171 (number)

← 170 171 172 →
← 170 171 172 173 174 175 176 177 178 179 → List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
Cardinal	one hundred seventy-one
Ordinal	171st (one hundred seventy-first)
Factorization	32 × 19
Divisors	1, 3, 9, 19, 57, 171
Greek numeral	ΡΟΑ´
Roman numeral	CLXXI
Binary	101010112
Ternary	201003
Senary	4436
Octal	2538
Duodecimal	12312
Hexadecimal	AB16

171 (one hundred [and] seventy-one) is the natural number following 170 and preceding 172.

In mathematics

171 is the 18th triangular number ^[1] and a Jacobsthal number. ^[2]

There are 171 transitive relations on three labeled elements, ^[3] and 171 combinatorially distinct ways of subdividing a cuboid by flat cuts into a mesh of tetrahedra, without adding extra vertices. ^[4]

The diagonals of a regular decagon meet at 171 points, including both crossings and the vertices of the decagon. ^[5]

There are 171 faces and edges in the 57-cell, an abstract 4-polytope with hemi-dodecahedral cells that is its own dual polytope. ^[6]

Within moonshine theory of sporadic groups, the friendly giant ${\displaystyle \mathbb {M} }$ is defined as having cyclic groups ⟨ ${\displaystyle m}$ ⟩ that are linked with the function,

${\displaystyle f_{m}(\tau )=q^{-1}+a_{1}q+a_{2}q^{2}+...,{\text{ }}a_{k}}$ ∈ ${\displaystyle \mathbb {Z} ,{\text{ }}q=e^{2\pi i\tau },{\text{ }}\tau >0;}$ where ${\displaystyle q}$ is the character of ${\displaystyle \mathbb {M} }$ at ${\displaystyle m}$.

This generates 171 moonshine groups within ${\displaystyle \mathbb {M} }$ associated with ${\displaystyle f_{m}}$ that are principal moduli for different genus zero congruence groups commensurable with the projective linear group ${\displaystyle \operatorname {PSL_{2}} (\mathbb {Z} )}$. ^[7]

See also

• The year AD 171 or 171 BC
• List of highways numbered 171
• All pages with titles containing 171

References

1. Sloane, N. J. A. (ed.). "SequenceA000217(Triangular numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
2. Sloane, N. J. A. (ed.). "SequenceA001045(Jacobsthal sequence)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
3. Sloane, N. J. A. (ed.). "SequenceA006905(Number of transitive relations on n labeled nodes)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
4. Pellerin, Jeanne; Verhetsel, Kilian; Remacle, Jean-François (December 2018). "There are 174 subdivisions of the hexahedron into tetrahedra". ACM Transactions on Graphics. 37 (6): 1–9. arXiv: 1801.01288 . doi:10.1145/3272127.3275037. S2CID   54136193.
5. Sloane, N. J. A. (ed.). "SequenceA007569(Number of nodes in regular n-gon with all diagonals drawn)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
6. McMullen, Peter; Schulte, Egon (2002). Abstract Regular Polytopes. Encyclopedia of Mathematics and its Applications. Vol. 92. Cambridge: Cambridge University Press. pp. 185–186, 502. doi:10.1017/CBO9780511546686. ISBN   0-521-81496-0. MR   1965665. S2CID   115688843.
7. Conway, John; Mckay, John; Sebbar, Abdellah (2004). "On the Discrete Groups of Moonshine" (PDF). Proceedings of the American Mathematical Society. 132 (8): 2233. doi: 10.1090/S0002-9939-04-07421-0 . eISSN   1088-6826. JSTOR   4097448. S2CID   54828343.