51 (fifty-one) is the natural number 51 following 50 and preceding 52.

In mathematics

Fifty-one is a pentagonal number as well as a centered pentagonal number (one of the few numbers to be both) and an 18-gonal number and a Perrin number. It is also the 6th Motzkin number, telling the number of ways to draw non-intersecting chords between any six points on a circle’s boundary, no matter where the points may be located on the boundary.

Since the greatest prime factor of 512 + 1 = 2602 is 1301, which is substantially more than 51 twice, 51 is a Størmer number. There are 51 different cyclic Gilbreath permutations on 10 elements, and therefore there are 51 different real periodic points of order 10 on the Mandelbrot set.

Since 51 is the product of the distinct Fermat primes 3 and 17, a regular polygon with 51 sides is constructible with compass and straightedge, the angle π/51 is constructible, and the number cos π/51 is expressible in terms of square roots.
In history

Alexander Hamilton wrote 51 of the 85 federalist papers.
In other fields

51 is:

The atomic number of antimony
The code for international direct dial phone calls to Peru
The last possible television channel number in the UHF bandplan for American terrestrial television since December 31, 2011, when channels 52—69 were withdrawn
The number of the laps of the Azerbaijan Grand Prix.
In the 2006 film Cars, 51 was Doc Hudson’s number.

See also

AD 51, a year in the Julian calendar
List of highways numbered 51
The model number of the P-51 Mustang World War II fighter aircraft
Area 51, a parcel of U.S. military-controlled land in southern Nevada, apparently containing a secret aircraft testing facility
Photo 51, an X-ray image of key importance in elucidating the structure of DNA in the 1950s
Fifty-One Tales, part of the title of a collection of stories by Lord Dunsany
«51st State», any future US state, usually referring to Washington D.C. or Puerto Rico. Occasionally used in commentary to refer to non-US entities (such as Alberta) adopting US-like policies or seeking to become part of the US.
51 (film), a 2011 American horror film
«Fifty-One», an episode of Breaking Bad
Pastis 51, often just called «51», is a brand of pastis owned by Pernod Ricard.
Greg Murphy used «#51» for most of the seasons he raced in V8 Supercars