Erdős Problem 6

References:

• erdosproblems.com/6
• [BFT15] Banks, William D. and Freiberg, Tristan and Turnage-Butterbaugh, Caroline L., Consecutive primes in tuples. Acta Arith. (2015), 261-266.
• [Ma15] Maynard, James, Small gaps between primes. Ann. of Math. (2) (2015), 383-413.

theorem Erdos6.erdos_6 : {n : ℕ | primeGap n < primeGap (n + 1) ∧ primeGap (n + 1) < primeGap (n + 2)}.Infinite

There are infinitely many $n$ such that $d_n < d_{n+1} < d_{n+2}$, where $d$ denotes the prime gap function.

theorem Erdos6.erdos_6.variants.increasing (m : ℕ) : {n : ℕ | ∀ i ∈ Finset.range m, primeGap (n + i) < primeGap (n + i + 1)}.Infinite

For all $m$, there are infinitely many $n$ such that $d_n < d_{n+1} < \dots < d_{n+m}$, where $d$ denotes the prime gap function.

Proved by Banks, Freiberg, and Turnage-Butterbaugh [BFT15] with an application of the Maynard-Tao machinery concerning bounded gaps between primes [Ma15]

theorem Erdos6.erdos_6.variants.decreasing (m : ℕ) : {n : ℕ | ∀ i ∈ Finset.range m, primeGap (n + i) > primeGap (n + i + 1)}.Infinite

For all $m$, there are infinitely many $n$ such that $d_n > d_{n+1} \dots > d_{n+m}$, where $d$ denotes the prime gap function.

Proved by Banks, Freiberg, and Turnage-Butterbaugh [BFT15] with an application of the Maynard-Tao machinery concerning bounded gaps between primes [Ma15]