The Ulam Spiral

… and many other spirals 
inspired by it …

findings3-fig1

Fig. -#1.

Tetrapolar
Ulam-type spiral.
Showing only
prime adjacents.

Spiral origin is at “0”.

“1” is represented on the spiral.
(“1” is adjacent to “0”, “0” is a 6-fold.)

“2” & “3” are not represented on the spiral.
(“2” & “3” are not adjacent to a 6-fold.)

Table – #1.

findings3-table1

Table – #2.

findings3-table2-2

Notes

In the drop-down menus of “polygonal spirals” & “polar Spirals”
other Ulam-like spirals are illustrated, and it is worth examining
to see if they have any ‘utility’, meaning if they reveal
interesting rows of prime numbers.

To look for such rows the tetrapolar spiral was chosen
because of all the polar spirals it is the closest to the
standard Ulam spiral, see in Fig’s. -#1 & -#2.
The tetrapolar spiral in Fig. -#1 reveals
interesting prime number rows,
and interestingly they are arcuate instead of linear.

2 prominent rows are immediately discernible.
The larger one is color-coded blue and
its numbers are also listed in Table -#1,
marked by the pointers.
The smaller one is color-coded red and
its numbers are also listed in Table -#2,
marked by the pointers.

These arcuate rows become harder to discern when one
tries to see them closer to the center of the spiral.
To discern these rows faultlessly near the center it is best to rely on
the numbers generated algebraically in the respective tables.

following the blue row from its highest number: “1597”
to the number just below sector-line -#1 is easy.
Beyond that sector-line one might expect the next two numbers to be
located in the two sand-colored locations shown in Fig. -#1,
but their actual locations are jogged over one step to the right.
This is because the spiral itself has a jog in its geometry,
that jog is designated by “A”.

following the red row from its highest number: “631”
to the number just below sector-line -#1 is easy.
Beyond that sector-line one might expect the next number to be
located in the sand-colored location shown in Fig. -#1,
but its actual location is jogged over one step to the right.
This is because the spiral itself has a jog in its geometry,
that jog is designated by “B”.

Starting with “5” each number on sector-line -#1 is
jogged over 1 step to the right due to
the jogs in the geometry of the spiral.

findings3-fig2

Fig. -#2.

Tetrapolar
Ulam-type spiral.
Showing only
composite adjacents.

Spiral origin is at “0”.

Conclusion

The tetrapolar spiral, -and other polar spirals- are
ideal places to find ‘lengthy’ rows of prime numbers
because the sector-lines do not disturb the geometry of
the rows when they cross over these lines.
This as also mentioned in
Findings (2).

Fig. -#2 is shown as a reminder that similar rows can be found
when searching thru spirals composed of composite adjacents.

A super-computer enabled search program of
maximally extensive polar spirals
should produce many interesting results indeed.