209 (two hundred ndnine) is the

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...

following

208 Year 208 ( CCVIII) was a leap year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Aurelius and Geta (or, less frequently, year 961 ''Ab urbe condita' ...

and preceding

210 Year 210 ( CCX) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Faustinus and Rufinus (or, less frequently, year 963 '' Ab urbe condit ...

In mathematics

*There are 209 spanning trees in a 2 × 5

grid graph In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space , forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a latti ...

, 209

partial permutation In combinatorial mathematics, a partial permutation, or sequence without repetition, on a finite set ''S'' is a bijection between two specified subsets of ''S''. That is, it is defined by two subsets ''U'' and ''V'' of equal size, and a one-to-one ...

s on four elements, and 209 distinct undirected simple graphs on 7 or fewer unlabeled vertices. *209 is the smallest number with six representations as a sum of three positive squares. These representations are: *:209 . :By

Legendre's three-square theorem In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers :n = x^2 + y^2 + z^2 if and only if is not of the form n = 4^a(8b + 7) for nonnegative integers and . The ...

, all numbers congruent to 1, 2, 3, 5, or 6 mod 8 have representations as sums of three squares, but this theorem does not explain the high number of such representations for 209. *, one less than the product of the first four prime numbers. Therefore, 209 is a

Euclid number In mathematics, Euclid numbers are integers of the form , where ''p'n''# is the ''n''th primorial, i.e. the product of the first ''n'' prime numbers. They are named after the ancient Greek mathematician Euclid, in connection with Euclid's theor ...

of the second kind, also called a Kummer number. One standard proof of

Euclid's theorem Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proved by Euclid in his work ''Elements''. There are several proofs of the theorem. Euclid's proof Euclid offered ...

that there are infinitely many primes uses the Kummer numbers, by observing that the prime factors of any Kummer number must be distinct from the primes in its product formula as a Kummer number. However, the Kummer numbers are not all prime, and as a

semiprime In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime ...

(the product of two smaller prime numbers ), 209 is the first example of a composite Kummer number.

References

Integers {{Number-stub