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For the avoidance of ambiguity, zero will always be a valid possible constituent of "sums of two squares", so for example every square of an integer is trivially expressible as the sum of two squares by setting one of them to be zero. 1. The product of two numbers, each of which is a sum of two squares, is itself a sum of two squares.
Therefore, the theorem states that it is expressible as the sum of two squares. Indeed, 2450 = 7 2 + 49 2. The prime decomposition of the number 3430 is 2 · 5 · 7 3. This time, the exponent of 7 in the decomposition is 3, an odd number. So 3430 cannot be written as the sum of two squares.
Legendre's three-square theorem states which numbers can be expressed as the sum of three squares; Jacobi's four-square theorem gives the number of ways that a number can be represented as the sum of four squares. For the number of representations of a positive integer as a sum of squares of k integers, see Sum of squares function.
The Brahmagupta–Fibonacci identity states that the product of two sums of two squares is a sum of two squares. Euler's method relies on this theorem but it can be viewed as the converse, given n = a 2 + b 2 = c 2 + d 2 {\displaystyle n=a^{2}+b^{2}=c^{2}+d^{2}} we find n {\displaystyle n} as a product of sums of two squares.
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]
In number theory, the sum of squares function is an arithmetic function that gives the number of representations for a given positive integer n as the sum of k squares, where representations that differ only in the order of the summands or in the signs of the numbers being squared are counted as different.
It may be used to prove Nicomachus's theorem that the sum of the first cubes equals the square of the sum of the first positive integers. [2] Summation by parts is frequently used to prove Abel's theorem and Dirichlet's test.
Writing integers as a sum of two squares [ edit ] When used in conjunction with one of Fermat's theorems , the Brahmagupta–Fibonacci identity proves that the product of a square and any number of primes of the form 4 n + 1 is a sum of two squares.