The (double) Primorial sieve & Prime spirals |
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The segmented prime spirals are an offspring of the Ulam spiral.

The counterclockwise Ulam spiral with startvalue 0 is a four-quarter spiral, and thus a prime spiral with four segments. Downwards is the prime spiral with two segments, which can be visualized as a two-quarter Ulam spiral.

The picture below shows the special factorable functions that contain no prime numbers > *p*2.

When a function intersects with a special factorable function, the natural number on the intersection is a composite number.

Clearly visible is the 29th NE diagonal intersecting with several special factorable functions.

When the intersection coincides with a lattice point the natural number is composite.

Note: for a divisor *d *applies that *d* | *f*(*n*) and also *d* | *f*(*n + d
· k*) with

Thus e.g. 29 |

The Ulam two-quarter spiral (see first picture) becomes a continuous spiral (see below) when the prime spiral is folded together. Clearly visible is the translation *n * to *n* - 1 at the original seam.

The Ulam two-quarter spiral can also be presented in a diamond shape.

Prime spirals: The Ulam two-quarter spiral as prime spiral with two segments.