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In mathematics and statistics, sums of powers occur in a number of contexts: . Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities.
In arithmetic and algebra, the seventh power of a number n is the result of multiplying seven instances of n together. So: n 7 = n × n × n × n × n × n × n.. Seventh powers are also formed by multiplying a number by its sixth power, the square of a number by its fifth power, or the cube of a number by its fourth power.
[3] The sum of Euler's totient function φ(x) over the first twenty integers is 128. [4] 128 can be expressed by a combination of its digits with mathematical operators, thus 128 = 2 8 − 1, making it a Friedman number in base 10. [5] A hepteract has 128 vertices. 128 is the only 3-digit number that is a 7th power (2 7).
Pierre de Fermat gave a criterion for numbers of the form 8a + 1 and 8a + 3 to be sums of a square plus twice another square, but did not provide a proof. [1] N. Beguelin noticed in 1774 [2] that every positive integer which is neither of the form 8n + 7, nor of the form 4n, is the sum of three squares, but did not provide a satisfactory proof. [3]
Two to the power of n, written as 2 n, is the number of values in which the bits in a binary word of length n can be set, where each bit is either of two values. A word, interpreted as representing an integer in a range starting at zero, referred to as an "unsigned integer", can represent values from 0 (000...000 2) to 2 n − 1 (111...111 2) inclusively.
According to Guy, Erdős has asked whether there are infinitely many pairs of consecutive powerful numbers such as (23 3, 2 3 3 2 13 2) in which neither number in the pair is a square. Walker (1976) showed that there are indeed infinitely many such pairs by showing that 3 3 c 2 + 1 = 7 3 d 2 has infinitely many solutions.
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The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial x + 1 is the quadratic polynomial (x + 1) 2 = x 2 ...