Consequences of binomial expansion's unexplored properties on fermat's triples and cosine law at the 5ecm in Amsterdam

The book reports integrally the intervention by Enzo Bonacci and Mario De Paz at the Fifth European Mathematical Congress held in Amsterdam in summer 2008. Presented on Monday 14th July during the Algebra Session, the poster 2.21 shows some unexplored properties of the binomial expansion with relevant influences on Fermat’s equation and Cosine Rule. It consists of three steps: 1) Proving unexplored properties of Pascal’s triangle; 2)Analysing the consequences of some binomial properties in limiting Fermat’s triple until an almost impossible condition of existence; 3) Using other binomial properties to extend the Law of Cosines to powers greater than two, i.e., on synclastic surfaces. The Appendix contains some papers on Number Theory written by the authors before they met.Mario De Paz is an Italian retired Professor of Physics from Genoa University, Director of the Educational Centre IMG, Professor of Didactics of Physics for the Master in Science, Professor of Didactics of Physics for the SISS and referee for the magazine “Biologi Italiani”.Enzo Bonacci is an Italian pre–university teacher of mathematics and physics in Latina, recently quoted among the 280 CBEL mathematicians for his new ideas on Number Theory and bestowed with the PhD Honoris Causa in Theoretical Physics after reviewing Einstein’s Relativity.