// // These magma functions may be used to verify the computation // details of Theorem 1 of the paper // // Perfect Powers Expressible as Sums of Two Curves // // by Samir Siksek. // last changed 4/7/2008 F107:=GF(107); E:=EllipticCurve([F107|0,0,0,6,-7]); //72A1 over $\F_{107} a107:=Trace(E); assert a107 eq 12; cE:={}; for a,b in [0..106] do if IsDivisibleBy(a^3+b^3,107) eq false then // note$107 \nmid c$by Lemma 2.2 Eab:=EllipticCurve([F107|0,0,0,3*a*b,b^3-a^3]); if Trace(Eab) eq 12 then epn:=(a^2-a*b+b^2)/(27*(a+b)^2); cE:=cE join {F107!epn}; end if; end if; end for; // The following proves Lemma 3.3 assert cE eq {F107!13, F107!14, F107!36, F107!37, F107!48, F107!57, F107!62}; Z106:=Integers(106); //$\Z/106\Z$// The following routine generates the Table 1 of the paper for R in [1..105] do if GCD(R,106) eq 1 then Rs:=Integers()!(1/Z106!R); //$R^*$SR:={alpha^Rs : alpha in cE}; //$S_R$SRd:={beta : beta in SR | IsSquare(beta-1)}; //$S_R^\prime$print "$", R,"$&$", Rs, "$&$ \\{", SR, "\\} $&$ \\{ " , SRd, "\\} \$ \\\\ \\hline"; end if; end for;