Problem #6: Let $d_n=p_{n+1}-p_n$

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Progress

Conjectured by Erdős and Turán [ErTu48]. Shockingly Erdős offered \$25000 for a disproof of this, but as he comments, it 'is certainly true'. (In [Er85c] he goes further and offers 'all the money I can earn, beg, borrow or steal for [a disproof]'.)

Indeed, the answer is yes, as proved by Banks, Freiberg, and Turnage-Butterbaugh [BFT15] with an application of the Maynard-Tao machinery concerning bounded gaps between primes [Ma15]. They in fact prove that, for any $m\geq 1$, there are infinitely many $n$ such that\[d_n<d_{n+1}<\cdots <d_{n+m}\]and infinitely many $n$ such that\[d_n> d_{n+1}>\cdots >d_{n+m}.\]This is discussed in problem A11 of Guy's collection [Gu04].