The Ulam Spiral is a simple way to visualize the prime numbers which reveals unexpected, irregular structure. Starting at the centre of a rectangular grid of numbers and spiralling outwards, mark the primes as `1`

and the composite numbers as `0`

.

The following code generates an image of the Ulam Spiral. The hard part is generating a "spiralled"-version of a two-dimensional array, which is the job of the `make_spiral`

. This function takes a two-dimensional array of *indices* into the array, `idx`

, and successively appends the top row to a new list of indices, `spiral_idx`

. After each append, the appended row is removed from `idx`

and the remainder of the array rotated by $90^\circ$. The concatenated array of `spiral_idx`

is then used to index into the array to be "spiralled".

*This code is also available on my github page.*

```
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.cm as cm
def make_spiral(arr):
nrows, ncols= arr.shape
idx = np.arange(nrows*ncols).reshape(nrows,ncols)[::-1]
spiral_idx = []
while idx.size:
spiral_idx.append(idx[0])
# Remove the first row (the one we've just appended to spiral).
idx = idx[1:]
# Rotate the rest of the array anticlockwise
idx = idx.T[::-1]
# Make a flat array of indices spiralling into the array.
spiral_idx = np.hstack(spiral_idx)
# Index into a flattened version of our target array with spiral indices.
spiral = np.empty_like(arr)
spiral.flat[spiral_idx] = arr.flat[::-1]
return spiral
# edge size of the square array.
w = 251
# Prime numbers up to and including w**2.
primes = np.array([n for n in range(2,w**2+1) if all(
(n % m) != 0 for m in range(2,int(np.sqrt(n))+1))])
# Create an array of boolean values: 1 for prime, 0 for composite
arr = np.zeros(w**2, dtype='u1')
arr[primes-1] = 1
# Spiral the values clockwise out from the centre
arr = make_spiral(arr.reshape((w,w)))
plt.matshow(arr, cmap=cm.binary)
plt.axis('off')
plt.show()
```

The prominent diagonal structures indicate that there are unexpectedly many primes of the form $f(n) = 4n^2 + bn + c$ for integer constants $b$ and $c$ and $n=1,2,3,\cdots$.

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## Comments

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## Derck van Schuylenburch 4 years, 4 months ago

All prime numbers -except 2 & 3- are adjacent to a multiple of 6. Many composite numbers are ALSO adjacent to a multiple of 6.

Link | ReplyWhen you populate an Ulam spiral with BOTH of these groups of numbers, the prime ones and the composite ones, you get

ENDLESS- UNINTERRUPTED- DIAGONAL rows of numbers in both diagonal directions.

All pairs of adjacent parallel rows of numbers are identically spaced. 2 subset-spirals can be derived from the foregoing:

one populated with ONLY the composite numbers adjacent to a multiple of 6 and one populated with ONLY the prime numbers adjacent to a multiple of 6.

This last subset-spiral is identical to the standard Ulam spiral except: 2 & 3 not shown, as they are not adjacent to a multiple of 6, while 1 is shown as it is adjacent to 0, which is amultiple of 6.

For more of this, please see:

https://www.ulamspiralmethod.com

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