## Fig. -1.

Fig. -1 shows an Ulam Spiral

and highlighted therein

8 rows of numbers color-coded red.

These rows can eachly be identified

in convenient cartographic fashion as:

“E” -row, starting at 1,

“NE” -row, starting at 2,

“N” -row, starting at 3,

“NW” -row, starting at 4,

“W” -row, starting at 5,

“SW” -row, starting at 6,

“S” -row, starting at 7,

“SE” -row, starting at 1.

Given the geometric prominence

of these rows in the spiral they can be

referred to as the “cardinal rows”.

Interspersed among these rows are

8 triangular areas of numbers, color-coded blue.

These areas can eachly be identified

in convenient cartographic fashion as:

“E/NE” -area,

“N/NE” -area,

“N/NW” -area,

“W/NW” -area,

“W/SW” -area,

“S/SW” -area,

“S/SE” -area,

“E/SE” -area.

These areas can be referred to as

the “inter-areas”.

## Fig. -2.

#### spiral origin at 0.

### Note -1.

Fig. -2 shows

the spiral with its origin at 0.

As can be seen, it does not have any

prime adjacent on any of the cardinal rows,

except for:

1 at the start of the “E” -row,

5 at the start of the “W” -row,

7 at the start of the “S” -ow.

All the other prime adjacent are

located in the inter-areas.

This remains true even as

the spiral grows to infinity.

## Fig. -3.

#### spiral origin at 1.

### Note -2.

Fig. -3 shows

the spiral with its origin at 1.

as can be seen, it has

prime adjacent located on

all the cardinal rows

except the “SE” -row.

These prime adjacent are

indicated by the red circles.

### Note -3.

From comparing the situations described in

Notes -1 & -2 one must conclude that

the Ulam Spiral with origin at 0

is vastly superior to

the Ulam Spiral with origin at 1.

This because of the

higher state of organization manifest in the latter,

where all the prime adjacents -except 1, 5, 7,- are

cleared from the cardinal rows and

herded into the inter-areas.

## Fig. -4.

### In Fig. -4:

The “E” -row consists of the following numbers:

1, 10, 27, 52, 85, 126,175, … definable as:

n(4n – 3), 1 ≤ n < ∞.

The “NE” -row consists of the following numbers:

2, 12, 30, 56, 90, 132, 182, … definable as:

(2n + 1)(2n + 2), 0 ≤ n < ∞

The “N” -row consists of the following numbers:

3, 14, 33, 60, 95, 138, 189, … definable as:

n(4n – 1), 1 ≤ n < ∞

The “NW” -row consists of the following numbers:

4, 16, 36, 64, 100, 144, 196, … definable as:

(2n)^2, 1 ≤ n < ∞

The “W” -row consists of the following numbers:

5, 18, 39, 68, 105, 150, 203, … definable as:

n(4n + 1), 1 ≤ n < ∞

The “SW” row consists of the following numbers:

6, 20, 42, 72, 110, 156, 210, … definable as:

2n(2n + 1), 1 ≤ n < ∞

The “S” -row consists of the following numbers:

7, 22, 45, 76, 115, 162, 217, … definable as:

n(4n + 3), 1 ≤ n < ∞

The “SE” -row consists of the following numbers:

1, 9, 25, 49, 81, 121, 169, 225, … definable as:

(2n + 1)^2, 0 ≤ n < ∞

### In Fig. -5:

Adjacent to the “SE” -row, to its left,

there is another row, starting at 8,

whose numbers consist of

numbers from the “SE” -row, eachly decreased by 1.

That row can be defined as the “SE-” -row.

Adjacent to the “SE” -row, to its right,

there is another row, starting at 2,

whose numbers consist of

numbers from the “SE” -row, eachly increased by 1.

That row can be defined as the “SE+” -row.

Adjacent to the “NW” -row, to its right,

there is another row, starting at 3,

whose numbers consist of

numbers from the “NW” -row, eachly decreased by 1.

That row can be defined as the “NW-” -row.

These 3 rows are color-coded yellow.

The “SE-” -row consists of the following numbers:

8, 24, 48, 80, 120, 168, 224, … definable as:

(2n + 1)^2 – 1, 1 ≤ n < ∞.

The “SE+” -row consists of the following numbers:

2, 10, 26, 50, 82, 122,170, 226, … definable as:

(2n + 1)^2 + 1, 1 ≤ n < ∞.

The “NW” -row consists of the following numbers:

3, 15, 35, 63, 99, 143,195, … definable as:

(2n)^2 – 1, 1 ≤ n < ∞.

The spiral does not

have any prime adjacent

on these 3 added rows.

This remains true even as

the spiral grows to infinity.

### In Fig. -6:

Adjacent to the “S” -row, to its left,

there is another row, starting at 6,

whose numbers consist of

numbers from the “S” -row, eachly decreased by 1.

That row can be identified as the “S-” -row.

Adjacent to the “E” -row, below it,

there is another row, starting at 9,

whose numbers consist of

numbers from the “E” -row, eachly decreased by 1.

That row can be identified as the “E-” -row.

These numbers are color-coded purple.

The “S-” -row consists of the following numbers:

6, 21,44, 75,114, 161, 216, … definable as:

(n + 2)(4n + 3), 0 ≤ n < ∞.

The “E-” -row consists of the following numbers:

9, 26, 51, 84, 125, 174, 231, … definable as:

n(4n + 5), 1 ≤ n < ∞.

The spiral does not

have any prime adjacents

on these 2 added rows.

This remains true even as

the spiral grows to infinity.

## Fig. -7.

### In Fig. -7:

the 3 added rows shown in Fig. -5 &

the 2 added rows shown in Fig. -6

are all represented.

These rows obviously diminish

the space available for

the prime adjacents in

some of the inter-areas.

The totality of space available

for the prime adjacents

is shown in Fig. -7,

color-coded blue.