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Later, in 1834, Carl Gustav Jakob Jacobi discovered a simple formula for the number of representations of an integer as the sum of four squares with his own four-square theorem. The formula is also linked to Descartes' theorem of four "kissing circles", which involves the sum of the squares of the curvatures of four circles.
The tallest Leyland cypress documented is about 40 m (130 ft) tall and still growing. [18] However, because their roots are relatively shallow, a large leylandii tends to topple over. The shallow root structure also means that it is poorly adapted to areas with hot summers, such as the southern half of the United States.
The harmonic mean of a set of positive integers is the number of numbers times the reciprocal of the sum of their reciprocals.; The optic equation requires the sum of the reciprocals of two positive integers a and b to equal the reciprocal of a third positive integer c.
Comment: The proof of Euler's four-square identity is by simple algebraic evaluation. Quaternions derive from the four-square identity, which can be written as the product of two inner products of 4-dimensional vectors, yielding again an inner product of 4-dimensional vectors: (a·a)(b·b) = (a×b)·(a×b).
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
The British flag theorem for rectangles equates two sums of two squares; The parallelogram law equates the sum of the squares of the four sides to the sum of the squares of the diagonals; Descartes' theorem for four kissing circles involves sums of squares; The sum of the squares of the edges of a rectangular cuboid equals the square of any ...
The number of ways to represent n as the sum of four squares is eight times the sum of the divisors of n if n is odd and 24 times the sum of the odd divisors of n if n is even (see divisor function), i.e.
Since 5 is odd, and has only itself as a divisor (or do we count the trivial divisor 1?) We should have 8(5) = 40 possible combinations of sums of squares. But there exists only one: 2 2 + 1 2 + 2(0 2) = 5 Assuming we allow all permutations of order, that only gives us 4!, and this is not unique to this decomposition either. Allowing squares of ...