Bruck–Ryser–Chowla theorem
The set of Bruck- Ryser- Chowla is a combinatorial statement about possible block plans that specifies necessary conditions for their existence.
The theorem says that if a symmetric block plan exists, then applies
- If v is even, then is a perfect square;
- If v is odd, then the Diophantine equation has
The sentence was in 1949 proved for the special case of projective planes by Richard Bruck and Herbert John Ryser and 1950 with generalized Chowla Sarvadaman to more general symmetric block plans.
- 2.1 Articles
- 2.2 textbooks that introduce the topic
Finite projective planes
In the special case of a minimal symmetric two - block plan with - so for finite projective planes - can be formulated as the set:
Summarizing a finite projective plane of order as special symmetric block diagram, then are the parameters that describe the block diagram
In this more specific formulation for projective planes of the sentence is also quoted as a set of Bruck and Ryser.
Inferences and examples
From the theorem then follows, for example, that there is to the orders 6 and 14 are not level, but he does not exclude the existence of levels of orders and out. It could be shown that no projective plane of the assembly 10 exists. It follows that the conditions in the theorem of Bruck- Ryser- Chowla are not sufficient condition for the existence of block diagrams.
- The systems fulfill the necessary condition of the theorem for projective planes. In fact, there are levels with these orders, since they are both prime powers.
- About levels of the order of the sentence makes no statement as is. Since 27 is a prime power, there exists a level with these regulations.
Excluded orders
The sequence of numbers that can not be orders of a projective plane due to the set of Bruck and Ruyser, so the numbers that are not the sum of two square numbers, form the sequence A046712 in OEIS.
The smallest so that orders are excluded: 6, 14, 21, 22, 30, 33, 38, 42, 46, 54, 57, 62, 66, 69, 70, 77, 78, 86, 93, 94, 102, 105, 110, 114, 118, 126, 129, 133, 134, 138, 141, 142, 150, 154, 158, 161, 165, 166, 174, 177, 182, 186, 189, 190, 198, 201, 206, 209, 210, 213, 214, 217, 222, 230, 237, 238 ..