168 (number)

168 (one hundred [and] sixty-eight) is the natural number following 167 and preceding 169.

It is the number of hours in a week, or 7 x 24 hours.

Mathematics

168 is the of fourth Dedekind number,^[1] and one of sixty-five idoneal numbers.^[2] It is one less than a square (13²) as well as the product of the first two perfect numbers,^[3]

${\displaystyle 168=6\times 28}$

168 is also coefficient four in the expansion of Eisenstein series ${\displaystyle E_{2}}$.^[4]

There are 168 primes less than 1000.^[a]

Composite index

The 128th composite number is 168,^[5] one of a few numbers ${\displaystyle n}$ in the list of composites whose indices are the product of strings of digits of ${\displaystyle n}$ in decimal representation.

The first nine ${\displaystyle n}$ with this property are the following:^[5]

The next such number is 198 where 19 × 8 = 152. The median between twenty-one integers [48, 68] is 58, where 148 is the median of forty-one integers [168, 128].

Totient values

For the Euler totient ${\displaystyle \varphi (n)}$ there is ${\displaystyle \varphi (168)=48}$,^[6] where ${\displaystyle \varphi (48)=16}$ is also equivalent to the number of divisors of 168;^[7] only eleven numbers have a totient of 48:{65, 104, 105, 112, 130, 140, 144, 156, 168, 180, 210}.^[6][d]

Idoneal number

Leonard Euler noted 65 idoneal numbers (the most known, of only a maximum possible of two more), such that ${\displaystyle x^{2}+Dy^{2}}$ for an integer ${\displaystyle D}$, expressible in only one way, yields a prime power or twice a prime power.^[2][10]

Of these, 168 is the forty-fourth, where the smallest number to not be idoneal is the fifth prime number 11.^[2] The largest such number 1848 (that is equivalent with the number of edges in the union of two cycle graphs of order 42)^[11] contains a total of thirty-two divisors whose arithmetic mean is 180^[12][13] (the second-largest number to have a totient of 48).^[6] Preceding 1848 in the list of idoneal numbers is 1365,^[e] whose arithmetic mean of divisors is equal to 168^[12][13] (while 1365 has a totient of 576 = 24²).

Where 48 is the 27th ideoneal number, 408 is the 58th.^[2][f] On the other hand, the total count of known idoneal numbers (65), that is also equal to the sum of ten integers [2, ..., 11], has a sum-of-divisors of 84.^[15]

Numbers of the form 2ⁿ

In base 10, 168 is the largest of ninety-two known ${\displaystyle n}$ such that ${\displaystyle 2^{n}}$ does not contain all numerical digits from that base (i.e. 0, 1, 2, ..., 9).^[16]

${\displaystyle 2^{68}}$ is the first number to have such an expression where between the next two ${\displaystyle n}$ is an interval of ten integers: [70, 79];^[16] the median value between these is 74, the composite index of 100.^[5][g]

Abstract algebra

168 is the number of maximal chains in the Bruhat order of symmetric group ${\displaystyle \mathrm {S} _{4},}$^[20] which is the largest solvable symmetric group with a total of ${\displaystyle 4!=24}$ elements.

168 is the order of the second smallest nonabelian simple group ${\displaystyle \mathrm {PSL} (2,7).}$ From Hurwitz's automorphisms theorem, 168 is the maximum possible number of automorphisms of a genus 3 Riemann surface, this maximum being achieved by the Klein quartic, whose symmetry group is ${\displaystyle \mathrm {PSL} (2,7)}$;^[21] the Fano plane, isomorphic to the Klein group, has 168 symmetries.

In other fields

Dominoes

In the game of dominoes, tiles are marked with a number of spots, or pips. A Double 6 set of 28 tiles contains a total of 168 pips.

Numerology

Some Chinese consider 168 a lucky number, because it is roughly homophonous with the phrase "一路發" which means "fortune all the way", or, as the United States Mint claims, "Prosperity Forever".^[22]

Notes

References

External links

Wikimedia Commons has media related to 168 (number).

• Number Facts and Trivia: 168
• The Number 168
• The Positive Integer 168
• Prime curiosities: 168

This page was last edited on 10 April 2024, at 18:20