[was@laptop compute_cong_numbers]$ [was@laptop compute_cong_numbers]$ Magma V2.9-25 Mon Dec 9 2002 01:21:34 on laptop [Seed = 872036444] Type ? for help. Type -D to quit. Loading startup file "/home/was/magma/local/emacs.m" Loading "/home/was/magma/local/init.m" > E := EC("11A"); > E := EC("389A"); > E![0,0]; (0 : 0 : 1) > Type($1); PtEll > load "code.m"; Loading "code.m" > c := [n : n in [1..50] | TunnelsCriterion(n)]; > c; [ 5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47 ] > CongruentTriangle(5); CongruentTriangle( n: 5 ) TriangleFromPoint( n: 5, P: (1681/144 : 62279/1728 : 1) ) abRepresentation( X: 1519, Y: 720, Z: 1681 ) In file "code.m", line 114, column 14: >> assert GCD([X, Y, Z]) eq 1; ^ Runtime error in 'GCD': GCD not defined for field > load "code.m"; Loading "code.m" > CongruentTriangle(5); 20/3 3/2 41/6 (-4 : -6 : 1) > c := [n : n in [1..70] | TunnelsCriterion(n)]; > c; [ 5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 69, 70 ] > c := [n mod 8 : n in [1..70] | TunnelsCriterion(n)]; > c; [ 5, 6, 7, 5, 6, 7, 4, 5, 6, 7, 0, 4, 5, 6, 7, 2, 5, 6, 7, 1, 5, 6, 7, 4, 5, 6, 7, 0, 4, 5, 6, 7, 1, 5, 6 ] > Set(c); { 0, 1, 2, 4, 5, 6, 7 } > c := [n mod 8 : n in [1..150] | TunnelsCriterion(n)]; > Set(c); { 0, 1, 2, 4, 5, 6, 7 } > c; [ 5, 6, 7, 5, 6, 7, 4, 5, 6, 7, 0, 4, 5, 6, 7, 2, 5, 6, 7, 1, 5, 6, 7, 4, 5, 6, 7, 0, 4, 5, 6, 7, 1, 5, 6, 7, 5, 6, 7, 0, 4, 5, 6, 7, 0, 4, 5, 6, 7, 0, 5, 6, 7, 5, 6, 7, 0, 4, 5, 6, 7, 0, 4, 5, 6, 7, 5, 6, 7, 0, 1, 2, 5, 6, 7, 1, 4, 5, 6 ] > time c := [n : n in [1..200] | TunnelsCriterion(n)]; Time: 14.080 > Set([x :x in c | x lt 50]); { 5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47 } > Set([x mod 8 :x in c | x lt 50]); { 0, 1, 2, 4, 5, 6, 7 } > Set([x mod 8 :x in c | x lt 200]); { 0, 1, 2, 4, 5, 6, 7 } > time c := c cat [n : n in [201..300] | TunnelsCriterion(n)]; Time: 24.870 > Set([x mod 8 :x in c | x lt 300]); { 0, 1, 2, 3, 4, 5, 6, 7 } > Set([x mod 8 :x in c | x lt 250]); { 0, 1, 2, 3, 4, 5, 6, 7 } > Set([x mod 8 :x in c | x lt 20]); { 5, 6, 7 } > Set([x mod 8 :x in c | x lt 230]); { 0, 1, 2, 3, 4, 5, 6, 7 } > Set([x mod 8 :x in c | x lt 210]); { 0, 1, 2, 4, 5, 6, 7 } > Set([x mod 8 :x in c | x lt 220]); { 0, 1, 2, 3, 4, 5, 6, 7 } > c; [ 5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 69, 70, 71, 77, 78, 79, 80, 84, 85, 86, 87, 88, 92, 93, 94, 95, 96, 101, 102, 103, 109, 110, 111, 112, 116, 117, 118, 119, 120, 124, 125, 126, 127, 133, 134, 135, 136, 137, 138, 141, 142, 143, 145, 148, 149, 150, 151, 152, 154, 156, 157, 158, 159, 161, 164, 165, 166, 167, 173, 174, 175, 180, 181, 182, 183, 184, 188, 189, 190, 191, 194, 197, 198, 199, 205, 206, 207, 208, 210, 212, 213, 214, 215, 216, 219, 220, 221, 222, 223, 224, 226, 229, 230, 231, 237, 238, 239, 240, 244, 245, 246, 247, 248, 252, 253, 254, 255, 257, 260, 261, 262, 263, 265, 269, 270, 271, 276, 277, 278, 279, 280, 284, 285, 286, 287, 291, 293, 294, 295, 299 ] > 210 mod 8; 2 > 212 mod 8; 4 > 213 mod 8; 5 > 219 mod 8; 3 > factor(219); [ <3, 1>, <73, 1> ] 1 > load "code.m"; time CongruentTriangle(157); Warning: rank computed (0) is only a lower bound (It may still be correct, though) 157 is probably a congruent number, but I can't find the representation. 0 Time: 0.020 > load "code.m"; time CongruentTriangle(157); Loading "code.m" Warning: rank computed (0) is only a lower bound (It may still be correct, though) 157 is probably a congruent number, but I can't find the representation. 0 Time: 0.000 > load "code.m"; time CongruentTriangle(157); Loading "code.m" Warning: rank computed (0) is only a lower bound (It may still be correct, though) Abelian Group isomorphic to Z/2 + Z/2 Defined on 2 generators Relations: 2*G.1 = 0 2*G.2 = 0 157 is probably a congruent number, but I can't find the representation. 0 Time: 0.000 > load "code.m"; time CongruentTriangle(157); Loading "code.m" Warning: rank computed (0) is only a lower bound (It may still be correct, though) 157 is probably a congruent number, but I can't find the representation. 0 Time: 0.010 > n := 157; E := CongruentNumberCurve(n); > E; Elliptic Curve defined by y^2 = x^3 - 24649*x over Rational Field > /* a = \frac{6803298487826435051217540}{411340519227716149383203}, \qquad b = \frac{411340519227716149383203}{21666555693714761309610} $$ > > > > > > */ > a := 6803298487826435051217540/411340519227716149383203; > b := 411340519227716149383203/21666555693714761309610; > _,c := IsSquare(a^2 + b^2); > c; 224403517704336969924557513090674863160948472041/8912332268928859588025535178967163570016480830 > a*b/2; 157 > K := FieldOfFractions(PolynomialRing(Q)); > K := FieldOfFractions(PolynomialRing(Q,2)); > R := PolynomialRing(K); > S := quo; > S; Univariate Quotient Polynomial Algebra in z over Multivariate Rational function field of rank 2 over Rational Field Variables: y, n with modulus z^2 + (-y^4 - 4*n^2)/y^2 > Sfrac := FieldOfFractions(S); Univariate Quotient Polynomial Algebra in z over Multivariate Rational function field of rank 2 over Rational Field Variables: y, n with modulus z^2 + (-y^4 - 4*n^2)/y^2 > Sfrac := FieldOfFractions(S); > Sfrac := FieldOfFractions(S); > (y^4 + z^2*y^2 + 4*n*z*y + 4*n^2)/(2*z*y^2 + 4*n*y); z > (-y^4 + z^2*y^2 + 4*n*z*y + 4*n^2)/(2*z*y^2 + 4*n*y) eq 2*n/y; true > > > jayce := [1,2]; > Explode(jayce); 1 2 > alpha, beta := Explode(jayce); > alpha; 1 > beta; 2 > > > > quit; Total time: 51.540 seconds, Total memory usage: 1.94MB [was@laptop compute_cong_numbers]$ me Magma V2.9-25 Mon Dec 9 2002 11:30:46 on laptop [Seed = 131006300] Type ? for help. Type -D to quit. Loading startup file "/home/was/magma/local/emacs.m" Loading "/home/was/magma/local/init.m" > R := FieldOfFractions(PolynomialRing(Rationals(),2)); S := PolynomialRing(R); T := quo; K := FieldOfFractions(T); U := FieldOfFractions(PolynomialRing(Rationals(),2)); > > > > > > > > U; Multivariate Rational function field of rank 2 over Rational Field Variables: r, s > function f(w) x,y,z := Explode(w); return [-n*y/(x+z), 2*n^2/(x+z)]; end function; function g(w) r,s := Explode(w); return [(n^2-r^2)/s, -2*r*n/s, (n^2+r^2)/s]; end function; function> function> function> > function> function> function> > > > g(f([2*n/y,y,z])); [ 2*n/y, y, z ] > f([2*n/y,y,z]); [ -n/y*z + 2*n^2/y^2, 2*n^2/y^2*z - 4*n^3/y^3 ] > g($1); [ 2*n/y, y, z ] > E := EllipticCurve([-25,0]); > E; Elliptic Curve defined by y^2 = x^3 - 25*x over Rational Field > G, f := MordellWeilGroup(E); > G; Abelian Group isomorphic to Z/2 + Z/2 + Z Defined on 3 generators Relations: 2*G.1 = 0 2*G.2 = 0 > f; Mapping from: GrpAb: G to Set of points of E with coordinates in Rational Field > f(G.1); (-5 : 0 : 1) > f(G.2); (5 : 0 : 1) > f(G.3); (-4 : -6 : 1) > g([-4,-6]); [ -1/6*n^2 + 8/3, -4/3*n, -1/6*n^2 - 8/3 ] > -25/6 + 8/3; -3/2 > -4/3*5; -20/3 > E;. Elliptic Curve defined by y^2 = x^3 - 25*x over Rational Field >> E;. ^ User error: bad syntax > E; Elliptic Curve defined by y^2 = x^3 - 25*x over Rational Field > P := f(G.3); > P; (-4 : -6 : 1) > 2*P; (1681/144 : 62279/1728 : 1) > g([$1[1],$1[2]]); [ 1728/62279*n^2 - 68921/18228, -984/1519*n, 1728/62279*n^2 + 68921/18228 ] > n:=5; 1728/62279*n^2 - 68921/18228; n:=5; 1728/62279*n^2 - 68921/18228; -1519/492 > load "code.m"; Loading "code.m" > TunnelsCriterion(1); false > time TunnelsCriterion(389); true Time: 0.610 > time TunnelsCriterion(32); false Time: 0.000 > time TunnelsCriterion(31); true Time: 0.020 > time TunnelsCriterion(8*5+3); false Time: 0.030 > time TunnelsCriterion(8*11+3); false Time: 0.080 > time TunnelsCriterion(8*13+3); false Time: 0.100 > 219 mod 8; 3 > time TunnelsCriterion(219); true Time: 0.240 > CongruentTriangle(219); Height bound (17.7189) on point search is too large -- reducing to 15.0000 This means that the computed group may only generate a group of finite index in the actual group. 264/13 949/44 16945/572 (507 : 10296 : 1) > quit; Total time: 136.570 seconds, Total memory usage: 1.70MB [was@laptop compute_cong_numbers]$ exit exit Process magma finished