Euler's prime formula, the Bunyakovsky Conjecture, and n^2 + n + 1

Euler's prime formula of n^2 + n + 41 produces prime numbers for all integers n from 1 to 40. 

If 41 is substituted for another larger prime, it produces a larger proportion of the first 1,000 primes.

Derpetology: The Archimedes, Ulam, and Sacks Spirals, Euler's Prime Formula, and the Non-random Distribution of Prime Numbers

n^2 + n + 1 produces prime numbers for all n that are congruent to n = 0 or 2 in modulo 6. 

Bunyakovsky conjecture - Wikipedia

Derpetology: spirals, primes, conjectures, and theorems

If we understood primes perfectly, we'd understand pi perfectly. Every prime is a little slice of pi. 

It all fits together somehow.