In arithmetic and algebra, the **fourth power** of a number *n* is the result of multiplying four instances of *n* together. So:

*n*^{4} =

*n* ×

*n* ×

*n* ×

*n*
Fourth powers are also formed by multiplying a number by its cube. Furthermore, they are squares of squares.

The sequence of fourth powers of integers (also known as **biquadratic numbers** or **tesseractic numbers**) is:

0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 390625, 456976, 531441, 614656, 707281, 810000, ... (sequence

A000583 in the OEIS)

The last two digits of a fourth power of an integer in base 10 can be easily shown (for instance, by computing the squares of possible last two digits of square numbers) to be restricted to only *twelve* possibilities:

if a number ends in 0, its fourth power ends in
00
(in fact in
0000
)
if a number ends in 1, 3, 7 or 9 its fourth power ends in
01
,
21
,
41
,
61
or
81
if a number ends in 2, 4, 6, or 8 its fourth power ends in
16
,
36
,
56
,
76
or
96
if a number ends in 5 its fourth power ends in
25
(in fact in
0625
)
These twelve possibilities can be conveniently expressed as 00, *e*1, *o*6 or 25 where *o* is an odd digit and e an even digit.

Every positive integer can be expressed as the sum of at most 19 fourth powers; every sufficiently large integer can be expressed as the sum of at most 16 fourth powers (see Waring's problem).

Euler conjectured a fourth power cannot be written as the sum of 3 smaller fourth powers, but 200 years later this was disproven (Elkies, Frye) with:

95800^{4} + 217519^{4} + 414560^{4} = 422481^{4}.

That the equation *x*^{4} + *y*^{4} = *z*^{4} has no solutions in nonzero integers (a special case of Fermat's Last Theorem), was known, see Fermat's right triangle theorem.