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# 4

Please do not rely on any information it contains.

4 is the square of 2.

## Membership in core sequences

 Even numbers 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, ... A005843 Composite numbers 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, ... A002808 Powers of 2 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ... A000079 Perfect squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, ... A000290 Lucas numbers 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, ... A000032

In Pascal's triangle, 4 occurs twice, surrounding the first instance of 6. (In Lozanić's triangle, 4 occurs five times).

## Sequences pertaining to 4

 Multiples of 4 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, ... A008586 Centered square numbers 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, ... A001844

## Partitions of 4

There are five partitions of 4, only one of which consists of distinct numbers: {1, 3}. The only possible prime partition is {2, 2}.

## Roots and powers of 4

Since 4 = 2 2, it follows from the basic properties of exponentiation that the powers of 4 are the even-indexed powers of 2. Whereas the powers of 2 are congruent to 2 or 1 modulo 3, all powers of 4 are congruent to 1 modulo 3. One consequence of this concerns the Collatz function: the powers of 4 can be accessed either from a halving step or from a "tripling" step, while all other powers of 2 can only be accessed from a halving step.

Another fact readily verified by the basic properties of exponentiation is that ${\displaystyle {\sqrt[{2x}]{4}}={\sqrt[{x}]{2}}}$.

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

 ${\displaystyle {\sqrt {4}}}$ 2.00000000 A000038 4 2 16 ${\displaystyle {\sqrt[{3}]{4}}}$ 1.58740105 A005480 4 3 64 ${\displaystyle {\sqrt[{4}]{4}}}$ 1.41421356 A002193 4 4 256 ${\displaystyle {\sqrt[{5}]{4}}}$ 1.31950791 A005533 4 5 1024 ${\displaystyle {\sqrt[{6}]{4}}}$ 1.25992104 A002580 4 6 4096 ${\displaystyle {\sqrt[{7}]{4}}}$ 1.21901365 A011186 4 7 16384 ${\displaystyle {\sqrt[{8}]{4}}}$ 1.18920711 A010767 4 8 65536 ${\displaystyle {\sqrt[{9}]{4}}}$ 1.16652903 A011188 4 9 262144 ${\displaystyle {\sqrt[{10}]{4}}}$ 1.14869835 A005531 4 10 1048576 ${\displaystyle {\sqrt[{11}]{4}}}$ 1.13431252 A011190 4 11 4194304 ${\displaystyle {\sqrt[{12}]{4}}}$ 1.12246204 A010768 4 12 16777216 ${\displaystyle {\sqrt[{13}]{4}}}$ 1.11253147 A011192 4 13 67108864 ${\displaystyle {\sqrt[{14}]{4}}}$ 1.10408951 A010769 4 14 268435456 ${\displaystyle {\sqrt[{15}]{4}}}$ 1.09682497 A011194 4 15 1073741824 ${\displaystyle {\sqrt[{16}]{4}}}$ 1.09050773 A010770 4 16 4294967296 A000302

## Logarithms and fourth 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.

From the basic properties of exponentiation, it follows that all fourth powers are squares, since ${\displaystyle b^{4}=(b^{2})^{2}}$. Hence fourth powers are sometimes called "biquadrates." And from Fermat's little theorem it follows that if ${\displaystyle b}$ is coprime to 5, then ${\displaystyle b^{4}\equiv 1\mod 5}$.

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

 ${\displaystyle \log _{4}2}$ 0.50000000 A020761 ${\displaystyle \log _{2}4}$ 2.00000000 2 4 16 ${\displaystyle \log _{4}e}$ 0.72134752 A133362 ${\displaystyle \log 4}$ 1.38629436 A016627 ${\displaystyle e^{4}}$ 54.5982 A092426 ${\displaystyle \log _{4}3}$ 0.79248125 A094148 ${\displaystyle \log _{3}4}$ 1.26185950 A100831 3 4 81 ${\displaystyle \log _{4}\pi }$ 0.82574806 ${\displaystyle \log _{\pi }4}$ 1.21102312 ${\displaystyle \pi ^{4}}$ 97.4091 A092425 ${\displaystyle \log _{4}4}$ 1.00000000 4 4 256 ${\displaystyle \log _{4}5}$ 1.16096404 A153201 ${\displaystyle \log _{5}4}$ 0.86135311 A153101 5 4 625 ${\displaystyle \log _{4}6}$ 1.29248125 A153460 ${\displaystyle \log _{6}4}$ 0.77370561 A153102 6 4 1296 ${\displaystyle \log _{4}7}$ 1.40367746 A153615 ${\displaystyle \log _{7}4}$ 0.71241437 A153103 7 4 2401 ${\displaystyle \log _{4}8}$ 1.50000000 ${\displaystyle \log _{8}4}$ 0.66666666 8 4 4096 ${\displaystyle \log _{4}9}$ 1.58496250 A020857 ${\displaystyle \log _{9}4}$ 0.63092975 A102525 9 4 6561 ${\displaystyle \log _{4}10}$ 1.66096404 A154155 ${\displaystyle \log _{10}4}$ 0.60205999 A114493 10 4 10000

(See A000583 for the fourth powers of integers).

## Values for number theoretic functions with 4 as an argument

 ${\displaystyle \mu (4)}$ 0 ${\displaystyle M(4)}$ –1 ${\displaystyle \pi (4)}$ 2 ${\displaystyle \sigma _{1}(4)}$ 7 ${\displaystyle \sigma _{0}(4)}$ 3 ${\displaystyle \phi (4)}$ 2 ${\displaystyle \Omega (4)}$ 2 ${\displaystyle \omega (4)}$ 1 ${\displaystyle \lambda (4)}$ 2 This is the Carmichael lambda function. ${\displaystyle \lambda (4)}$ 1 This is the Liouville lambda function. ${\displaystyle \zeta (2)={\frac {\pi ^{4}}{6}}}$ 1.082323233711138191516... (see A013662). 4! 24 ${\displaystyle \Gamma (4)}$ 6

## Factorization of 4 in some quadratic integer rings

As was mentioned above, 4 is the square of 2 in ${\displaystyle \mathbb {Z} }$. But it has different factorizations in some quadratic integer rings. We could just take the factorizations of 2 and stick in a bunch of 2s as exponents, change some exponent 2s to exponent 4s. That works, at least for those rings that are unique factorization domains, but it seems to ignore rings that are not UFDs.

And yet, it is possible for an integer that is composite in ${\displaystyle \mathbb {Z} }$ to still have only one factorization in a non-UFD. For example, one might think that ${\displaystyle (38-12{\sqrt {10}})(38+12{\sqrt {10}})}$ constitutes a factorization of 4 in ${\displaystyle \mathbb {Z} [{\sqrt {10}}]}$ apart from 2 2, but it isn't, since ${\displaystyle 19-6{\sqrt {10}}}$ is a unit and ${\displaystyle 2(19-6{\sqrt {10}})=(38-12{\sqrt {10}})}$. A very similar thing happens in ${\displaystyle \mathbb {Z} [{\sqrt {15}}]}$: suffice it to say that ${\displaystyle 4-{\sqrt {15}}}$ is a unit... In general, if both parts are even, the factorization is probably not distinct.

The table below would have to go at least up to 65 to show a real quadratic ring in which 4 has more than one distinct factorization: in ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {65}})}}$, 2 is irreducible, yet ${\displaystyle (-1)\left({\frac {7}{2}}-{\frac {\sqrt {65}}{2}}\right)\left({\frac {7}{2}}+{\frac {\sqrt {65}}{2}}\right)=4}$.

Despite the fact that very few imaginary quadratic rings are UFDs, 4 has only one distinct factorization in all imaginary rings except one, namely ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}}$, as the table below shows. In all other imaginary rings besides the ones listed below, 2 is irreducible and consequently 4 can only be factored as 2 2.

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

## Representation of 4 in various bases

 Base 2 3 4 5 through 36 Representation 100 11 10 4

The negabinary representation of 4 is the same as in binary, and its balanced ternary numeral system representation is the same as its ternary. In ${\displaystyle {\sqrt {2}}}$-base, 4 is 10000. The quater-imaginary base representation of 4 is somewhat more interesting, being 10300, since ${\displaystyle (-2)^{4}+3(2i)^{2}=16-3\times (-4)=4}$.

The table above shows 4 is a palindromic number in ternary, and trivially palindromic in all higher integer bases. And it turns out that 4 is a Harshad number in all positional positive integer bases.

 ${\displaystyle -1}$ 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