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In other words, a Gaussian integer m is a Gaussian prime if and only if either its norm is a prime number, or m is the product of a unit (±1, ±i) and a prime number of the form 4n + 3. It follows that there are three cases for the factorization of a prime natural number p in the Gaussian integers:
A Gaussian integer is either the zero, one of the four units (±1, ±i), a Gaussian prime or composite.The article is a table of Gaussian Integers x + iy followed either by an explicit factorization or followed by the label (p) if the integer is a Gaussian prime.
Fermat's theorem on sums of two squares is strongly related with the theory of Gaussian primes.. A Gaussian integer is a complex number + such that a and b are integers. The norm (+) = + of a Gaussian integer is an integer equal to the square of the absolute value of the Gaussian integer.
where p 1 < p 2 < ... < p k are primes and the n i are positive integers. This representation is commonly extended to all positive integers, including 1, by the convention that the empty product is equal to 1 (the empty product corresponds to k = 0). This representation is called the canonical representation [10] of n, or the standard form [11 ...
Of the form an + d for fixed integers a and d. Also called primes congruent to d modulo a. The primes of the form 2n+1 are the odd primes, including all primes other than 2. Some sequences have alternate names: 4n+1 are Pythagorean primes, 4n+3 are the integer Gaussian primes, and 6n+5 are the Eisenstein primes
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.
Consider again the case of the Gaussian integers. We take θ to be the imaginary unit i {\displaystyle i} , with minimal polynomial H ( X ) = X 2 + 1. Since Z [ i {\displaystyle i} ] is the whole ring of integers of Q ( i {\displaystyle i} ), the conductor is the unit ideal, so there are no exceptional primes.
If we regard the ring of Gaussian integers, we get the case b = 1 + i and b = 1 − i, and can ask for which n the number (1 + i) n − 1 is a Gaussian prime which will then be called a Gaussian Mersenne prime.