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The Ulam Spiral

… and many other spirals 
inspired by it …

Ulam Spiral with
origin at 1.

Ulam Spiral with
origin at 0.

The spirals above are distinct as follows:
the top spiral has “1” as its origin and
the bottom spiral has “0” as its origin.

Along each of the
8 cardinal directions on both spirals
starting at the origin of the spiral
a series of 4 numbers are shown.
Also shown: the respective
formulations generating these numbers.

The formulations in the bottom spiral
are simpler as follows:
a)
In the case of the numbers
along the diagonal directions:
In the top spiral their respective formulations
contain “1” as a 2nd. term,
in the bottom spiral their respective formulations
contain no 2nd. term.
b)
In the case of the numbers
along the horizontal- & vertical directions:
in both spirals their respective formulations
contain a 2nd. term,
but in the bottom spiral it is less by “1”,
-even to the point of vanishing by becoming “0”.

thus greater simplicity and clarity is obtained by
having “0” as the spiral origin.

Other benefit are the following:
with “0” as the spiral origin:
a)
none of the prime numbers are
located along the diagonal directions.
(Except “2”.)
b)
none of the prime numbers are
located along the horizontal- & vertical directions.
(Except: “1”, “3”, “5”, “7”.) *
——————————————-
* In case of the horizontal row: “1”, “10”, “27”, “52”, ….
The uneven numbered entries “1”, “27”, ….
always consist of uneven numbers.
these numbers are always (?) composite according as
(4n+1)^2 + 2n is always (?) composite
for any n, 0 > n > ∞.

I have tried a long sequence of these numbers and
found them always to be composite, but a proof of
(4n+1)^2 + 2n is composite for 0 > n > ∞
would be needed to fully validate
the relevant part of the assertion under b).