An Erdős-Straus divisor–lattice refinement for 4/n

ABSTRACT.We give a proof by analysis-synthesis for the Erdos-Straus equation 4/n = 1/x + 1/y + 1/z with x,y,z in N.In the analysis step, we introduce an integer parameter D = delta0(n) + 4m, set x = (n + D)/4, and apply a Euclidean remainder correction that produces a rational epsilon in [1,2) such that y = n(n + D)/(4D) + epsilon and z = n^2 x^2/(D^2 epsilon) + n x / D.In the synthesis step, these explicit formulas are recombined to obtain the identity 4/n = 1/x + 1/y + 1/z and to define dyadic-style closure bands I_D(n) for the divisor parameter d3 = L_n / z, where L_n = lcm(1,2,...,n).Using the 2-density of L_n, we show that whenever I_D(n) is contained in [1, L_n], the intersection Div(L_n) intersect I_D(n) is nonempty, guaranteeing a divisor in the target scale window.As an application, we provide an explicit catalogue of unit-fraction decompositions for primes p