Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers AmnonYekutieli DepartmentofMathematics BenGurionUniversity email:[email protected] Notesavailableat http://www.math.bgu.ac.il/~amyekut/lectures written7June2015 AmnonYekutieli(BGU) PythagoreanTriples 1/28 c b a 1.PythagoreanTriples 1. PythagoreanTriples APythagoreantripleisatriple(a,b,c)ofpositiveintegers,satisfying (1.1) a2+b2 = c2. Thereasonforthenameis,ofcourse,becausethesearethesidesofaright angledtriangle: AmnonYekutieli(BGU) PythagoreanTriples 2/28 c b a 1.PythagoreanTriples 1. PythagoreanTriples APythagoreantripleisatriple(a,b,c)ofpositiveintegers,satisfying (1.1) a2+b2 = c2. Thereasonforthenameis,ofcourse,becausethesearethesidesofaright angledtriangle: AmnonYekutieli(BGU) PythagoreanTriples 2/28 c b a 1.PythagoreanTriples 1. PythagoreanTriples APythagoreantripleisatriple(a,b,c)ofpositiveintegers,satisfying (1.1) a2+b2 = c2. Thereasonforthenameis,ofcourse,becausethesearethesidesofaright angledtriangle: AmnonYekutieli(BGU) PythagoreanTriples 2/28 1.PythagoreanTriples 1. PythagoreanTriples APythagoreantripleisatriple(a,b,c)ofpositiveintegers,satisfying (1.1) a2+b2 = c2. Thereasonforthenameis,ofcourse,becausethesearethesidesofaright angledtriangle: c b a AmnonYekutieli(BGU) PythagoreanTriples 2/28 1.PythagoreanTriples Wesaythatthetriples(a,b,c)and(a(cid:48),b(cid:48),c(cid:48))areequivalentifthe correspondingtrianglesaresimilar. Thismeansthatthereisapositivenumberr,suchthat (a(cid:48),b(cid:48),c(cid:48)) = (ra,rb,rc) or (a(cid:48),b(cid:48),c(cid:48)) = (rb,ra,rc). Clearlyr isrational. Wesaythatthetriple(a,b,c)isreducedifthegreatestcommondivisorof thesenumbersis1. Thetripleiscalledorderedifa ≤ b. Itiseasytoseethatanytriple(a,b,c)isequivalenttoexactlyonereduced orderedtriple(a(cid:48),b(cid:48),c(cid:48)). Exercise1.2. Let(a,b,c)beareducedorderedtriple. Thenc isodd,and a < b. AmnonYekutieli(BGU) PythagoreanTriples 3/28 1.PythagoreanTriples Wesaythatthetriples(a,b,c)and(a(cid:48),b(cid:48),c(cid:48))areequivalentifthe correspondingtrianglesaresimilar. Thismeansthatthereisapositivenumberr,suchthat (a(cid:48),b(cid:48),c(cid:48)) = (ra,rb,rc) or (a(cid:48),b(cid:48),c(cid:48)) = (rb,ra,rc). Clearlyr isrational. Wesaythatthetriple(a,b,c)isreducedifthegreatestcommondivisorof thesenumbersis1. Thetripleiscalledorderedifa ≤ b. Itiseasytoseethatanytriple(a,b,c)isequivalenttoexactlyonereduced orderedtriple(a(cid:48),b(cid:48),c(cid:48)). Exercise1.2. Let(a,b,c)beareducedorderedtriple. Thenc isodd,and a < b. AmnonYekutieli(BGU) PythagoreanTriples 3/28 1.PythagoreanTriples Wesaythatthetriples(a,b,c)and(a(cid:48),b(cid:48),c(cid:48))areequivalentifthe correspondingtrianglesaresimilar. Thismeansthatthereisapositivenumberr,suchthat (a(cid:48),b(cid:48),c(cid:48)) = (ra,rb,rc) or (a(cid:48),b(cid:48),c(cid:48)) = (rb,ra,rc). Clearlyr isrational. Wesaythatthetriple(a,b,c)isreducedifthegreatestcommondivisorof thesenumbersis1. Thetripleiscalledorderedifa ≤ b. Itiseasytoseethatanytriple(a,b,c)isequivalenttoexactlyonereduced orderedtriple(a(cid:48),b(cid:48),c(cid:48)). Exercise1.2. Let(a,b,c)beareducedorderedtriple. Thenc isodd,and a < b. AmnonYekutieli(BGU) PythagoreanTriples 3/28 1.PythagoreanTriples Wesaythatthetriples(a,b,c)and(a(cid:48),b(cid:48),c(cid:48))areequivalentifthe correspondingtrianglesaresimilar. Thismeansthatthereisapositivenumberr,suchthat (a(cid:48),b(cid:48),c(cid:48)) = (ra,rb,rc) or (a(cid:48),b(cid:48),c(cid:48)) = (rb,ra,rc). Clearlyr isrational. Wesaythatthetriple(a,b,c)isreducedifthegreatestcommondivisorof thesenumbersis1. Thetripleiscalledorderedifa ≤ b. Itiseasytoseethatanytriple(a,b,c)isequivalenttoexactlyonereduced orderedtriple(a(cid:48),b(cid:48),c(cid:48)). Exercise1.2. Let(a,b,c)beareducedorderedtriple. Thenc isodd,and a < b. AmnonYekutieli(BGU) PythagoreanTriples 3/28 1.PythagoreanTriples Wesaythatthetriples(a,b,c)and(a(cid:48),b(cid:48),c(cid:48))areequivalentifthe correspondingtrianglesaresimilar. Thismeansthatthereisapositivenumberr,suchthat (a(cid:48),b(cid:48),c(cid:48)) = (ra,rb,rc) or (a(cid:48),b(cid:48),c(cid:48)) = (rb,ra,rc). Clearlyr isrational. Wesaythatthetriple(a,b,c)isreducedifthegreatestcommondivisorof thesenumbersis1. Thetripleiscalledorderedifa ≤ b. Itiseasytoseethatanytriple(a,b,c)isequivalenttoexactlyonereduced orderedtriple(a(cid:48),b(cid:48),c(cid:48)). Exercise1.2. Let(a,b,c)beareducedorderedtriple. Thenc isodd,and a < b. AmnonYekutieli(BGU) PythagoreanTriples 3/28