# Gaussian Prime Spirals

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Start a loop with a point having integer coordinates in the complex plane (called a Gaussian integer) and trace a path as follows:

[more]
Contributed by: Joseph O'Rourke and Stan Wagon (April 2012)

(Smith College and Macalester College)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The problem of whether all initial values lead to closed cycles was first raised in [1]. It is related to the open question: Does every horizontal line contain infinitely many Gaussian primes? This is a consequence of the far-reaching hypothesis H of Schinzel [2]. But even if the horizontal line conjecture is true, it is not clear that starting points always cycle.

Within the square bounded by ±100, the largest cycle has length 4820 and starts at . The largest cycle in a search bounded by ±500 starts at and has length 316628. The starting value leads to a walk with 3,900,404 steps.

References

[1] Gaussian prime spirals, MathOverflow question 91423

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