Sieve dimension and search depth for the Erd\H{o}s-Straus conjecture, $n \equiv 1 \pmod{24}$

Abstract

For primes $n\equiv1\pmod{24}$ we study how deep an explicit, factorization-free search for a decomposition of $4/n$ into three unit fractions has to go. Write $E_2(N;J)$ for the set of such primes $n\le N$ at which no witness of depth at most $J$ exists, in the sense of the two-parameter criterion of Theorem 3.9 with coprime parameters $u,a\le J$. We prove that for every fixed $J$ $$|E_2(N;J)| \ll_J \frac{N}{(\log N)^{1+\mathfrak{A}(J)/2}},$$ the exponent being the exact dimension of the covering on which the proof rests. The proof replaces the subgroup generated by the prime factors, an approach that breaks down as soon as $(\mathbb{Z}/4m)^{\times}$ has exponent greater than $2$, by a fixed-point-free involution, and is unconditional at every $J$. Second, we exhibit an unconditional obstruction. At the shift $c=7$ there are $\asymp N(\log N)^{-3/2}$ primes $n\le N$, $n\equiv1\pmod{24}$, at which both branches of the divisor criterion fail. The representation of $K_7=(n+7)/4$ by the principal form of discriminant $-7$ has to be primitive, so that the relevant input is the primitive-representation theorem of Fuchs, Hsu, Rickards, Schindler and Stange [25] rather than the classical results of Iwaniec; a fixed shift therefore cannot leave a finite residual set. Third, a factorization-free procedure decides the conjecture for all primes of an interval $[N,2N]$. Its Type II pass costs $\mathcal{O}(N(\log N)^{3})$ while its Type I pass costs $\Theta(N^{2})$, which locates the whole quadratic cost in the extraction of the divisors of $4u^{2}d+1$ and exhibits an asymmetry between the two halves of the Type I/Type II dichotomy. The conjecture itself remains open.