Fourth power
Encyclopedia
In arithmetic
and algebra
, the fourth power
of a number n is the result of multiplying n by itself four times. So:
Fourth powers are also formed by multiplying a number by its cube
. Furthermore, they are squares
of squares.
The sequence of fourth powers of integer
s (also known as biquadratic numbers or tesseract
ic numbers) is:
The last two digits of a fourth power of an integer 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:
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 with:
958004 + 2175194 + 4145604 = 4224814.
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
and algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, the fourth power
Exponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
of a number n is the result of multiplying n by itself four times. So:
- n4 = n × n × n × n
Fourth powers are also formed by multiplying a number by its cube
Cube (arithmetic)
In arithmetic and algebra, the cube of a number n is its third power — the result of the number multiplying by itself three times:...
. Furthermore, they are squares
Square number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
of squares.
The sequence of fourth powers of integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s (also known as biquadratic numbers or tesseract
Tesseract
In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...
ic numbers) is:
- 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, ...
The last two digits of a fourth power of an integer 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:
- 00, 01, 16, 21, 25, 36, 41, 56, 61, 76, 81, 96
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
Waring's problem
In number theory, Waring's problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers...
).
Euler conjectured a fourth power cannot be written as the sum of 3 smaller fourth powers, but 200 years later this was disproven with:
958004 + 2175194 + 4145604 = 4224814.