A Lipschitz integer is a Quaternion with integer coefficients. The norm is defined as $N(a+ib+jc+kd)=a^2+b^2+c^2+d^2$ which is a multiplicative function

$N:\mathbb H\to\mathbb R$, $N(\alpha\beta)=N(\alpha)N(\beta)$

Now since any natural number can be written as the sum of four squares of integers, all prime numbers in $\mathbb N$ are composite Lipschitz integers


Obviously, if $\alpha$ is a composite, then so is $N(\alpha)$.

In the ring of Gaussian integers it also holds that if both the real and the imaginary terms are non-zero, then the norm is prime if the Gaussian number is prime. My question is

If a Lipschitz integer is prime and all four terms are non-zero, is then the norm a prime ?


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