Remarks on the Egyptian 2/D table in favor of a global approach (D prime number)

Abstract

For h=3 or 4, Egyptian decompositions into h unit fractions, like 2/D = 1/D1 + ... +1/Dh, were given by using (h-1) divisors (di) of D1. This ancient modus operandi, well recognized today, provides Di=DD1/di for i greater than 1. Decompositions selected (depending on di) have generally been studied by modern researchers through the intrinsic features of di itself. An unconventional method is presented here without considering the di properties but just the differences d(h-1)-dh. In contrast to widespread ideas about the last denominator like 'Dh smaller than 1000', it is more appropriate to adopt a global boundary of the form 'Dh smaller or equal to 10D', where 10 comes from the Egyptian decimal system. Singular case 2/53 (with 15 instead of 10) is explained. The number of preliminary alternatives before the final decisions is found tobe so low (71) for h=3 or 4 that a detailed overview is possible. A simple additive method of trials, independent of any context, can be carried out, namely 2n+1= d2+ ... + dh. Clearly the decisions fit with a minimal value of the differences d(h-1)-dh, independently of any di values.