arXiv:2005.03273v3 [math.NT] 24 Aug 2021 A SHORT PROOF OF ERD ¨ OS–STRAUS CONJECTURE FOR EVERY n ≡ 13 mod 24 MARIO GIONFRIDDO AND ELENA GUARDO Abstract. The Erd¨os–Straus conjecture states that the equation 4 n = 1 x + 1 y + 1 z has positive integer solutions x, y, z for every postive integers n ≥ 2. In this short note we find explicity the solutions of the famous conjecture for the case n ≡ 13 mod 24. 1. Introduction In Number Theory there are many unsolvable problems that still attract a lot of attention. Among all, there is a famous conjecture of Erd¨os–Straus which states that, for all positive integers n ≥ 2, the rational number 4 n can be expressed as the sum of three positive unit fractions. Specifically, in 1948 P. Erd¨os and E. G. Straus formulated the following: Conjecture 1.1. For every positive integer n ≥ 2 there exist positive inte- gers x,y,z such that: (1) 4 n = 1 x + 1 y + 1 z . This conjecture has attracted a lot of attention not only among reseachers in Number Theory but also among many people involved among the different areas in Mathematics, such as L. Bernstein [1], M. B. Crawford [3], M. Di Giovanni, S. Gallipoli, M. Gionfriddo [4], C. Elsholtz and T. Tao [5], J. Guanouchi [6, 7], S. K. Jeswal and S. Chakraverty [8], L. J. Mordell [9], D. J. Negash [10], R. Obl´ ath [11], L. A. Rosati [12], J. W. Sander [14], R. C. Vaughan [16], K. Yamamoto [18], just to cite some of them. For example, Swett in [15] has established validity of the conjecture for all n ≤ 10 14 and it also appears in The Penguin Dictionary of Curious and Interesting Numbers, [17], or S. E. Salez [13] that proved that the conjecture holds for all n ≤ 10 17 . It is not clear if Conjecture 1.1 is true or not since there are many papers in which some authors adfirm to have proved that the conjecture is true and others in which the authors proved that it is false (see [19]). Since we have found some mistakes in some of these published papers, it is our opinion that this conjecture is still open. L. J. Mordell, in [9], Chapter 30, Section 1, has only proven that the conjecture is true for all n except possibly cases in which n is congruent 2010 Mathematics Subject Classification. 11D72, 11D45, 11P81 . Key words and phrases. Conjecture Erd¨o–Straus, Diophantine equation . Last updated: March 10, 2021. 1