51 (number)

← 50 51 52 →
← 50 51 52 53 54 55 56 57 58 59 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	fifty-one
Ordinal	51st (fifty-first)
Factorization	3 × 17
Divisors	1, 3, 17, 51
Greek numeral	ΝΑ´
Roman numeral	LI, li
Binary	1100112
Ternary	12203
Senary	1236
Octal	638
Duodecimal	4312
Hexadecimal	3316

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

In mathematics

Fifty-one is

• a pentagonal number ^[1] as well as a centered pentagonal number ^[2] and an 18-gonal number ^[3]
• 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. ^[4]
• a Perrin number, ^[5] coming after 22, 29, 39 in the sequence (and the sum of the first two)
• a Størmer number, since the greatest prime factor of 51² + 1 = 2602 is 1301, which is substantially more than 51 twice. ^[6]
• There are 51 different cyclic Gilbreath permutations on 10 elements, ^[7] and therefore there are 51 different real periodic points of order 10 on the Mandelbrot set. ^[8]
• 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.

References

1. "Sloane's A000326 : Pentagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
2. "Sloane's A005891 : Centered pentagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
3. "Sloane's A051870 : 18-gonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
4. "Sloane's A001006 : Motzkin numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
5. "Sloane's A001608 : Perrin sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
6. "Sloane's A005528 : Størmer numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
7. Sloane, N. J. A. (ed.). "SequenceA000048". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
8. Diaconis, Persi; Graham, Ron (2012), "Chapter 5: From the Gilbreath Principle to the Mandelbrot Set", Magical Mathematics: the mathematical ideas that animate great magic tricks, Princeton University Press, pp. 61–83.