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Gaussian Prime — Definition, Formula & Examples

A Gaussian prime is a Gaussian integer whose only divisors are units (1, −1, ii, −i-i) and associates of itself. It plays the same role among Gaussian integers that ordinary prime numbers play among the regular integers.

A nonzero Gaussian integer π∈Z[i]\pi \in \mathbb{Z}[i] is a Gaussian prime if, whenever π=αβ\pi = \alpha\beta with α,β∈Z[i]\alpha, \beta \in \mathbb{Z}[i], at least one of α\alpha or β\beta is a unit. Equivalently, π\pi is a Gaussian prime if and only if one of the following holds: (1) π=a+bi\pi = a + bi where a2+b2a^2 + b^2 is an ordinary prime, or (2) π\pi is an associate of a rational prime p≡3(mod4)p \equiv 3 \pmod{4}.

Key Formula

N(a+bi)=a2+b2N(a + bi) = a^2 + b^2
Where:
  • a+bia + bi = A Gaussian integer with integer real part a and integer imaginary part b
  • NN = The norm function on Gaussian integers; a Gaussian integer with prime norm is always a Gaussian prime

How It Works

To determine whether a Gaussian integer a+bia + bi is a Gaussian prime, compute its norm N(a+bi)=a2+b2N(a+bi) = a^2 + b^2. If both aa and bb are nonzero and a2+b2a^2 + b^2 is an ordinary prime, then a+bia + bi is a Gaussian prime. If one of aa or bb is zero, say b=0b = 0, then a+0ia + 0i is a Gaussian prime exactly when ∣a∣|a| is an ordinary prime congruent to 3(mod4)3 \pmod{4}. The prime 22 is not a Gaussian prime because 2=−i(1+i)22 = -i(1+i)^2, and primes p≡1(mod4)p \equiv 1 \pmod{4} split into two conjugate Gaussian primes a+bia + bi and a−bia - bi where a2+b2=pa^2 + b^2 = p.

Worked Example

Problem: Determine whether 3+2i3 + 2i is a Gaussian prime.
Compute the norm: Find the norm of 3+2i3 + 2i.
N(3+2i)=32+22=9+4=13N(3 + 2i) = 3^2 + 2^2 = 9 + 4 = 13
Check if the norm is prime: 13 is a prime number in Z\mathbb{Z}.
Conclude: Since both real and imaginary parts are nonzero and the norm is an ordinary prime, 3+2i3 + 2i is a Gaussian prime.
Answer: 3+2i3 + 2i is a Gaussian prime because its norm, 13, is a rational prime.

Why It Matters

Gaussian primes are essential in algebraic number theory and provide the first example of unique factorization in a ring beyond Z\mathbb{Z}. They appear in proofs of Fermat's two-square theorem (every prime p≡1(mod4)p \equiv 1 \pmod{4} is a sum of two squares) and in the study of quadratic reciprocity and integer lattice problems.

Common Mistakes

Mistake: Assuming every ordinary prime remains prime as a Gaussian integer.
Correction: Only primes p≡3(mod4)p \equiv 3 \pmod{4} stay prime. Primes p≡1(mod4)p \equiv 1 \pmod{4} factor into two conjugate Gaussian primes, and 2=−i(1+i)22 = -i(1+i)^2 ramifies.