Number theory edit

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

${\displaystyle 168=6\times 28.}$ 

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

Composite index edit

The 128th composite number is 168,^[4] 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:^[4]

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 edit

For the Euler totient ${\displaystyle \varphi (n)}$  there is ${\displaystyle \varphi (168)=48}$ ,^[5] where ${\displaystyle \varphi (48)=16}$  is also equivalent to the number of divisors of 168;^[6] only eleven numbers have a totient of 48:{65, 104, 105, 112, 130, 140, 144, 156, 168, 180, 210}.^[5][d]

408,^[e] with a different permutation of the digits {0, 4, 8} where 048 is 48, has an totient of 128. So does the sum-of-divisors of 168,^[9]

${\displaystyle \sigma (168)=480=2^{5}\times 3\times 5}$ 

as one of nine numbers total to have a totient of 128.^[5]

Idoneal number edit

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).^[5] Preceding 1848 in the list of idoneal numbers is 1365,^[f] 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][g] 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 (or, one-half of 168).^[9]

Numbers of the form 2ⁿ edit

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).^[15]

${\displaystyle 2^{68}}$  is the first number to have such an expression where between the next two ${\displaystyle n}$  is an interval of ten integers: [70, 79];^[15] the median value between these is 74, the composite index of 100.^[4][h]

Cunningham number edit

As a number of the form ${\displaystyle b^{n}\pm 1}$  for positive integers ${\displaystyle n}$ , ${\displaystyle b}$  and ${\displaystyle b}$  not a perfect power, 168 is the thirty-second Cunningham number,^[19] where it is one less than a square:

${\displaystyle 168=13^{2}-1.}$ 

On the other hand, 168 is one more than the third member of the fourth chain of nearly doubled primes of the first kind {41, 83, 167},^[20][21] where 167 represents the thirty-ninth prime^[22] (with 39 × 2 = 78). The smallest such chain is {2, 5, 11, 23, 47}.

Eisenstein series edit

168 is also coefficient four in the expansion of Eisenstein series ${\displaystyle E_{2}}$ ,^[23] which also includes 144 and 96 (or 48 × 2) as the fifth and third coefficients, respectively — these have a sum of 240, which follows 144 and 187 in the list of successive composites ${\displaystyle A(n+1)=A(n)}$ ;^cf.[4] the latter holds a sum-of-divisors of 216 = 6³,^[9] which is the 168th composite number.^[4]

Abstract algebra edit

168 is the number of maximal chains in the Bruhat order of symmetric group ${\displaystyle \mathrm {S} _{4},}$ ^[24] 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 ${\displaystyle \mathrm {PSL} (2,7)}$ ;^[25] the Fano plane, isomorphic to the Klein group, has 168 symmetries.

Dominoes edit

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 edit

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".^[26]

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