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Lagrange's four-square theorem, also known as Bachet's conjecture, states that every nonnegative integer can be represented as a sum of four non-negative integer squares. [1] That is, the squares form an additive basis of order four. = + + + where the four numbers ,,, are integers.
The sum of the reciprocals of the first n primes is not an integer for any n. There are 14 distinct combinations of four integers such that the sum of their reciprocals is 1, of which six use four distinct integers and eight repeat at least one integer. An Egyptian fraction is the sum of a finite number of reciprocals of positive integers.
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.
For the sum of squares of consecutive integers, see Square pyramidal number; For representing an integer as a sum of squares of 4 integers, see Lagrange's four-square theorem; Legendre's three-square theorem states which numbers can be expressed as the sum of three squares
8000 is the cube of 20, as well as the sum of four consecutive integers cubed, 11 3 + 12 3 + 13 3 + 14 3. The fourteen tallest mountains on Earth, which exceed 8000 meters in height, are sometimes referred to as eight-thousanders. [1]
A Young diagram representing visually a polite expansion 15 = 4 + 5 + 6. In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. A positive integer which is not polite is called impolite.
In particular, for a prime number p we have the explicit formula r 4 (p) = 8(p + 1). [2] Some values of r 4 (n) occur infinitely often as r 4 (n) = r 4 (2 m n) whenever n is even. The values of r 4 (n) can be arbitrarily large: indeed, r 4 (n) is infinitely often larger than . [2]
Subset sum problem, an algorithmic problem that can be used to find the shortest representation of a given number as a sum of powers; Pollock's conjectures; Sums of three cubes, discusses what numbers are the sum of three not necessarily positive cubes; Sums of four cubes problem, discusses whether every integer is the sum of four cubes of integers