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# 3

Please do not rely on any information it contains.

3 is the smallest odd prime number, and the only prime number that is one more than another prime (namely, 2).

## Membership in core sequences

 Odd numbers 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, ... A005408 Prime numbers 2, 3, 5, 7, 11, 13, 17, 19, ... A000040 Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, ... A000045 Lucas numbers 2, 1, 3, 4, 7, 11, 18, 29, 47, ... A000032 Triangular numbers 1, 3, 6, 10, 15, 21, 28, 36, 45, ... A000217

In Pascal's triangle, 3 occurs twice. (In Lozanić's triangle, 3 occurs four times).

## Sequences pertaining to 3

 Multiples of 3 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ... A008585 Inert rational primes in ${\displaystyle \mathbb {Q} ({\sqrt {3}})}$ 5, 7, 17, 19, 29, 31, 41, 43, 53, 67, 79, 89, ... A003630 Positive numbers of the form ${\displaystyle x^{2}-3y^{2}}$ 1, 4, 6, 9, 13, 16, 22, 24, 25, 33, 36, 37, 46, ... A084916 Negative numbers of the form ${\displaystyle x^{2}-3y^{2}}$ –2, –3, –8, –11, –12, –18, –23, –26, –27, –32, ... A084917 ${\displaystyle 3x+1}$ sequence beginning at 3 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, ... A033478 Fermat pseudoprimes to base 3 91, 121, 286, 671, 703, 949, 1105, 1541, 1729, ... A005935

## Partitions of 3

There are only three partitions of 3: {1, 1, 1}, {1, 2} and {3}. Thus the only partition of 3 into primes is a trivial partition.

## Roots and powers of 3

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

 ${\displaystyle {\sqrt {3}}}$ 1.73205080 A002194 3 2 9 ${\displaystyle {\sqrt[{3}]{3}}}$ 1.44224957 A002581 3 3 27 ${\displaystyle {\sqrt[{4}]{3}}}$ 1.31607401 A011002 3 4 81 ${\displaystyle {\sqrt[{5}]{3}}}$ 1.24573093 A005532 3 5 243 ${\displaystyle {\sqrt[{6}]{3}}}$ 1.20093695 A246708 3 6 729 ${\displaystyle {\sqrt[{7}]{3}}}$ 1.16993081 A246709 3 7 2187 ${\displaystyle {\sqrt[{8}]{3}}}$ 1.14720269 A246710 3 8 6561 ${\displaystyle {\sqrt[{9}]{3}}}$ 1.12983096 A011446 3 9 19683 ${\displaystyle {\sqrt[{10}]{3}}}$ 1.11612317 A246711 3 10 59049 ${\displaystyle {\sqrt[{11}]{3}}}$ 1.10503150 3 11 177147 ${\displaystyle {\sqrt[{12}]{3}}}$ 1.09587269 3 12 531441 A000244

## Logarithms and cubes

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.

If ${\displaystyle n}$ is not a multiple of 7, then either ${\displaystyle n^{3}-1}$ or ${\displaystyle n^{3}+1}$ is. Hence the formula for the Legendre symbol ${\displaystyle \left({\frac {a}{7}}\right)=a^{3}\mod 7}$.

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

 ${\displaystyle \log _{3}2}$ 0.63092975 A102525 ${\displaystyle \log _{2}3}$ 1.58496250 A102525 2 3 8 ${\displaystyle \log _{3}e}$ 0.91023922 A121935 ${\displaystyle \log 3}$ 1.09861228 A002162 ${\displaystyle e^{3}}$ 20.0855 A091933 ${\displaystyle \log _{3}3}$ 1.00000000 3 3 27 ${\displaystyle \log _{3}\pi }$ 1.04197804 ${\displaystyle \log _{\pi }3}$ 0.95971311 ${\displaystyle \pi ^{3}}$ 31.0063 A091925 ${\displaystyle \log _{3}4}$ 1.26185950 A100831 ${\displaystyle \log _{4}3}$ 0.79248125 A094148 4 3 64 ${\displaystyle \log _{3}5}$ 1.46497352 A113209 ${\displaystyle \log _{5}3}$ 0.68260619 A152914 5 3 125 ${\displaystyle \log _{3}6}$ 1.63092975 A153459 ${\displaystyle \log _{6}3}$ 0.61314719 A152935 6 3 216 ${\displaystyle \log _{3}7}$ 1.77124374 A152565 ${\displaystyle \log _{7}3}$ 0.56457503 A152945 7 3 343 ${\displaystyle \log _{3}8}$ 1.89278926 A113210 ${\displaystyle \log _{8}3}$ 0.52832083 A152956 8 3 512 ${\displaystyle \log _{3}9}$ 2.00000000 ${\displaystyle \log _{9}3}$ 0.50000000 9 3 729 ${\displaystyle \log _{3}10}$ 2.09590327 A152566 ${\displaystyle \log _{10}3}$ 0.47712125 A114490 10 3 1000

(See A000578 for integer cubes).

## Values for number theoretic functions with 3 as an argument

 ${\displaystyle \mu (3)}$ –1 ${\displaystyle M(3)}$ –1 ${\displaystyle \pi (3)}$ 2 ${\displaystyle \sigma _{1}(3)}$ 4 ${\displaystyle \sigma _{0}(3)}$ 2 ${\displaystyle \phi (3)}$ 2 ${\displaystyle \Omega (3)}$ 1 ${\displaystyle \omega (3)}$ 1 ${\displaystyle \lambda (3)}$ 2 This is the Carmichael lambda function. ${\displaystyle \lambda (3)}$ –1 This is the Liouville lambda function. ${\displaystyle \zeta (3)}$ 1.2020569031595942853997381615... (see A002117) 3! 6 ${\displaystyle \Gamma (3)}$ 2

## Factorization of some small integers in a quadratic integer ring adjoining the square roots of −3, 3

${\displaystyle \mathbb {Z} [{\sqrt {-3}}]}$ is actually a subdomain because it is not integrally closed. Since ${\displaystyle -3\equiv 1\mod 4}$, it is not enough to consider algebraic integers of the form ${\displaystyle a+b{\sqrt {-3}}}$ (with ${\displaystyle a,b\in \mathbb {Z} }$); we must also consider algebraic integers of the form ${\displaystyle {\frac {a}{2}}+{\frac {b{\sqrt {-3}}}{2}}}$ (with ${\displaystyle ab}$ odd).

Define ${\displaystyle \omega =-{\frac {1}{2}}+{\frac {\sqrt {-3}}{2}}}$. This number is of great importance not only because the ring of algebraic integers ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-3}})}}$ is ${\displaystyle \mathbb {Z} [\omega ]}$, but also because it is a complex cubic root of unity, meaning that ${\displaystyle \omega ^{3}=1}$. And since its norm is 1, this means that ${\displaystyle \omega }$ is a unit of ${\displaystyle \mathbb {Z} [\omega ]}$, and so are ${\displaystyle \omega ^{2}}$, ${\displaystyle -\omega }$ and ${\displaystyle -\omega ^{2}}$. Thus, together with 1 and –1, this means that ${\displaystyle \mathbb {Z} [\omega ]}$ has six units, more than any other imaginary quadratic integer ring.

That's very little compared to ${\displaystyle \mathbb {Z} [{\sqrt {3}}]}$, which has infinitely many units of the form ${\displaystyle (2-{\sqrt {3}})^{n}}$. Both ${\displaystyle \mathbb {Z} [\omega ]}$ (also called the domain of Eisenstein integers) and ${\displaystyle \mathbb {Z} [{\sqrt {3}}]}$ are unique factorization domains. The inertial primes in ${\displaystyle \mathbb {Z} [\omega ]}$ are the primes congruent to 2 modulo 3 (see A003627). If a prime ${\displaystyle p|\left({\frac {3^{p-1}}{2}}+1\right)}$, then it is inertial in ${\displaystyle \mathbb {Z} [{\sqrt {3}}]}$ (see A003630).

In the table below, some factorizations in ${\displaystyle \mathbb {Z} [\omega ]}$ will also be expressed using ${\displaystyle {\sqrt {-3}}}$; this does not constitute a distinct factorization since ${\displaystyle \mathbb {Z} [\omega ]}$ has unique factorization and ${\displaystyle {\sqrt {-3}}=1+2\omega }$. This is done for the sake of clarification, to facilitate comparison to other domains not usually expressed in terms of multiples of a unit that has both a real and an imaginary part.

 ${\displaystyle n}$ ${\displaystyle \mathbb {Z} [\omega ]}$ ${\displaystyle \mathbb {Z} [{\sqrt {3}}]}$ 1 Unit 2 Prime ${\displaystyle (-1)(1-{\sqrt {3}})(1+{\sqrt {3}})}$ 3 ${\displaystyle (-1)(1+2\omega )^{2}}$. Since ${\displaystyle \Re (2\omega )=-1}$, this is tantamount to ${\displaystyle (-1)({\sqrt {-3}})^{2}}$. ${\displaystyle ({\sqrt {3}})^{2}}$ 4 2 2 ${\displaystyle (1-{\sqrt {3}})^{2}(1+{\sqrt {3}})^{2}}$ 5 Prime 6 ${\displaystyle (-1)2(1+2\omega )^{2}}$ ${\displaystyle (-1)(1-{\sqrt {3}})(1+{\sqrt {3}})({\sqrt {3}})^{2}}$ 7 ${\displaystyle (-2+\omega )(-2+\omega ^{2})}$. This may seem clearer expressed as ${\displaystyle \left({\frac {5}{2}}-{\frac {\sqrt {-3}}{2}}\right)\left({\frac {5}{2}}+{\frac {\sqrt {-3}}{2}}\right)}$. Prime 8 2 3. Note that ${\displaystyle (-1+{\sqrt {-3}})^{3}}$ is not a distinct factorization. See 8 for a full explanation. ${\displaystyle (-1)(1-{\sqrt {3}})^{3}(1+{\sqrt {3}})^{3}}$ 9 ${\displaystyle (1+2\omega )^{4}}$ ${\displaystyle ({\sqrt {3}})^{4}}$ 10 2 × 5 ${\displaystyle (-1)(1-{\sqrt {3}})(1+{\sqrt {3}})5}$ 11 Prime ${\displaystyle (-1)(1-2{\sqrt {3}})(1+2{\sqrt {3}})}$ 12 ${\displaystyle (-1)(1+2\omega )^{2}2^{2}}$ ${\displaystyle (1-{\sqrt {3}})^{2}(1+{\sqrt {3}})^{2}({\sqrt {3}})^{2}}$ 13 ${\displaystyle (3-\omega )(4+\omega )}$ ${\displaystyle (5-2{\sqrt {3}})(5+2{\sqrt {3}})}$ 14 ${\displaystyle 2(-2+\omega )(-2+\omega ^{2})}$ ${\displaystyle (-1)(1-{\sqrt {3}})(1+{\sqrt {3}})7}$ 15 ${\displaystyle (-1)(1+2\omega )^{2}5}$ ${\displaystyle ({\sqrt {3}})^{2}5}$ 16 2 4 ${\displaystyle (1-{\sqrt {3}})^{4}(1+{\sqrt {3}})^{4}}$ 17 Prime 18 ${\displaystyle (-1)2(1+2\omega )^{4}}$ ${\displaystyle (-1)(1-{\sqrt {3}})(1+{\sqrt {3}})({\sqrt {3}})^{4}}$ 19 ${\displaystyle (5+2\omega )(5+2\omega ^{2})}$ Prime 20 2 2 × 5 ${\displaystyle (1-{\sqrt {3}})^{2}(1+{\sqrt {3}})^{2}5}$

## Factorization of 3 in some quadratic integer rings

As was mentioned above, 3 is a prime number in ${\displaystyle \mathbb {Z} }$. But it is composite in some quadratic integer rings.

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

For ${\displaystyle \mathbb {Z} [{\sqrt {-13}}]}$ through ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-163}})}}$, 3 is irreducible and possibly prime. And beyond ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-163}})}}$, without reference to a specific ring, it can only be guaranteed to be irreducible, as there are no more imaginary quadratic rings with unique factorization after that.

## Representation of 3 in various bases

In binary, 3 is represented as 11 and in ternary as 10 (as well as in balanced ternary). For all ordinary higher integer bases, 3 is 3, as well as in quater-imaginary base. In negabinary, 3 is 111, since ${\displaystyle (-2)^{2}+(-2)^{1}+(-2)^{0}=4-2+1=3}$.

In base 10, as well as in any other case in which the base is congruent to 1 modulo 3, there is a simple divisibility test for 3: if the digital root of a given number is 3 or any multiple thereof, then the number is also divisible by 3. In base 10, this means that all integers having a digital root of 3, 6 or 9 are multiples of 3 (see A008585).

 ${\displaystyle -1}$ 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