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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.
This implementation has the downside that it requires an additional floating-point division, which can double the cost of the naive implementation, as multiplication and addition are typically far faster than division and square root. Typically, the implementation is slower by a factor of 2.5 to 3. [11]
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 sum of one odd square and one even square is congruent to 1 mod 4, but there exist composite numbers such as 21 that are 1 mod 4 and yet cannot be represented as sums of two squares. Fermat's theorem on sums of two squares states that the prime numbers that can be represented as sums of two squares are exactly 2 and the odd primes congruent ...
The prime decomposition of the number 2450 is given by 2450 = 2 · 5 2 · 7 2. Of the primes occurring in this decomposition, 2, 5, and 7, only 7 is congruent to 3 modulo 4. Its exponent in the decomposition, 2, is even. Therefore, the theorem states that it is expressible as the sum of two squares. Indeed, 2450 = 7 2 + 49 2.
The problem is uninteresting for K of characteristic 2, since over such fields every sum of squares is a square, and we exclude this case. It is believed that otherwise admissibility is independent of the field of definition. [1]: 137
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To divide a given square into a sum of two squares. To divide 16 into a sum of two squares. Let the first summand be , and thus the second . The latter is to be a square. I form the square of the difference of an arbitrary multiple of x diminished by the root [of] 16, that is, diminished by 4. I form, for example, the square of 2x − 4.