2020/12/30 by S. Maiti, Maiti, S
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Benford’s Law and Fraud Detection #FOS: Mathematics #General Mathematics (math.GM)
paper · pdf · doi:10.48550/arxiv.2101.00975
openalex publication_date 2020/12/30 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
The Erdős-Straus conjecture is a renowned problem which describes that for every natural number n~(≥ 2), (4)/(n) can be represented as the sum of three unit fractions. The main purpose of this study is to show that the Erdős-Straus conjecture is true. The study also re-demonstrates Mordell theorem which states that (4)/(n) has a expression as the sum of three unit fractions for every number n except possibly for those primes of the form n≡ r (mod 780) with r=12,112,132,172,192,232. For l,r,a∈ℕ; (4)/(24l+1)-(1)/(6l+r)=(4r-1)/((6l+r)(24l+1)) with 1≤ r≤ 12l, if at least one of the sums in right side of the expression, say, a+(4r-a-1),~1≤ a≤ 2r-1 for at least one of the possible value of r such that a,(4r-a-1) divide (6l+r)(24l+1); then the conjecture is valid for the corresponding l. However, in this way the conjecture can not be proved only twelve values of l for l up to l=105.