Editing Finite geometry (section)

=== Order of planes ===
A finite plane of '''order''' ''n'' is one such that each line has ''n'' points (for an affine plane), or such that each line has ''n'' + 1 points (for a projective plane). One major open question in finite geometry is:
:''Is the order of a finite plane always a prime power?''
This is conjectured to be true.

Affine and projective planes of order ''n'' exist whenever ''n'' is a [[prime power]] (a [[prime number]] raised to a [[Positive number|positive]] [[integer]] [[exponent]]), by using affine and projective planes over the finite field with {{nowrap|1=''n'' = ''p''<sup>''k''</sup>}} elements. Planes not derived from finite fields also exist (e.g. for <math>n=9</math>), but all known examples have order a prime power.<ref>{{Cite book|url=https://books.google.com/books?id=VwqN86g68sIC&pg=PA146|title=Discrete Mathematics Using Latin Squares|last1=Laywine|first1=Charles F.|last2=Mullen|first2=Gary L.|date=1998-09-17|publisher=John Wiley & Sons|isbn=9780471240648|language=en}}</ref>

The best general result to date is the [[Bruck–Ryser theorem]] of 1949, which states:
:If ''n'' is a [[positive integer]] of the form {{nowrap|4''k'' + 1}} or {{nowrap|4''k'' + 2}} and ''n'' is not equal to the sum of two integer [[Square (algebra)|square]]s, then ''n'' does not occur as the order of a finite plane.

The smallest integer that is not a prime power and not covered by the Bruck–Ryser theorem is 10; 10 is of the form {{nowrap|4''k'' + 2}}, but it is equal to the sum of squares {{nowrap|1<sup>2</sup> + 3<sup>2</sup>}}. The non-existence of a finite plane of order 10 was proven in a [[computer-assisted proof]] that finished in 1989 – see {{Harv|Lam|1991}} for details.

The next smallest number to consider is 12, for which neither a positive nor a negative result has been proved.