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Statement

The paper's solution-counting function is $f(p)=|\{(x,y,z):4/p=1/x+1/y+1/z\}|$ for $p\in\mathbb N^*$ (p. 2); the paper does not say whether the triples are ordered. Following Bradford, it searches $\lceil p/4\rceil\le x\le\lceil p/2\rceil$ and builds $y$ and $z$ from $x$ and a divisor $d\mid x^2$ satisfying one of two conditions (the "Bradford conditions", p. 2), one for Type-1 solutions ($p\nmid y$) and one for Type-2 solutions ($p\mid y$). The paper states that the conjecture is equivalent to $f(p)>0$ for all $p\in\mathbb N^*$ (p. 2).

Section 3 (pp. 2--3), the reported computation. Let $\mathcal P$ be the increasing sequence of primes $p\equiv r\pmod{840}$ with $r\in\{1,121,169,289,361,529\}$, the classes Mordell left open. The authors evaluated $f(\mathcal P_i)$ for $1\le i\le N$, $N=66737$, which they describe as the "difficult" primes $p\le3.5\cdot10^7$ (p. 3). Over these primes they checked $T=29860049601808$ divisors of squares of admissible $x$ and found $S=18601583$ satisfying a Bradford condition, of which $S_1=12763383$ are of Type-1 and $S_2=5838200$ of Type-2 (p. 3), so Type-1 solutions are more than twice as common as Type-2 in this range. Figure 1 (p. 3) plots $f(\mathcal P_i)$ by type on a logarithmic horizontal axis. The authors read the data as $f(p)$ appearing to increase, consistently with the Elsholtz--Tao upper bound (p. 3).

Source. Mihnea and Dumitru, arXiv:2509.00128v1 (29 August 2025), 4 pp.; Section 3 runs from p. 2 to p. 3, read on the page images.

Read depth. Claims checked: the statements of Section 3 were read clause by clause. This is an empirical computation with no theorem label; the counts are the authors' report, not rerun here, and the code (footnote 1, p. 2) was not fetched. The paper attributes to Elsholtz and Tao a polylogarithmic upper bound for $f(p)$ (p. 2); their pointwise bound is $f(p)\ll p^{3/5+O(1/\log\log p)}$ (Proposition 1.7), and a polylogarithmic size is known only on average (Theorem 1.1). The primes below $3.5\cdot10^7$ are also covered by the verification of Section 2, so the count adds no new case of the conjecture.

Dependencies

Bradford's search range and divisor construction (Bradford, Elemental patterns from the Erdős–Straus conjecture, 2024; not consulted); Mordell's residue classes modulo $840$.

Bears on

• Problem 242: numerical data on the number of solutions for primes $p\le3.5\cdot10^7$ in the six classes modulo $840$; it settles no $n$ beyond Section 2's range and proves no bound on $f(p)$.