The (double) Primorial sieve & Prime spirals |
---|

The Ulam spiral (1963) is named after the Polish mathematician Stanislaw Ulam.

The graphical display shows that prime numbers have the tendency to appear on certain diagonals within the spiral. These clear patterns continue even when the spiral grows bigger. The counterclockwise spiral can start with the initial value 1 as used by Ulam (see above), or with any other natural number like 0, 41 or 59.

The unraveling of the Ulam spiral.pdf is an offspring of the (double) primorial sieve.pdf project.

The mystery is solved by the find of the eight families of functions that fully capture all natural numbers in the spiral.

With the families of functions ANY area of the spiral can be examined, without building the spiral from the ground up.

The unraveled Ulam spiral offers a wide range of elegant features to further analyze prime numbers.

The picture below shows the density of prime numbers in some functions, amongst them the 59th SE function that played an important role in the unraveling of Ulam spiral mystery. The ratio of prime numbers in the 59th SE function is 6.3 times higher than the ratio of prime numbers in* f*(*n*) =*n*. It supports the conjecture that diagonal lines that are rich with prime numbers contain infinite many prime numbers, see also the unraveling of the Ulam spiral.pdf.

The Ulam spiral: The Ulam spiral unraveled