
|-
|align="right" | 2 × 3 × 7 × 47 || + 1 = || 1975, || &nbsp; || which is divisible by but unequal to 395.
|}