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Claim. Theorem 1.3 of Kyle Pratt, The irrationality of an infinite series involving $\omega(n)$ under a prime tuples conjecture, J. Number Theory 276 (2025), 57--71 (arXiv:2409.15185, posted 2024-09-23 under the title The irrationality of a prime factor series under a prime tuples conjecture), states that under the paper's Conjecture 1.2 the number $\sum_{n\ge1}\omega(n)/t^n$ is irrational for every integer $t\ge2$. The case $t=2$ is Problem 69. The claim is conditional: Conjecture 1.2 is a quantitative prime $K$-tuples conjecture with uniformity, asserting that for admissible linear forms $L_k(n)=a_kn+b_k$ with coefficients at most $(\log\log x)^{100}$ and $K\le100\log\log\log x$, the number of $n\le x$ with every $L_k(n)$ prime is $(1+o(1))\,\mathfrak S(L)\,x/(\log x)^K$. That conjecture is unproven, so this page derives nothing for the problem's standing; the unconditional proof is Tao and Teräväinen. The argument assumes the sum is $a/b$, so that $bt^N$ times the tail after $N$ is an integer; the prime-tuples input (Proposition 2.1) supplies $N=n_0Q$ with $n_0Q/k+1$ prime for $k\le K$, which makes that integer equal $a+b/(t-1)$ plus a small nonzero quantity, a contradiction (arXiv v1, p. 3); the source card pratt_2024_irrationality_prime_factor_series_under_prime digests the arXiv version; the journal version is not held. This outline is a reading aid, not proof coverage.

Acceptance. The refereed evidence is the journal publication cited above (Journal of Number Theory, volume 276, print date November 2025, as Crossref records it). The site's remarks record the result as conditional on a uniform version of the prime tuples conjecture, while the site's label PROVED credits the unconditional proof of Tao and Teräväinen, so the curator's remark is context, not acceptance of this claim. The corpus holds no compiled or reviewed proof.

Depends on. No page of this wiki.