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Claim. Theorem 1.1 of Lasse Grimmelt and Jori Merikoski, On the Greatest Prime Factor and Uniform Equidistribution of Quadratic Polynomials (arXiv:2505.00493, v2 of 30 May 2025, p. 2), states: there is a small $\varepsilon>0$ such that for every real $X>\varepsilon^{-1}$ and integers $a,h$ with $1\le h\le X^{1+\varepsilon}$ squarefree, $1\le a\le X^\varepsilon$ and $\gcd(a,h)=1$, if a prime-sum hypothesis on the root counts of $a\nu^2+h$ holds, then some integer $m\in[X,2X]$ has $P^+(am^2+h)>X^{1.312}$. The paragraph after the theorem states that the hypothesis holds unconditionally when $ah\le X^{\varepsilon^2}$. For fixed $a\ge1$ and squarefree $h\ge1$ with $\gcd(a,h)=1$ this holds for every large $X$, so taking $X=n/2$ gives $F_f(n)>2^{-1.312}n^{1.312}$ eventually for $f(t)=at^2+h$, in the notation of Problem 976. The theorem and the case $a=h=1$ are recorded on the theorem card.

Covers. The first question for $f(t)=at^2+h$ with fixed $a\ge1$ and squarefree $h\ge1$ coprime to $a$, including $t^2+1$. It does not give $F_f(n)\gg n^2$.

Depends on. No page of this wiki.

Standing. An arXiv preprint, first posted on 1 May 2025, with no journal publication recorded. The proof imports an automorphic-kernel bound from a companion paper of the same authors and the sieve calculations of Merikoski's 2023 paper on $n^2+1$. The site labels the problem OPEN.