# 163 (number)

__: Natural number__

**Short description**
| ||||
---|---|---|---|---|

Cardinal | one hundred sixty-three | |||

Ordinal | 163rd (one hundred sixty-third) | |||

Factorization | prime | |||

Prime | 38th | |||

Divisors | 1, 163 | |||

Greek numeral | ΡΞΓ´ | |||

Roman numeral | CLXIII | |||

Binary | 10100011_{2} | |||

Ternary | 20001_{3} | |||

Quaternary | 2203_{4} | |||

Quinary | 1123_{5} | |||

Senary | 431_{6} | |||

Octal | 243_{8} | |||

Duodecimal | 117_{12} | |||

Hexadecimal | A3_{16} | |||

Vigesimal | 83_{20} | |||

Base 36 | 4J_{36} |

**163** (**one hundred [and] sixty-three**) is the natural number following 162 and preceding 164.

## In mathematics

163 is a strong prime in the sense that it is greater than the arithmetic mean of its two neighboring primes.

163 is a lucky prime^{[1]} and a fortunate number.^{[2]}

163 is a strictly non-palindromic number, since it is not palindromic in any base between base 2 and base 161.

Given 163, the Mertens function returns 0, it is the fourth prime with this property, the first three such primes are 2, 101 and 149.^{[3]}

As approximations, [math]\displaystyle{ \pi \approx {2^9 \over 163} \approx 3.1411... }[/math], and [math]\displaystyle{ e \approx {163 \over 3\cdot4\cdot5} \approx 2.7166\dots }[/math]

163 is a permutable prime in base 12, which it is written as 117, the permutations of its digits are 171 and 711, the two numbers in base 12 are 229 and 1021 in base 10, both of which are prime.

The function [math]\displaystyle{ f(n) = n^2 - n + 41 }[/math] gives prime values for all values of [math]\displaystyle{ n }[/math] between 0 and 39, while for [math]\displaystyle{ n \lt 10^7 }[/math] approximately half of all values are prime. 163 appears as a result of solving [math]\displaystyle{ f(n)=0 }[/math], which gives [math]\displaystyle{ n = (-1+ \sqrt{-163} ) / 2 }[/math].

163 is a Heegner number, the largest of the nine such numbers. That is, the ring of integers of the field [math]\displaystyle{ \mathbb{Q}(\sqrt{-a}) }[/math] has unique factorization for [math]\displaystyle{ a=163 }[/math]. The only other such integers are [math]\displaystyle{ a = 1, 2, 3, 7, 11, 19, 43, 67 }[/math]. (sequence A003173 in the OEIS)

163 is the number of linearly **Z**-independent McKay-Thompson series for the monster group, which also represent their collective maximum dimensional representation. This fact about 163 might be a clue for understanding monstrous moonshine.^{[4]}

[math]\displaystyle{ \sqrt{163} }[/math]
appears in the Ramanujan constant, since -163 is a quadratic nonresidue to modulo all the primes 3, 5, 7, ..., 37. In which [math]\displaystyle{ e^{\pi \sqrt{163}} }[/math] almost equals the integer 262537412640768744 = 640320^{3} + 744. Martin Gardner famously asserted that this identity was exact in a 1975 April Fools' hoax in *Scientific American*; in fact the value is 262537412640768743.99999999999925007259...

## In other fields

**163** is also:

- The number of days following the first day of Passover (Pesach), used to calculate the date of Rosh Hashanah, the Jewish New Year.
- The Burj Khalifa in Dubai, United Arab Emirates, which is the tallest building in the world, has 163 floors.
- In darts, 163 is the lowest number that cannot be shot with three darts on a standard dart board.

## See also

- 163 (disambiguation)

## References

- ↑ Sloane, N. J. A., ed. "Sequence A031157 (Numbers that are both lucky and prime)". OEIS Foundation. https://oeis.org/A031157. Retrieved 2016-05-28.
- ↑ Sloane, N. J. A., ed. "Sequence A005235 (Fortunate numbers)". OEIS Foundation. https://oeis.org/A005235. Retrieved 2016-05-28.
- ↑ Sloane, N. J. A., ed. "Sequence A100669 (Zeros of the Mertens function that are also prime)". OEIS Foundation. https://oeis.org/A100669. Retrieved 2016-05-29.
- ↑ He, Yang-Hui; McKay, John (2015). "Sporadic and exceptional". pp. 1–49. arXiv:1505.06742 [math.AG]. (See p. 13)

- Wells, D. (1987).
*The Penguin Dictionary of Curious and Interesting Numbers*(pp. 141–142). London: Penguin Group.

## External links

- Clark, Alex. "163 and Ramanujan Constant".
*Numberphile*. Brady Haran. http://www.numberphile.com/videos/163.html. - The Number 163
- The Positive Integer 163
- Prime curiosities: 163
- 163.com website from NetEase
- The coolest number is 163
- 163, the Monster and Number Theory
- VirtueScience: 163
- International Brotherhood of Electrical Workers Local Union 163
- US Highway 1633?], from Mathematics Stack Exchange

Original source: https://en.wikipedia.org/wiki/163 (number).
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