64

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64 is the square of 8.

Membership in core sequences

Even numbers	..., 58, 60, 62, 64, 66, 68, 70, ...	A005843
Composite numbers	..., 60, 62, 63, 64, 65, 66, 68, ...	A002808
Cubes	1, 8, 27, 64, 125, 216, 343, 512, ...	A000578
Powers of 2	1, 2, 4, 8, 16, 32, 64, 128, 256, ...	A000079

In Pascal's triangle, 64 occurs only twice, namely in row 64, in the second and next to last positions. But it also appears in a subtler way, as the sum of the numbers in row 6: 1 + 6 15 + 20 + 15 + 6 + 1 = 64.

Sequences pertaining to 64

Multiples of 64	0, 64, 128, 192, 256, 320, 384, 448, 512, ...	A152691
$3x-1$ sequence beginning at 87	..., 86, 43, 128, 64, 32, 16, 8, 4, 2, 1, 2, ...	A008899
$3x+1$ sequence beginning at 21	21, 64, 32, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, ...	A033481

Partitions of 64

There are 1741630 partitions of 64.

The Goldbach representations of 64 are: 3 + 61 = 5 + 59 = 11 + 53 = 17 + 47 = 23 + 41.

Roots and powers of 64

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

${\sqrt[{2}]{64}}$	8.00000000		64 2	4096
${\sqrt[{3}]{64}}$	4.00000000		64 3	262144
${\sqrt[{4}]{64}}$	2.82842712	A010466	64 4	16777216
${\sqrt[{5}]{64}}$	2.29739670	A011149	64 5	1073741824
${\sqrt[{6}]{64}}$	2.00000000		64 6	68719476736
${\sqrt[{7}]{64}}$	1.81144732	A011246	64 7	4398046511104
${\sqrt[{8}]{64}}$	1.68179283	A011006	64 8	281474976710656
${\sqrt[{9}]{64}}$	1.58740105	A005480	64 9	18014398509481984
${\sqrt[{10}]{64}}$	1.51571656	A011093	64 10	1152921504606846976
				A089357

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

• 8, –8 (both real)
• 4, $-2\pm {\sqrt {-12}}$ (the two complex roots are the same except for the sign of the imaginary part)
• ${\sqrt[{4}]{64}}$, $-{\sqrt[{4}]{64}}$, $i{\sqrt[{4}]{64}}$, $-i{\sqrt[{4}]{64}}$
• ${\sqrt[{5}]{64}}$, etc.

Logarithms and 64th 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.

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

TABLE GOES HERE

Values for number theoretic functions with 64 as an argument

TABLE GOES HERE

Factorization of 64 in some quadratic integer rings

In $\mathbb {Z}$, 64 has the prime factorization of 2 6. But it has different factorizations in some quadratic integer rings.

$\mathbb {Z} [i]$	$(1-i)^{6}(1+i)^{6}$
$\mathbb {Z} [{\sqrt {-2}}]$	$({\sqrt {-2}})^{12}$	$\mathbb {Z} [{\sqrt {2}}]$	$({\sqrt {2}})^{12}$
$\mathbb {Z} [\omega ]$	2 6	$\mathbb {Z} [{\sqrt {3}}]$	$(1-{\sqrt {3}})^{6}(1+{\sqrt {3}})^{6}$
$\mathbb {Z} [{\sqrt {-5}}]$		$\mathbb {Z} [\phi ]$	2 6
$\mathbb {Z} [{\sqrt {-6}}]$		$\mathbb {Z} [{\sqrt {6}}]$	$(2-{\sqrt {6}})^{6}(2+{\sqrt {6}})^{6}$
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}$	$\left({\frac {1}{2}}-{\frac {\sqrt {-7}}{2}}\right)^{6}\left({\frac {1}{2}}+{\frac {\sqrt {-7}}{2}}\right)^{6}$	$\mathbb {Z} [{\sqrt {7}}]$	$(3-{\sqrt {7}})^{6}(3+{\sqrt {7}})^{6}$
$\mathbb {Z} [{\sqrt {-10}}]$	2 6	$\mathbb {Z} [{\sqrt {10}}]$	2 6
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}$		$\mathbb {Z} [{\sqrt {11}}]$	$(3-{\sqrt {11}})^{6}(3+{\sqrt {11}})^{6}$
$\mathbb {Z} [{\sqrt {-13}}]$		${\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}$	2 6
$\mathbb {Z} [{\sqrt {-14}}]$		$\mathbb {Z} [{\sqrt {14}}]$	$(4-{\sqrt {14}})^{6}(4+{\sqrt {14}})^{6}$
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}$	2 6 OR $\left({\frac {1}{2}}-{\frac {\sqrt {-15}}{2}}\right)^{3}\left({\frac {1}{2}}+{\frac {\sqrt {-15}}{2}}\right)^{3}$	$\mathbb {Z} [{\sqrt {15}}]$	2 6
$\mathbb {Z} [{\sqrt {-17}}]$	2 6	${\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}$	$\left({\frac {3}{2}}-{\frac {\sqrt {17}}{2}}\right)^{6}\left({\frac {3}{2}}+{\frac {\sqrt {17}}{2}}\right)^{6}$
${\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}$		$\mathbb {Z} [{\sqrt {19}}]$	$(13-3{\sqrt {19}})^{6}(13+3{\sqrt {19}})^{6}$

Representation of 64 in various bases

Base	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Representation	1000000	2101	1000	224	144	121	100	71	64	59	54	4C	48	44	40	3D	3A	37	34

See also

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