Prime numbers in complex domains are actually quite simple

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Some primes in the ring of algebraic integers of Q(√−35). Hollow green dot at the left is −5, leftmost full cyan dot is −9/2 + (√−35)/2. This diagram was produced by version 0.95 of a Java program by Alonso del Arte.

Prime numbers

Algebraic integers

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“black scientific calculator beside black headphones” by Charles Deluvio 🇵🇭🇨🇦 on Unsplash

Imaginary and complex numbers

Irreducible does not always mean prime

Drawing the diagrams

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The magenta dot gives the approximate location of √−2.
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The magenta dot gives the approximate location of 1 + √−2.

Programming the computer to draw the diagrams

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Diagram of primes in Z[√−5].
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Diagram of primes in Z[√−5], zoomed out to 2 pixels per unit interval.
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Diagram of primes in the ring of algebraic integers of Q(√−7).
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Diagram of primes in the ring of algebraic integers of Q(√−11).
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Diagram of primes in the ring of algebraic integers of Q(√−19).
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Diagram of primes in the ring of algebraic integers of Q(√−43).
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Diagram of primes in the ring of algebraic integers of Q(√−67).
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Diagram of primes in the ring of algebraic integers of Q(√−163).
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“Manual” attempt at a diagram of prime numbers in Z[√2].

is a composer and photographer from Detroit, Michigan. He has been working on a Java program to display certain mathematical diagrams.

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