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### Gaussian Integer

A Gaussian integer is a complex number where and are integers. The Gaussian integers are members of the imaginary quadratic field and form a ring often denoted , or sometimes (Hardy and Wright 1979, p. 179). The sum, difference, and product of two Gaussian integers are Gaussian integers, but only if there is an such that(1)(Shanks 1993).Gaussian integers can be uniquely factored in terms of other Gaussian integers (known as Gaussian primes) up to powers of and rearrangements.The units of are and .One definition of the norm of a Gaussian integer is its complexmodulus(2)Another common definition (e.g., Herstein 1975; Hardy and Wright 1979, p. 182; Artin 1991; Dummit and Foote 2004) defines the norm of a Gaussian integer to be(3)the square of the above quantity. (Note that the Gaussian integers form a Euclidean ring, which is what makes them particularly of interest, only under the latter definition.) Because of the two possible definitions,..

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