The Ulam Spiral

… and many other spirals¬†
inspired by it …

spiral-origins-1-2

Ulam Spiral with
origin at 1.

spiral-origins-0-2

Ulam Spiral with
origin at 0.

In both spirals above:
Along each of the 8 cardinal directions starting at the origin of the spiral
a series of 4 numbers are shown.
Also Shown:
the formulas generating these numbers.

The number-formulas in the bottom spiral are simpler as follows:
a)
In case of the numbers along the diagonal directions:
In the top spiral their formulas each contain “1” as a 2nd. term,
in the bottom spiral these formulas contain no such term.
b)
In case of the numbers along the horizontal- & vertical directions:
In the top spiral their formulas each contain 2nd. term,
in the bottom spiral these 2nd. terms are reduced by “1”.

For these reasons the bottom spiral presents itself as a
more “fortuitous” arrangement with which to investigate
various properties of the Ulam spiral.

Another unique benefit of the bottom spiral is the following:
none of the prime numbers are
located on any of the 4 diagonal sector-lines.