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Claim. Assume Hypothesis 19.2 of the paper. Then there are infinitely many primary pseudoperfect numbers, so the answer to Problem 313 would be yes. This is Wang's Theorem 19.5 in the preprint Port fillings for primary pseudoperfect numbers (arXiv:2605.21518v1, 18 May 2026). A terminal port is a triple $(R,c,p)$ with $cp-R=1$ and $p$ prime. The proof starts from the ambient terminal port $(N_9,1,N_9+1)$, where $N_9=5998279018951962402$ is the nine-factor primary pseudoperfect number of the paper's Theorem 9.1 and $N_9+1$ is prime by its Theorem 10.1. At each stage the hypothesis supplies five distinct primes above the terminal prime that solve $c\,x_1\cdots x_5-R\sum_i\prod_{j\ne i}x_j=1$; the product of $R$ with these primes is primary pseudoperfect, and four of the primes join $R$ to form a new ambient terminal port whose terminal prime is the fifth. The terminal primes strictly increase, so the numbers produced are distinct.

Hypothesis. Hypothesis 19.2 says: for a terminal port $(R,c,p)$ with $p>3$, if the equation $c\,x_1\cdots x_5-R\sum_i\prod_{j\ne i}x_j=1$ has an unbounded smooth positive real component on which every coordinate exceeds $p$, and has a solution in $(\mathbb F_\ell^\times)^5$ for every prime $\ell$, then that component contains a point whose coordinates are pairwise distinct primes above $p$. The paper's Lemmas 19.3 and 19.4 verify the local and real conditions for the ports of the construction. The paper says that the hypothesis "is not a theorem and is not a formal consequence of the classical one-variable Bateman–Horn conjecture" (Section 19, p. 17). The hypothesis is unproved, and the claim gives no unconditional answer.

Depends on.

• Theorem 9.1, the starting number $N_9$.
• Theorem 11.1, whose result page also states Theorem 10.1, the primality of $N_9+1$.

No claim page of this wiki is needed.

Standing. Claimed. The paper is an unrefereed preprint, with no journal record and no outside review found, and its Section 20 states that "No unconditional proof of infinitude is claimed." A conditional claim leaves the problem open. The author announced the preprint and its two new numbers in the site's discussion thread on 16 May 2026, describing it as partial progress rather than a solution; the thread post links the arXiv record, which was first posted on 18 May 2026, and the page is dated by that posting.