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All primes except $2$ and $3$ are of the form $6x-1$ and $6x+1$. For every prime of the form $6x-1$ are there comparable number of primes of the form $6x+1$ in the first $10000$ primes or is there an excess of one form over the other? Thanks

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2 Answers

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Among the first 10000 primes, there are $4988$ of the form $6k + 1$ and $5010$ of the form $6k-1$. Got these numbers using dumb computer search.

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Going beyond $10000$ (which, as Dan Shved pointed out, you could simply count), the following fact illustrates that the primes are "equally" spread between $6n-1$ and $6n+1$ forms. Define $\chi(p)=-1$ if $p\equiv 1\bmod 3$ and $\chi(p)=1$ if $p\equiv -1 \bmod 3$. Then the series $$ \sum_{p\text{ prime}} \frac{\chi(p)}{p} \tag1 $$ converges. Since the series $\sum_{p\text{ prime}} \frac{1}{p}$ diverges, the finiteness of (1) demonstrates that the positive contribution of the primes of the form $6n-1$ is evenly matched by the negative contributions of the primes of the form $6n+1$.

For more, see this answer by Bruno Joyal.

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