Some research of Erdős–Straus

This preprint develops a structural and sieve-theoretic study of the Erdős–Straus equation 4/n=1/x+1/y+1/z through two-term completions at a fixed first denominator, interval decision procedures, and the large sieve. It establishes sharp Type I/II envelopes for the smallest denominator with intrinsic common-gcd character restrictions, a complete factorization-free decision procedure on (N,2N] in O_\varepsilon(N^{5/4+\varepsilon}) unit-cost operations, uniform bounds for the seven polynomial certificate families and an all-Q form of the (\log N)^{2/3} ceiling, plus an exact classification of square and twice-square denominator pairs and a norm-form characterization of every reduced completion. Complete self-contained proofs are supplied throughout, including the inverse theorems for dense divisor-residue obstructions; composite witnesses further show that the leading constant in the Type II exceptional-set estimate is unbounded as a smooth auxiliary level grows, while the exponent ceiling persists. All exceptional-set bounds are average statements: they do not empty the exceptional set, and the Erdős–Straus conjecture remains open.