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# 8

Please do not rely on any information it contains.

8 is the cube of 2, the largest cube in the sequence of Fibonacci numbers.

## Membership in core sequences

 Even numbers 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, ... A005843 Composite numbers 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, ... A002808 Cubes 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, ... A000578 Powers of 2 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ... A000079 Primes and powers of primes 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, ... A000961 Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ... A000045

In Pascal's triangle, 8 occurs only twice, namely in row 8, in the second and next to last positions.

## Sequences pertaining to 8

 Multiples of 8 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, ... A008590 Octagonal numbers 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, ... A000567 Centered octagonal numbers 1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, ... A016754 Octagonal pyramidal numbers 1, 9, 30, 70, 135, 231, 364, 540, 765, 1045, ... A002414

## Partitions of 8

There are twenty-two partitions of 8. The partitions of 8 into primes are 2 + 3 + 3 and 3 + 5.

## Roots and powers of 8

In the table below, irrational numbers are given truncated to eight decimal places.

 ${\displaystyle {\sqrt {8}}}$ 2.82842712 A010466 8 2 64 ${\displaystyle {\sqrt[{3}]{8}}}$ 2.00000000 A000038 8 3 512 ${\displaystyle {\sqrt[{4}]{8}}}$ 1.68179283 A011006 8 4 4096 ${\displaystyle {\sqrt[{5}]{8}}}$ 1.51571656 A011093 8 5 32768 ${\displaystyle {\sqrt[{6}]{8}}}$ 1.41421356 A002193 8 6 262144 ${\displaystyle {\sqrt[{7}]{8}}}$ 1.34590019 A011246 8 7 2097152 ${\displaystyle {\sqrt[{8}]{8}}}$ 1.29683955 A011247 8 8 16777216 ${\displaystyle {\sqrt[{9}]{8}}}$ 1.25992104 A002580 8 9 134217728 ${\displaystyle {\sqrt[{10}]{8}}}$ 1.23114441 A011249 8 10 1073741824 A001018

Of course the roots given above are the principal real roots. There are also negative real roots and complex roots.

• ${\displaystyle {\sqrt {8}}}$, ${\displaystyle -{\sqrt {8}}}$ (both real)
• ${\displaystyle {\sqrt[{3}]{8}}}$, ${\displaystyle -1\pm {\sqrt {-3}}}$ (the two complex roots are the same except for the sign of the imaginary part)
• ${\displaystyle {\sqrt[{4}]{8}}}$, ${\displaystyle -{\sqrt[{4}]{8}}}$, ${\displaystyle i{\sqrt[{4}]{8}}}$, ${\displaystyle -i{\sqrt[{4}]{8}}}$
• ${\displaystyle {\sqrt[{5}]{8}}}$, etc.

## Logarithms and eighth powers

In the OEIS specifically and mathematics in general, ${\displaystyle \log x}$ refers to the natural logarithm of ${\displaystyle x}$, whereas all other bases are specified with a subscript. Octal logarithms are only occasionally considered.

As above, irrational numbers in the following table are truncated to eight decimal places.

 ${\displaystyle \log _{8}2}$ 0.33333333 A010701 ${\displaystyle \log _{2}8}$ 3.00000000 2 8 256 ${\displaystyle \log _{8}e}$ 0.48089834 ${\displaystyle \log 8}$ 2.07944154 A016631 ${\displaystyle e^{8}}$ 2980.96 ${\displaystyle \log _{8}3}$ 0.52832083 A152956 ${\displaystyle \log _{3}8}$ 1.89278926 A113210 3 8 6561 ${\displaystyle \log _{8}\pi }$ 0.55049870 ${\displaystyle \log _{\pi }8}$ 1.81653468 ${\displaystyle \pi ^{8}}$ 9488.53 ${\displaystyle \log _{8}4}$ 0.66666666 ${\displaystyle \log _{4}8}$ 1.50000000 4 8 65536 ${\displaystyle \log _{8}5}$ 0.77397603 A153204 ${\displaystyle \log _{5}8}$ 1.29202967 A153739 5 8 390625 ${\displaystyle \log _{8}6}$ 0.86165416 A153493 ${\displaystyle \log _{6}8}$ 1.16055842 A153754 6 8 1.67962e+06 ${\displaystyle \log _{8}7}$ 0.93578497 A153618 ${\displaystyle \log _{7}8}$ 1.06862156 A153755 7 8 5.7648e+06 ${\displaystyle \log _{8}8}$ 1.00000000 8 8 1.67772e+07 ${\displaystyle \log _{8}9}$ 1.05664166 A154010 ${\displaystyle \log _{9}8}$ 0.94639463 A153756 9 8 4.30467e+07 ${\displaystyle \log _{8}10}$ 1.10730936 A154159 ${\displaystyle \log _{1}08}$ 0.90308998 A153790 10 8 1e+08

(See A001016 for the eighth powers of integers).

## Values for number theoretic functions with 8 as an argument

 ${\displaystyle \mu (8)}$ 0 ${\displaystyle M(8)}$ –2 ${\displaystyle \pi (8)}$ 4 ${\displaystyle \sigma _{1}(8)}$ 15 ${\displaystyle \sigma _{0}(8)}$ 4 ${\displaystyle \phi (8)}$ 4 ${\displaystyle \Omega (8)}$ 3 ${\displaystyle \omega (8)}$ 1 ${\displaystyle \lambda (8)}$ -1 This is the Carmichael lambda function. ${\displaystyle \lambda (8)}$ –1 This is the Liouville lambda function. ${\displaystyle \zeta (8)={\frac {\pi ^{8}}{9450}}}$ 1.0040773561979443393786852385... (see A013666). 8! 40320 ${\displaystyle \Gamma (8)}$ 5040

## Factorization of 8 in some quadratic integer rings

In ${\displaystyle \mathbb {Z} }$, 8 has the prime factorization of 2 3. But it has different factorizations in some quadratic integer rings.

 ${\displaystyle \mathbb {Z} [i]}$ ${\displaystyle (1-i)^{3}(1+i)^{3}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-2}}]}$ ${\displaystyle (-1)({\sqrt {-2}})^{6}}$ ${\displaystyle \mathbb {Z} [{\sqrt {2}}]}$ ${\displaystyle ({\sqrt {2}})^{6}}$ ${\displaystyle \mathbb {Z} [\omega ]}$ 2 3 ${\displaystyle \mathbb {Z} [{\sqrt {3}}]}$ ${\displaystyle (-1)(1-{\sqrt {3}})^{3}(1+{\sqrt {3}})^{3}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-5}}]}$ ${\displaystyle \mathbb {Z} [\phi ]}$ 2 3 ${\displaystyle \mathbb {Z} [{\sqrt {-6}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {6}}]}$ ${\displaystyle (-1)(2-{\sqrt {6}})^{3}(2+{\sqrt {6}})^{3}}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}}$ ${\displaystyle \left({\frac {1}{2}}-{\frac {\sqrt {-7}}{2}}\right)^{3}\left({\frac {1}{2}}+{\frac {\sqrt {-7}}{2}}\right)^{3}}$ ${\displaystyle \mathbb {Z} [{\sqrt {7}}]}$ ${\displaystyle (3-{\sqrt {7}})^{3}(3+{\sqrt {7}})^{3}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-10}}]}$ 2 3 ${\displaystyle \mathbb {Z} [{\sqrt {10}}]}$ 2 3 ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {11}}]}$ ${\displaystyle (-1)(3-{\sqrt {11}})^{3}(3+{\sqrt {11}})^{3}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-13}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}}$ 2 3 ${\displaystyle \mathbb {Z} [{\sqrt {-14}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {14}}]}$ ${\displaystyle (4-{\sqrt {14}})^{3}(4+{\sqrt {14}})^{3}}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {15}}]}$ 2 3 ${\displaystyle \mathbb {Z} [{\sqrt {-17}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}}$ ${\displaystyle (-1)\left({\frac {3}{2}}-{\frac {\sqrt {17}}{2}}\right)^{3}\left({\frac {3}{2}}+{\frac {\sqrt {17}}{2}}\right)^{3}}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {19}}]}$ ${\displaystyle (-1)(13-3{\sqrt {19}})^{3}(13+3{\sqrt {19}})^{3}}$

The astute reader might wonder if ${\displaystyle (-1+{\sqrt {-3}})^{3}}$ counts as a factorization apart from 2 3 in ${\displaystyle \mathbb {Z} [{\sqrt {-3}}]}$. But that is not a complete ring of algebraic integers, as ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-3}})}=\mathbb {Z} [\omega ]}$, where ${\displaystyle \omega =-{\frac {1}{2}}+{\frac {\sqrt {-3}}{2}}}$. So, as it turns out, ${\displaystyle -1+{\sqrt {-3}}=2\omega }$, thus ${\displaystyle 8=(2\omega )^{3}=2^{3}\omega ^{3}}$. But ${\displaystyle \omega }$ is a unit, and therefore ${\displaystyle 8=(-1+{\sqrt {-3}})^{3}}$ is no more a distinct factorization than, say, ${\displaystyle 8=(-1)^{2}2^{3}}$, in which –1 is a unit and therefore does not turn 2 3 into anything substantially different.

## The relationship of −8, 8 to quadratic integer rings adjoining the square roots of −2, 2

There really isn't such a thing as ${\displaystyle \mathbb {Z} [{\sqrt {-8}}]}$ or ${\displaystyle \mathbb {Z} [{\sqrt {8}}]}$. Any factorization that we can come up with in those "rings" essentially boils down to a factorization in ${\displaystyle \mathbb {Z} [{\sqrt {-2}}]}$ or ${\displaystyle \mathbb {Z} [{\sqrt {2}}]}$.

For example, ${\displaystyle 9=(1-{\sqrt {-8}})(1+{\sqrt {-8}})}$. Note that ${\displaystyle (1-{\sqrt {-2}})(1+{\sqrt {-2}})=3}$, and that ${\displaystyle (1-{\sqrt {-2}})^{2}=-1-2{\sqrt {-2}}}$; likewise ${\displaystyle (1+{\sqrt {-2}})^{2}=-1+2{\sqrt {-2}}}$. Also note that ${\displaystyle (2{\sqrt {-2}})^{2}=-8}$.

Our example for ${\displaystyle \mathbb {Z} [{\sqrt {8}}]}$ will be much more direct. This time, in ${\displaystyle 7=(-1)(1-{\sqrt {8}})(1+{\sqrt {8}})}$, we immediately substitute ${\displaystyle 2{\sqrt {2}}}$ for ${\displaystyle {\sqrt {8}}}$, giving us ${\displaystyle 7=(-1)(1-2{\sqrt {2}})(1+2{\sqrt {2}})}$.

Essentially "${\displaystyle \mathbb {Z} [{\sqrt {-8}}]}$" consists of those algebraic integers of the form ${\displaystyle n+2m{\sqrt {-2}}}$. Since ${\displaystyle 2m}$ must be even, we are ignoring algebraic integers for which ${\displaystyle {\sqrt {-2}}}$ is multiplied by an odd integer, and so "${\displaystyle \mathbb {Z} [{\sqrt {-8}}]}$" is a "sub-domain" of ${\displaystyle \mathbb {Z} [{\sqrt {-2}}]}$. Likewise for "${\displaystyle \mathbb {Z} [{\sqrt {8}}]}$" and ${\displaystyle \mathbb {Z} [{\sqrt {2}}]}$.

## Representation of 8 in various bases

 Base 2 3 4 5 6 7 8 9 through 36 Representation 1000 22 20 13 12 11 10 8

In the balanced ternary numeral system, 8 is {1, 0, –1}, meaning ${\displaystyle 3^{2}-3^{0}}$. In negabinary, 8 is 11000, since ${\displaystyle (-2)^{4}+(-2)^{3}=16-8=8}$. In quater-imaginary base, 8 is 10200. In the factorial numeral system, 8 is 110, since ${\displaystyle 3!+2!=8}$. And in ${\displaystyle {\sqrt {2}}}$-base, 8 is 1000000.

The octal numeral system, which is occasionally used by computer programmers, can be thought of as a shorthand for binary, whereby three binary digits correspond to one octal digit:

 000 0 001 1 010 2 011 3 100 4 101 5 110 6 111 7

Thus, the powers of 2 in octal are then 1, 2, 4, 10, 20, 40, 100, ... (A004647), and the Mersenne numbers are 1, 3, 7, 17, 37, 77, ...

 ${\displaystyle -1}$ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1729