How do you prove that a Gaussian integer is prime?
Table of Contents
- 1 How do you prove that a Gaussian integer is prime?
- 2 How is the fundamental theorem of arithmetic for prime numbers represented?
- 3 Is the ring of Gaussian integers a PID?
- 4 Are there complex prime numbers?
- 5 Is a prime number squared prime?
- 6 Is 29 a Gaussian prime?
- 7 What are the units in the ring of Gaussian integers Zi?
How do you prove that a Gaussian integer is prime?
A Gaussian integer a + bi is a Gaussian prime if and only if either:
- 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.
- 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}.