4

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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 $\sqrt[2x]{4}=\sqrt[x]{2}$.

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

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

Logarithms and fourth powers

In the OEIS specifically and mathematics in general, $\log x$ refers to the natural logarithm of $x$, whereas all other bases are specified with a subscript.

From the basic properties of exponentiation, it follows that all fourth powers are squares, since $b^4=(b^2{)}^2$. Hence fourth powers are sometimes called "biquadrates." And from Fermat's little theorem it follows that if $b$ is coprime to 5, then $b^4\equiv 1mod5$.

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

${\log }_{4}2$	0.50000000	A020761	${\log }_{2}4$	2.00000000		2 4	16
${\log }_{4}e$	0.72134752	A133362	$\log 4$	1.38629436	A016627	$e^4$	54.59815003	A092426
${\log }_{4}3$	0.79248125	A094148	${\log }_{3}4$	1.26185950	A100831	3 4	81
${\log }_4\pi$	0.82574806		${\log }_{\pi }4$	1.21102312		${\pi }^4$	97.40909103	A092425
${\log }_{4}4$			1.00000000			4 4	256
${\log }_{4}5$	1.16096404	A153201	${\log }_{5}4$	0.86135311	A153101	5 4	625
${\log }_{4}6$	1.29248125	A153460	${\log }_{6}4$	0.77370561	A153102	6 4	1296
${\log }_{4}7$	1.40367746	A153615	${\log }_{7}4$	0.71241437	A153103	7 4	2401
${\log }_{4}8$	1.50000000		${\log }_{8}4$	0.66666666		8 4	4096
${\log }_{4}9$	1.58496250	A020857	${\log }_{9}4$	0.63092975	A102525	9 4	6561
${\log }_{4}10$	1.66096404	A154155	${\log }_{10}4$	0.60205999	A114493	10 4	10000

(See A000583 for the fourth powers of integers).

Values for number theoretic functions with 4 as an argument

$\mu (4)$	0
$M(4)$	–1
$\pi (4)$	2
${\sigma }_1(4)$	7
${\sigma }_0(4)$	3
$\varphi (4)$	2
$\mathrm{\Omega }(4)$	2
$\omega (4)$	1
$\lambda (4)$	2	This is the Carmichael lambda function.
$\lambda (4)$	1	This is the Liouville lambda function.
$\zeta (2)=\frac{{\pi }^4}{6}$ 1.082323233711138191516... (see A013662).
4!	24
$\mathrm{\Gamma }(4)$	6

Factorization of 4 in some quadratic integer rings

As was mentioned above, 4 is the square of 2 in $\mathbb{\mathbb{Z}}$. 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 $\mathbb{\mathbb{Z}}$ to still have only one factorization in a non-UFD. For example, one might think that $(38-12\sqrt{10})(38+12\sqrt{10})$ constitutes a factorization of 4 in $\mathbb{\mathbb{Z}}[\sqrt{10}]$ apart from 2 2, but it isn't, since $19-6\sqrt{10}$ is a unit and $2(19-6\sqrt{10})=(38-12\sqrt{10})$. A very similar thing happens in $\mathbb{\mathbb{Z}}[\sqrt{15}]$: suffice it to say that $4-\sqrt{15}$ 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 ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{65})}$, 2 is irreducible, yet $(-1)(\frac{7}{2}-\frac{\sqrt{65}}{2})(\frac{7}{2}+\frac{\sqrt{65}}{2})=4$.

Despite the fact that very few imaginary quadratic rings are UFDs, 4 has only one distinct factorization in all imaginary rings except one, namely ${{𝒪}}_{\mathbb{\mathbb{Q}}(\sqrt{-15})}$, 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.

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

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 $\sqrt{2}$-base, 4 is 10000. The quater-imaginary base representation of 4 is somewhat more interesting, being 10300, since $(-2{)}^4+3(2i{)}^2=16-3\times (-4)=4$.

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.

See also

• 2^n mod n

Some integers

							$-1$
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