2025/08/10 by Ghermoul, Bilal
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2508.07367
The generalized Erdős-Straus conjecture, proposed by Wacław Sierpiński in 1956, asks whether the Diophantine equation (5)/(a) = (1)/(b) + (1)/(c) + (1)/(d) admits positive integer solutions b,c,d ∈ ℕ for every integer a ≥ 2. In this work we present explicit solutions for all integers a ≥ 2. We begin with the simplest known cases where a ≡ i \pmod5 for i ∈ \0,2,3,4\, providing direct decompositions. The remaining open case, a ≡ 1 \pmod5, is addressed for a = 5q + 1 with q \not≡ 0 \pmod252, where we give explicit decompositions, often with q expressed as three-variable polynomials. For q ≡ 0 \pmod252, we conjecture that a specific polynomial p1(x,y,z)=z (x (5 y-1)-y)-x,~ x,y,z ∈ ℕ^*, which exactly satisfies the generalized Erdős--Straus equation, generates all such multiples of 252. This conjecture has been verified computationally for 5q+1 up to approximately 1010, and the corresponding Mathematica implementation is included.