How do you prove that a Gaussian integer is prime?

A Gaussian integer a + bi is a Gaussian prime if and only if either:

1. one of a, b is zero and absolute value of the other is a prime number of the form 4n + 3 (with n a nonnegative integer), or.
2. both are nonzero and a2 + b2 is a prime number (which will not be of the form 4n + 3).

How is the fundamental theorem of arithmetic for prime numbers represented?

Fundamental Theorem of Arithmetic Proof We will prove that for every integer, n ≥ 2, it can be expressed as the product of primes in a unique way: n = p1 × p2 ×⋯ × pi. We will prove this using mathematical induction.

Is 17 a Gaussian prime?

The number 17 is a regular prime, but it is not a Gaussian prime.

Is the ring of Gaussian integers a PID?

The ring of Gaussian integers, , is a PID because it is a Euclidean domain. (Proof; its Euclidean function is “take the norm”.) This is an example of a unique factorisation domain which is not a PID. The ring is not a PID, because it is not an integral domain.

Are there complex prime numbers?

A complex prime or Gaussian prime is a Gaussian integer z such that |z| > 1 and is divisible only by its units and associates in Z[i].

Is 13 a Gaussian prime?

Note that there are rational primes which are not Gaussian primes. A simple example is the rational prime 5, which is factored as 5=(2+i)(2−i) in the table, and therefore not a Gaussian prime….Factorizations.

norm	integer	factors
10	1+3i 3+i	(1+i)·(2+i) (1+i)·(2−i)
13	3+2i 2+3i	(p) (p)
16	4	−(1+i)4
17	1+4i 4+i	(p) (p)

Is a prime number squared prime?

A prime number by definition has exactly 2 factors – 1 and itself. Therefore no prime number is a square and no square number is prime.

Is 29 a Gaussian prime?

Note that there are rational primes which are not Gaussian primes. A simple example is the rational prime 5, which is factored as 5=(2+i)(2−i) in the table, and therefore not a Gaussian prime….Factorizations.

norm	integer	factors
29	2+5i 5+2i	(p) (p)
32	4+4i	−(1+i)5
34	3+5i 5+3i	(1+i)·(4+i) (1+i)·(4−i)
36	6	−i·(1+i)2·3

Is 11 a Gaussian prime?

This is because we do not know efficient integer factorization for huge numbers. Since 11 is a Gaussian prime, we can divide the original number by 11 and get 40 − 5i. For the factor 13 we have to divide the result of the previous step 3 − 2i by 3 + 2i or 3 − 2i.

What are the units in the ring of Gaussian integers Zi?

Let (Z[i],+,×) be the ring of Gaussian integers. The set of units of (Z[i],+,×) is {1,i,−1,−i}.