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# 29

Please do not rely on any information it contains.

29 is an integer.

## Membership in core sequences

 Odd numbers ..., 23, 25, 27, 29, 31, 33, 35, ... A005408 Prime numbers ..., 17, 19, 23, 29, 31, 37, 41, ... A000040 Squarefree numbers ..., 22, 23, 26, 29, 30, 31, 33, ... A005117 Lucas numbers ..., 7, 11, 18, 29, 47, 76, 123, ... A000032 Pell numbers ..., 2, 5, 12, 29, 70, 169, 408, ... A000129 Central polygonal numbers ..., 11, 16, 22, 29, 37, 46, 56, ... A000124

In Pascal's triangle, 29 occurs twice. (In Lozanić's triangle, 29 occurs four times).

## Sequences pertaining to 29

 Multiples of 29 0, 29, 58, 87, 116, 145, 174, 203, 232, 261, 290, 319, 348, ... A195819 29-gonal numbers 1, 29, 84, 166, 275, 411, 574, 764, 981, 1225, 1496, 1794, ... A255187 29-gonal pyramidal numbers 1, 30, 114, 280, 555, 966, 1540, 2304, 3285, 4510, 6006, ... A256649 Primes with primitive root 29 2, 3, 11, 17, 19, 41, 43, 47, 73, 79, 89, 97, 101, 113, 127, ... A019355 ${\displaystyle 3x+1}$ sequence starting at 51 51, 154, 77, 232, 116, 58, 29, 88, 44, 22, 11, 34, 17, 52, ... A033479 ${\displaystyle 5x+1}$ sequence starting at 7 7, 36, 18, 9, 46, 23, 116, 58, 29, 146, 73, 366, 183, 916, ... A028389

## Partitions of 29

There are 4565 partitions of 29.

The Goldbach representations of 29 using distinct primes are: 3 + 7 + 19 = 5 + 7 + 17 = 5 + 11 + 13 = 29.

## Roots and powers of 29

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

 ${\displaystyle {\sqrt {29}}}$ 5.38516480 A010484 29 2 841 ${\displaystyle {\sqrt[{3}]{29}}}$ 3.07231682 A010600 29 3 24389 ${\displaystyle {\sqrt[{4}]{29}}}$ 2.32059578 A011024 29 4 707281 ${\displaystyle {\sqrt[{5}]{29}}}$ 1.96100905 A011114 29 5 20511149 ${\displaystyle {\sqrt[{6}]{29}}}$ 1.75280256 29 6 594823321 ${\displaystyle {\sqrt[{7}]{29}}}$ 1.61775965 29 7 17249876309 ${\displaystyle {\sqrt[{8}]{29}}}$ 1.52335018 29 8 500246412961 ${\displaystyle {\sqrt[{9}]{29}}}$ 1.45374644 29 9 14507145975869 ${\displaystyle {\sqrt[{10}]{29}}}$ 1.40036033 29 10 420707233300201 A009973

## Logarithms and 29th 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.

If ${\displaystyle n}$ is not a multiple of 59, then either ${\displaystyle n^{29}-1}$ or ${\displaystyle n^{29}+1}$ is. Hence the formula for the Legendre symbol ${\displaystyle \left({\frac {a}{59}}\right)=a^{29}{\pmod {59}}}$.

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

 ${\displaystyle \log _{29}2}$ 0.205847 ${\displaystyle \log _{2}29}$ 4.85798 2 29 5.36871e+08 ${\displaystyle \log _{29}e}$ 0.296974 ${\displaystyle \log 29}$ 3.3673 A016652 ${\displaystyle e^{29}}$ 3.93133e+12 ${\displaystyle \log _{29}3}$ 0.32626 ${\displaystyle \log _{3}29}$ 3.06504 3 29 6.86304e+13 ${\displaystyle \log _{29}4}$ 0.411694 ${\displaystyle \log _{4}29}$ 2.42899 4 29 2.8823e+17 ${\displaystyle \log _{29}5}$ 0.477962 ${\displaystyle \log _{5}29}$ 2.09222 5 29 1.86265e+20 ${\displaystyle \log _{29}6}$ 0.532106 ${\displaystyle \log _{6}29}$ 1.87932 6 29 3.68457e+22 ${\displaystyle \log _{29}7}$ 0.577885 ${\displaystyle \log _{7}29}$ 1.73045 7 29 3.21991e+24 ${\displaystyle \log _{29}8}$ 0.61754 ${\displaystyle \log _{8}29}$ 1.61933 8 29 1.54743e+26 ${\displaystyle \log _{29}9}$ 0.652519 ${\displaystyle \log _{9}29}$ 1.53252 9 29 4.71013e+27 ${\displaystyle \log _{29}10}$ 0.683808 ${\displaystyle \log _{10}29}$ 1.4624 10 29 1e+29

See A122970 for the 29th powers of integers.

## Values for number theoretic functions with 29 as an argument

 ${\displaystyle \mu (29)}$ −1 ${\displaystyle M(29)}$ −2 ${\displaystyle \pi (29)}$ 10 ${\displaystyle \sigma _{1}(29)}$ 30 ${\displaystyle \sigma _{0}(29)}$ 2 ${\displaystyle \phi (29)}$ 28 ${\displaystyle \Omega (29)}$ 1 ${\displaystyle \omega (29)}$ 1 ${\displaystyle \lambda (29)}$ 28 This is the Carmichael lambda function. ${\displaystyle \lambda (29)}$ −1 This is the Liouville lambda function. 29! 8841761993739701954543616000000 ${\displaystyle \Gamma (29)}$ 304888344611713860501504000000

## Factorization of some small integers in a quadratic integer ring adjoining square roots of −29, 29

${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {29}})}}$ is a unique factorization domain, ${\displaystyle \mathbb {Z} [{\sqrt {-29}}]}$ is not. Units in ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {29}})}}$ are of the form ${\displaystyle \left({\frac {5}{2}}+{\frac {\sqrt {29}}{2}}\right)^{n}}$. Units in ${\displaystyle \mathbb {Z} [{\sqrt {-29}}]}$ are just 1 and −1.

 ${\displaystyle n}$ ${\displaystyle \mathbb {Z} [{\sqrt {-29}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {29}})}}$ 2 Irreducible Prime 3 4 2 2 5 Irreducible ${\displaystyle (-1)\left({\frac {3}{2}}-{\frac {\sqrt {29}}{2}}\right)\left({\frac {3}{2}}+{\frac {\sqrt {29}}{2}}\right)}$ 6 2 × 3 7 Prime ${\displaystyle (-1)\left({\frac {1}{2}}-{\frac {\sqrt {29}}{2}}\right)\left({\frac {1}{2}}+{\frac {\sqrt {29}}{2}}\right)}$ 8 2 3 9 3 2 10 2 × 5 ${\displaystyle (-1)2\left({\frac {3}{2}}-{\frac {\sqrt {29}}{2}}\right)\left({\frac {3}{2}}+{\frac {\sqrt {29}}{2}}\right)}$ 11 Irreducible Prime 12 2 2 × 3 13 Irreducible ${\displaystyle \left({\frac {9}{2}}-{\frac {\sqrt {29}}{2}}\right)\left({\frac {9}{2}}+{\frac {\sqrt {29}}{2}}\right)}$ 14 2 × 7 ${\displaystyle (-1)2\left({\frac {1}{2}}-{\frac {\sqrt {29}}{2}}\right)\left({\frac {1}{2}}+{\frac {\sqrt {29}}{2}}\right)}$ 15 3 × 5 ${\displaystyle (-1)3\left({\frac {3}{2}}-{\frac {\sqrt {29}}{2}}\right)\left({\frac {3}{2}}+{\frac {\sqrt {29}}{2}}\right)}$ 16 2 4 17 Prime 18 2 × 3 2 19 Irreducible Prime 20 2 2 × 5 ${\displaystyle (-1)2^{2}\left({\frac {3}{2}}-{\frac {\sqrt {29}}{2}}\right)\left({\frac {3}{2}}+{\frac {\sqrt {29}}{2}}\right)}$ 21 3 × 7 ${\displaystyle (-1)3\left({\frac {1}{2}}-{\frac {\sqrt {29}}{2}}\right)\left({\frac {1}{2}}+{\frac {\sqrt {29}}{2}}\right)}$ 22 2 × 11 23 Prime ${\displaystyle \left({\frac {11}{2}}-{\frac {\sqrt {29}}{2}}\right)\left({\frac {11}{2}}+{\frac {\sqrt {29}}{2}}\right)}$ 24 2 3 × 3 25 5 2 ${\displaystyle \left({\frac {3}{2}}-{\frac {\sqrt {29}}{2}}\right)^{2}\left({\frac {3}{2}}+{\frac {\sqrt {29}}{2}}\right)^{2}}$ 26 2 × 13 ${\displaystyle 2\left({\frac {9}{2}}\pm {\frac {\sqrt {29}}{2}}\right)}$ 27 3 3 28 2 2 × 7 ${\displaystyle (-1)2^{2}\left({\frac {1}{2}}-{\frac {\sqrt {29}}{2}}\right)\left({\frac {1}{2}}+{\frac {\sqrt {29}}{2}}\right)}$ 29 ${\displaystyle (-1)({\sqrt {-29}})^{2}}$ ${\displaystyle ({\sqrt {29}})^{2}}$ 30 2 × 3 × 5 OR ${\displaystyle (1-{\sqrt {-29}})(1+{\sqrt {-29}})}$ ${\displaystyle (-1)2\times 3\left({\frac {3}{2}}\pm {\frac {\sqrt {29}}{2}}\right)}$ 31 Prime 32 2 5 33 3 × 11 OR ${\displaystyle (2-{\sqrt {-29}})(2+{\sqrt {-29}})}$ 3 × 11 34 2 × 17 35 5 × 7 ${\displaystyle \left({\frac {3}{2}}\pm {\frac {\sqrt {29}}{2}}\right)\left({\frac {1}{2}}\pm {\frac {\sqrt {29}}{2}}\right)}$ 36 2 2 × 3 2 37 38 2 × 19 OR ${\displaystyle (3-{\sqrt {-29}})(3+{\sqrt {-29}})}$ 2 × 19 39 3 × 13 ${\displaystyle 3\left({\frac {9}{2}}\pm {\frac {\sqrt {29}}{2}}\right)}$ 40 2 3 × 5 ${\displaystyle (-1)2^{3}\left({\frac {3}{2}}-{\frac {\sqrt {29}}{2}}\right)\left({\frac {3}{2}}+{\frac {\sqrt {29}}{2}}\right)}$

${\displaystyle \mathbb {Z} [{\sqrt {-29}}]}$ has class number 6. Here we will exhibit a few more examples of numbers with more than one distinct factorization in ${\displaystyle \mathbb {Z} [{\sqrt {-29}}]}$ in which the factorizations have differing numbers of irreducible factors.

 ${\displaystyle n}$ ${\displaystyle \mathbb {Z} [{\sqrt {-29}}]}$ 45 3 2 × 5 OR ${\displaystyle (4-{\sqrt {-29}})(4+{\sqrt {-29}})}$ 54 2 × 3 3 OR ${\displaystyle (5-{\sqrt {-29}})(5+{\sqrt {-29}})}$ 78 2 × 3 × 13 OR ${\displaystyle (7-{\sqrt {-29}})(7+{\sqrt {-29}})}$ 110 2 × 5 × 11 OR ${\displaystyle (9-{\sqrt {-29}})(9+{\sqrt {-29}})}$ 117 3 2 × 13 OR ${\displaystyle (1-2{\sqrt {-29}})(1+2{\sqrt {-29}})}$ 120 2 3 × 3 × 5 OR ${\displaystyle (2-2{\sqrt {-29}})(2+2{\sqrt {-29}})}$ 125 5 3 OR ${\displaystyle (3-2{\sqrt {-29}})(3+2{\sqrt {-29}})}$

Ideals really help us make sense of multiple distinct factorizations in ${\displaystyle \mathbb {Z} [{\sqrt {-29}}]}$, while raising some questions about ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {29}})}}$.

 ${\displaystyle p}$ Factorization of ${\displaystyle \langle p\rangle }$ In ${\displaystyle \mathbb {Z} [{\sqrt {-29}}]}$ In ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {29}})}}$ 2 ${\displaystyle \langle 2,1+{\sqrt {-29}}\rangle ^{2}}$ Prime 3 ${\displaystyle \langle 3,1-{\sqrt {-29}}\rangle \langle 3,1+{\sqrt {-29}}\rangle }$ 5 ${\displaystyle \langle 5,1-{\sqrt {-29}}\rangle \langle 5,1+{\sqrt {-29}}\rangle }$ 7 Prime 11 ${\displaystyle \langle 11,2-{\sqrt {-29}}\rangle \langle 11,2+{\sqrt {-29}}\rangle }$ Prime 13 ${\displaystyle \langle 13,6-{\sqrt {-29}}\rangle \langle 13,6+{\sqrt {-29}}\rangle }$ ${\displaystyle \langle 4-{\sqrt {29}}\rangle \langle 13,4+{\sqrt {29}}\rangle }$ 17 Prime 19 ${\displaystyle \langle 19,3-{\sqrt {29}}\rangle \langle 19,3+{\sqrt {29}}\rangle }$ Prime 23 Prime 29 ${\displaystyle \langle {\sqrt {-29}}\rangle ^{2}}$ ${\displaystyle \langle {\sqrt {29}}\rangle ^{2}}$ 31 37 41 43 47

## Factorization of 29 in some quadratic integer rings

As was mentioned above, 29 is a prime number in ${\displaystyle \mathbb {Z} }$. But it is composite in some quadratic integer rings.

 ${\displaystyle \mathbb {Z} [i]}$ ${\displaystyle (2-5i)(2+5i)}$ ${\displaystyle \mathbb {Z} [{\sqrt {-2}}]}$ Prime ${\displaystyle \mathbb {Z} [{\sqrt {2}}]}$ Prime ${\displaystyle \mathbb {Z} [\omega ]}$ ${\displaystyle \mathbb {Z} [{\sqrt {3}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {-5}}]}$ ${\displaystyle (3-2{\sqrt {-5}})(3+2{\sqrt {-5}})}$ ${\displaystyle \mathbb {Z} [\phi ]}$ ${\displaystyle (-1)(20-13\phi )(7+13\phi )}$ ${\displaystyle \mathbb {Z} [{\sqrt {-6}}]}$ Irreducible ${\displaystyle \mathbb {Z} [{\sqrt {6}}]}$ ${\displaystyle (-1)(11-5{\sqrt {6}})(11+5{\sqrt {6}})}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}}$ Prime ${\displaystyle \mathbb {Z} [{\sqrt {7}}]}$ ${\displaystyle (27-10{\sqrt {7}})(27+10{\sqrt {7}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {-10}}]}$ Irreducible ${\displaystyle \mathbb {Z} [{\sqrt {10}}]}$ Irreducible ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}}$ Prime ${\displaystyle \mathbb {Z} [{\sqrt {11}}]}$ Prime ${\displaystyle \mathbb {Z} [{\sqrt {-13}}]}$ Irreducible ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}}$ ${\displaystyle (-1)\left({\frac {1}{2}}-{\frac {3{\sqrt {13}}}{2}}\right)\left({\frac {1}{2}}+{\frac {3{\sqrt {13}}}{2}}\right)}$ ${\displaystyle \mathbb {Z} [{\sqrt {-14}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {14}}]}$ Prime ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {15}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {-17}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}}$ Prime ${\displaystyle \mathbb {Z} [{\sqrt {19}}]}$

## Representation of 29 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 11101 1002 131 104 45 41 35 32 29 27 25 23 21 1E 1D 1C 1B 1A 19

 ${\displaystyle -1}$ 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