# 171 (number)

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

 ← 170 171 172 →
Cardinalone hundred seventy-one
Ordinal171st
(one hundred seventy-first)
Factorization32 × 19
Divisors1, 3, 9, 19, 57, 171
Greek numeralΡΟΑ´
Roman numeralCLXXI
Binary101010112
Ternary201003
Senary4436
Octal2538
Duodecimal12312
HexadecimalAB16

## In mathematics

171 is a 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]

It is the number of 3-digit Duck Numbers, i.e., Numbers with at least one digit zero, in Decimal Base.

## References

1. ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
2. ^ Sloane, N. J. A. (ed.). "Sequence A001045 (Jacobsthal sequence)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
3. ^ Sloane, N. J. A. (ed.). "Sequence A006905 (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. ^
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.