## 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).