It is a very well-known result that the series of the reciprocals of the prime numbers diverges (see, for instance, [1, pp. 187-188] or [2]). On the other hand, according to the so-called internal model of OpenAI, the Erdős-Turán conjecture holds true (cf. Corollary 1.2 in [3, p. 4]): that is, every set $A$ of positive integers for which $\sum_{a \in A} 1/a$ diverges contains arithmetic progressions of every finite length. It does follow that there are arbitrarily long arithmetic progressions of prime numbers. Q.E.D.
References
[1] L. Euler, Variae observationes circa series infinitas. Commentarii academiae scientiarum Petropolitanae, Vol. 9, 1744, pp. 160-188.
[2] P. Erdős, Über die Reihe $\sum \frac{1}{p}$. Mathematica, Zutphen B, Vol. 7, 1938, pp. 1-2.
[3] OpenAI, Quasipolynomial bounds for arithmetic progressions. Preprint released on September 23, 2026.
* Postdata. While preparing the list of references for this entry I noticed that the identity $$\frac{\pi}{4} = \frac{3}{4}\cdot \frac{5}{4} \cdot \frac{7}{8} \frac{11}{12} \cdot \frac{13}{12} \cdots $$ is the subject matter of the eleventh theorem in [1, p. 178]. So, I suppose I have settle the doubt I expressed when I blogged about this infinite product for $\frac{\pi}{4}$ on March 14, 2023. Yet, I must confess that for a certain lapse I was starting to dellude myself into thinking that I had seen this identity for the first time in the paper "Infinite Sums, Diophantine Equations, and Fermat's Last Theorem" by H. Darmon and C. Levesque (S. Kanemitsu & C. Jia (eds.), Number theoretic methods. Developments in Mathematics, Vol. 8, Springer Verlag, Boston, MA, USA, 2002, pp. 73-95).
