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import FormalConjecturesUtilErdős Problem 218
namespace Erdos218
The set of indices $n$ for which a prime gap is followed by a larger or equal prime gap has a natural density of $\frac 1 2$.
@[category research open, AMS 11]
theorem erdos_218.variants.le : {n | primeGap n ≤ primeGap (n + 1)}.HasDensity <| 1 / 2 := ⊢ {n | primeGap n ≤ primeGap (n + 1)}.HasDensity (1 / 2)
All goals completed! 🐙
The set of indices $n$ for which a prime gap is preceeded by a larger or equal prime gap has a natural density of $\frac 1 2$.
@[category research open, AMS 11]
theorem erdos_218.variants.ge : {n | primeGap (n + 1) ≤ primeGap n}.HasDensity <| 1 / 2 := ⊢ {n | primeGap (n + 1) ≤ primeGap n}.HasDensity (1 / 2)
All goals completed! 🐙
There are infintely many indices $n$ such that the prime gap at $n$ is equal to the prime gap
at $n+1$. This is equivalent to the existence of infinitely many arithmetic progressions of
length $3$, see erdos_141.variants.infinite_three.
@[category research open, AMS 11]
theorem erdos_218.variants.infinite_equal_prime_gap : {n | primeGap n = primeGap (n + 1)}.Infinite := ⊢ {n | primeGap n = primeGap (n + 1)}.Infinite
All goals completed! 🐙
end Erdos218