The Goldbach Comet

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Goldbach's conjecture is probably one of the most famous unsolved problems in mathematics. It states that

Every even integer $n>2$ can be expressed as the sum of two primes.

The number of different ways that $n$ can be expressed as the sum of two primes is known as the Goldbach function, $g(n)$, and a plot of $g(n)$ against $n$ is called the Goldbach comet:

enter image description here

To explain the shape of this plot, first note that all prime numbers $p > 3$ are of the form $6q \pm 1$ ($q=1,2,\cdots$) (that is, $p \;\mathrm{mod}\;6$ is 1 or 5). To see this, write an arbitrary integer as $m=6q+r$: then if $r$ is 0, 2 or 4 then certainly $m$ is even and if $r$ is 3 then $m$ is divisible by 3. So if $m>3$ is prime it must be that case that $r$ is 1 or 5.

Now consider the possible ways of forming an even integer $n>6$ from two primes:

\begin{align*} (6a-1) + (6b-1) &= 6(a+b) - 2 = 4\;(\mathrm{mod}\;6)\\ (6a-1) + (6b+1) &= 6(a+b) = 0\;(\mathrm{mod}\;6)\\ (6a+1) + (6b-1) &= 6(a+b) = 0\;(\mathrm{mod}\;6)\\ (6a+1) + (6b+1) &= 6(a+b) + 2 = 2\;(\mathrm{mod}\;6)\\ (6a-1) + 3 &= 6a + 2 = 2\;(\mathrm{mod}\;6)\\ (6a+1) + 3 &= 6a + 4 = 4\;(\mathrm{mod}\;6)\\ \end{align*}

That is, we can expect that there are more ways of creating an even integer equal to 0 (mod 6) from two primes (the blue points above) than there are of creating an even integer equal to 2 or 4 (mod 6) (the green and red points).

The Python code used to create the above plot is below.

import numpy as np
import matplotlib.pyplot as plt

nmax = 2000
# Odd prime numbers up to nmax.
odd_primes = np.array([n for n in range(3, nmax) if all(
                        (n % m) != 0 for m in range(2,int(np.sqrt(n))+1))])

def get_g(n):
    """Return the value of the Goldbach function for even integer n>2."""
    g = 0
    for p in odd_primes:
        if p > n//2:
            break
        if n-p in odd_primes:
            g += 1
    return g

# Array of indexes into our Goldbach function array.
imax = nmax//2 - 1
idx = np.arange(imax)
def get_n_from_index(i):
    return 2*(i+2)
# The values of even n>2 from the index array.
n = get_n_from_index(np.arange(imax))

# Get an array of Goldbach function values corresponding to the n array.
g = np.zeros(imax, dtype=int)
for i in idx:
    g[i] = get_g(n[i])

# Indexes into the arrays for the cases n = 0, 2 or 4 (mod 6)
i_0 = idx[((n%6)==0)]
i_2 = idx[((n%6)==2)]
i_4 = idx[((n%6)==4)]

# Make the scatter plot for these three cases with different colour markers.
plt.scatter(n[i_0], g[i_0], marker='+', c='b', alpha=0.5,
            label=r'$n=0\;(\mathrm{mod}\;6)$')
plt.scatter(n[i_2], g[i_2], marker='+', c='g', alpha=0.5,
            label=r'$n=2\;(\mathrm{mod}\;6)$')
plt.scatter(n[i_4], g[i_4], marker='+', c='r', alpha=0.5,
            label=r'$n=4\;(\mathrm{mod}\;6)$')

# Set the plot limits and tidy.
plt.xlim(0, nmax)
plt.ylim(0, np.max(g[i_0]))
plt.xlabel(r'$n$')
plt.ylabel(r'$g(n)$')
plt.legend(loc='upper left', scatterpoints=1)
plt.savefig('goldbach_comet.png')
plt.show()
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