# The Ulam Spiral

## Fig. -1

All prime numbers -except “2” & “3”-
By not being adjacent to a 6-fold
the little group consisting of “2” & “3”
constitutes an entirely different species of primes.

For this reason none of the spirals in this site
contain these numbers, since their presence
muddles the geometric pattern.
This can be seen in the spiral at left
where they are included, color-coded purple.

All the spirals in this site, including the one in Fig. -1,
have “0” as the spiral origin instead of “1” because of
Spiral Origins”.

All these spirals
include “1” as an entry
since it is an adjacent of “0”,
which is a 6-fold since 0 x 6 = 0.
The presence of “1”
maintains the wholeness of the geometric pattern,
as can be seen by inspecting the spiral at left.

## Fig.-2

The numbers-stack in Fig. -2 also demonstrates
the detriments of including “2” & “3” and
the benefits of including “1” mentioned above.

The numbers-stack in Fig. -2 also demonstrates
the detriments of including “2” & “3” and
the benefits of including “1” mentioned above.

Researchers wishing to find patterns in
the distribution of prime- or composite adjacents
-in a spiral or elsewhere-
might want to consider these concerns.

###### Notes.

A formula to generate primes
-assuming such a formula exists-
would probably generate
only those primes that are
starting with “1” as the 1st. prime.
(“1” is adjacent to “0”, which is a 6-fold.)
It probably won’t include “2” & “3”
since neither are adjacent to a 6-fold,
thus being a different species.

###### Suggested number-definitions:

Prime number:
Any whole number larger than 4
that cannot be evenly divided by
any whole number besides itself
that is larger than 1.

Composite number:
Any whole number larger than 3
that can be evenly divided by
one- or more whole numbers besides itself
that is/are larger than 1.

Quasi prime:
(P = 1, 2, 3.)
A prime small enough
such that there is
no whole number “f”
such that 1 < f < P/2.

Actual prime:
(5 ≤ P < ∞)
A prime “P” large enough
such that there is at least
1 whole number “f”
such that 1 < f < P/2.

(With each prime “P” the inability
of each of the available
whole numbers such as “f”
to evenly divide into “P”
establishes the actual
primehood of “P”.)