\\ \\ x400.gp \\ \\ find integer x,y,z,w such that A^4+B^4+2*n^2*C^4=4*D^4 and gcd(A,B,C,D)=1. \\ \\ x+y=A/D, x-y=B/D, t=C/D \\ 2*(x^4+6*^2*y^2+y^4)+2*n^2*t^4-4 \\ x^4+6*^2*y^2+y^4+n^2*t^4-2 \\ \\ {(u^2+1)}*y^2 = -(u^2+4*u+5)*x^2-(u^2+2*u-1) \\ \pm{n*(u^2+1)}*t^2 = 2*(u-2+2*u^1)*x^2+(u^2-2*u-1) \\ \\ \\ \\ A(u)=u^2+4*u+5 \\ B(u)=u^2+2*u-1 \\ C(u)=u^2-2*u-1 \\ \\ \pm{(u^2+1)}*y^2 = A(u)*x~2+B(u) \\ \pm{n*(u-2+1)}*t^2 = 2*B(u)*x^2+C(u) \\ \\ \\ \r syzygy3.gp A(u)=u^2+4*u+5; B(u)=u^2+2*u-1; C(u)=u^2-2*u-1; YY2(u,x)=(-A(u)*x^2-B(u))/(u^2+1); TT2(n,u,x)=(2*B(u)*x^2+C(u))/(n*(u^2+1)); \\ \\ utau : u ---> (u-2)/(u-1) \\ utau(u)=if(u=0,[0],-1/(n1*n2*u)); \\ TT2(u)=751689*u^4 - 1326510*u^3 + 1307436*u^2 - 456042*u + 97971; \\ v=[751689, -1326510, 1307436, -456042, 97971]; gg(n,u)= { local(d,d2,y2,t2); y2=YY2(u,x); t2=TT2(n,u,x); d=denominator(polcoeff(y2,0)); d2=denominator(polcoeff(t2,0)); y2=y2*d^2; t2=t2*d2^2; print("ratpoints '",polcoeff(y2,0)," ",polcoeff(y2,1)," ",polcoeff(y2,2),"' 10000"); print("ratpoints '",polcoeff(t2,0)," ",polcoeff(t2,1)," ",polcoeff(t2,2),"' 10000"); print("ratpoints '",polcoeff(-t2,0)," ",polcoeff(-t2,1)," ",polcoeff(-t2,2),"' 10000"); [x2,t2] } \\ h(r)=[sqrt(numerator(r)),sqrt(denominator(r))]; h(a)=local(n,d);n=numerator(a);d=denominator(a);if(issquare(n) && issquare(d),sqrtint(n)/sqrtint(d),[]); h2(a)=local(b);b=core(a);[b,sqrtint(a/b)]; YY(u,x0)=h(abs(YY2(u,x0))); TT(n,u,x0)=h(abs(TT2(n,u,x0))); NN(x,y,z,t)=(4*t^4-x^4-y^4)/(2*z^4); nn(x,y,z,t)= { local(N); N=NN(x,y,z,t); h(N) } uu(n,x,y,z,t)= { local(i,j,x2,y2,m1,m2,m3,m4,k1,k2,k3,k4,v,cc); v=vector(10); cc=0; x2=(x+y)/2; y2=(x-y)/2; m1=factor((y2/t)^2-YY2(u,x2/t)); m2=factor((z/t)^2-TT2(n,u,x2/t)); m3=factor((z/t)^2+TT2(n,u,x2/t)); print(m1); print(m2); print(m3); k1=matsize(m1)[1]; k2=matsize(m2)[1]; k3=matsize(m3)[1]; \\ print(k1); \\ print(k2); \\ print(k3); for(i=1,k1, if(poldegree(m1[i,1])==1 && m1[i,2]>0, for(j=1,k2, if(poldegree(m2[j,1])==1 && m2[j,2]>0 && m1[i,1]==m2[j,1], cc=cc+1; v[cc]=-polcoeff(m1[i,1],0)/polcoeff(m1[i,1],1) ) ); for(j=1,k3, if(poldegree(m3[j,1])==1 && m3[j,2]>0 && m1[i,1]==m3[j,1], cc=cc+1; v[cc]=-polcoeff(m1[i,1],0)/polcoeff(m1[i,1],1) ) ) ) ); v=v[1..cc]; v=vecsort(v,(x,y)->ht(x)-ht(y)) } xx(u,x0,y0)= { local(P); \\ print("[u,x0,y0]=",[u,x0,y0]); if(YY2(u,x0)==y0^2, print("(3a+) YY2(x)=",YY2(u,x)); P=(YY2(u,x)-(k*(x-x0)+y0)^2)/(x-x0), print("(3a-) YY2(x)=",-YY2(u,x)); P=(-YY2(u,x)-(k*(x-x0)+y0)^2)/(x-x0) ); -polcoeff(P,0,'x)/polcoeff(P,1,'x) } xxt(n,u,x0,t0)= { local(P); print("n1=",n); if(TT2(n,u,x0)==t0^2, P=(TT2(n,u,x)-(k*(x-x0)+t0)^2)/(x-x0); print("(3b+) TT2(x)=",TT2(n,u,x)), P=(-TT2(n,u,x)-(k*(x-x0)+t0)^2)/(x-x0); print("(3b-) TT2(x)=",-TT2(n,u,x)) ); -polcoeff(P,0,'x)/polcoeff(P,1,'x) } rp4(p)= { local(v); v=vector(5,i,polcoeff(p,5-i,'x)); v=sign(v[1])*v; print(v); print("ratpoints '",v[5]," ",v[4]," ",v[3]," ",v[2]," ",v[1],"' 10000"); print(-v); print("ratpoints '",-v[5]," ",-v[4]," ",-v[3]," ",-v[2]," ",-v[1],"' 10000"); v } sss4(n,u0,K)= { local(m,k,x,y,x2,y2,z,t,tt,u,w,v,vv,c); m=length(K); vv=vector(m); c=0; print("[u0,x0,y0]=",[u0,x0,y0]); for(i=1,m, k=K[i]; x=xk(k); y=k*(x-x0)+y0; t=TT(n,u0,x); \\ print("[x,y,t]=",[x,y,t]); x2=x+y; y2=x-y; z=lcm(lcm(denominator(x2),denominator(x2)),denominator(t)); \\ print("[x2,y2,t,z]=",[x2,y2,t,z]); v=[abs(x2),abs(y2),abs(t),1]*z; if(v[1]>v[2], tt=v[1]; v[1]=v[2]; v[2]=tt ); \\ print("v=",v); c=c+1; vv[c]=v ); vv=vv[1..c]; vv=vecsort(vv,(x,y)->x[4]-y[4]); w=[0,0,0,0]; c=0; for(i=1,#vv, v=vv[i]; if(v != w, c=c+1; print(v[1],"^4+",v[2],"^4+",2*n^2,"*",v[3],"^4=4*",v[4],"^4"); w=v ) ); c } sss4t(n,u0,K)= { local(m,k,x,y,x2,y2,z,t,tt,u,w,v,vv,c); m=#K; vv=vector(m); c=0; print("[u0,x0,t0]=",[u0,x0,t0]); for(i=1,m, k=K[i]; x=xkt(k); t=k*(x-x0)+t0; y=YY(n,u0,x); x2=x+y; y2=x-y; z=lcm(lcm(denominator(x2),denominator(y2)),denominator(t)); v=[abs(x2),abs(y2),abs(t),1]*z; if(v[1]>v[2], tt=v[1]; v[1]=v[2]; v[2]=tt ); c=c+1; vv[c]=v ); vv=vv[1..c]; vv=vecsort(vv,(x,y)->x[4]-y[4]); w=[0,0,0,0]; c=0; for(i=1,#vv, v=vv[i]; if(v != w, c=c+1; print(v[1],"^4+",v[2],"^4+",2*n^2,"*",v[3],"^4=4*",v[4],"^4"); w=v ) ); c } QQ(e,x)=[x,ellordinate(e,x)[1]]; \\ v,e0,T0,P1,P2,P3: fixed r1(m,e0,P1)=ellpow(e0,P1,m); r1t(m,e0,P1,T0)=elladd(e0,T0,r1(m,e0,P1)); r2(m,n,e0,P1,P2)=elladd(e0,ellpow(e0,P1,m),ellpow(e0,P2,n)); r2t(m,n,e0,P1,P2,T0)=elladd(e0,T0,r2(m,n,e0,P1,P2)); r3(m,n,k,e0,P1,P2,P3)=elladd(e0,ellpow(e0,P1,m),elladd(e0,ellpow(e0,P2,n),ellpow(e0,P3,k))); r3t(m,n,k,e0,P1,P2,P3,T0)=elladd(e0,T0,r3(m,n,k,e0,P1,P2,P3)); r4(m,n,k,j,e0,P1,P2,P3,P4)=elladd(e0,elladd(e0,ellpow(e0,P1,m),ellpow(e0,P2,n)),elladd(e0,ellpow(e0,P3,k),ellpow(e0,P4,j))); r4t(m,n,k,j,e0,P1,P2,P3,P4,T0)=elladd(e0,T0,r4(m,n,k,j,e0,P1,P2,P3,P4)); ppp(L,MM)= { local(s,k,w,cc); s=length(L); w=vector(s); cc=0; for(i=1,s, k=L[i]; if(lcm(abs(numerator(k)),abs(denominator(k)))<=MM, cc=cc+1; w[cc]=k ) ); w=w[1..cc] } ht(r)=max(abs(numerator(r)),abs(denominator(r))); hsort(L,MM)= { local(s,k,k2,w,cc); s=length(L); w=vector(s,i,[ht(L[i]),L[i]]); w=vecsort(w,(x,y)->x[1]-y[1]); cc=0; k2=0; for(i=1,s, k=w[i]; if(k[1]<=MM && k[2]!=k2, cc=cc+1; w[cc]=k[2]; k2=k[2] ) ); w=w[1..cc] } ss4(n,u,K)= { local(m,k,x,y,z,t,w,v,v2); m=length(K); for(i=1,m, x=xk(K[i]); y=YY(n,u,x); t=TT(n,u,x); \\ print([x,y,z,t]); z=lcm(lcm(denominator(x),denominator(y)),denominator(t)); w=[abs(x),abs(y),1,abs(t)]*z; \\ print(v); print(w[1],"^4+",n,"*",w[2],"^4+",4*n,"*",w[3],"^4=",w[4],"^4") ); m } RR(x)= { local(m,k); m=factor(core(abs(x))); k=matsize(m)[1]; prod(i=1,k,m[i,1]%8==1); } R(m,n)=RR(2*m^2+n^2)*RR(2*m^2-4*m*n+n^2); R2(m,n)=RR(2*m^2*m*n+n^2)*RR(2*m^2+n^2)*RR(2*m^3+2*m*n+n^2); R3(m,n)=RR(2*m^2-4*m*n+n^2)*RR(2*m^2-2*m*n+n^2)*RR(2*m^2+n^2)*RR(2*m^2+2*m*n+n^2); \\ \\ square check G(v,x) (mod p) \\ zz(v,n1,n2)= { local(w,P,d); w=[]; d=gcd(gcd(gcd(v[1],v[2]),gcd(v[3],v[4])),v[5]); print("gcd(v[1]..v[5])=",d,"; ",factor(d)); P=G(v,'k); forprime(p=n1,n2, if(lift(Mod(P,p))!=0, w=zzp(P,p); if(w!=[], break ) ) ); w } zz2(P,n1,n2)= { local(m,k,a); P=G(v,'k); a=polcoeff(P,4); forprime(p=n1,n2, if(kronecker(a,p)==-1 || kronecker(-a,p)==-1, m=factor(Mod(P,p)); k=matsize(m)[1]; if((k==1 && m[1,2]%2==0 && polcoeff(m[1,1],0)!=0) || (k==2 && m[1,2]%2==0 && m[2,2]%2==0 && polcoeff(m[1,1],0)!=0 && polcoeff(m[2,1],0)!=0), print(p,":",m); break ) ) ) } zzv(v,p)= { local(m,d,a,c,P,k,w); w=[]; P=Mod(G(v,'k),p); d=poldegree(P); a=lift(polcoeff(P,d)); c=lift(polcoeff(P,0)); if(c!=0 && kronecker(a,p)==-1, if(d==4 || d==2, m=factor(Mod(P,p)); k=matsize(m)[1]; if((k==1 && m[1,2]%2==0) || (k==2 && m[1,2]%2==0 && m[2,2]%2==0), print(lift(P)," (mod ",p,")"); print(lift(m)," (mod ",p,")"); print("kronecker(",a,",",p,")=-1"); w=[p,lift(P),lift(m)] ), if(d==0, print(lift(P)," (mod ",p,")"); print("kronecker(",a,",",p,")=-1"); w=[p,lift(P),lift(m)] ) ) ); w } zzp(PP,p)= { local(m,d,a,c,k,w,P); w=[]; P=Mod(PP,p); d=poldegree(P); a=lift(polcoeff(P,d)); c=lift(polcoeff(P,0)); if(c!=0 && kronecker(a,p)==-1, if(d==4 || d==2, m=factor(Mod(P,p)); k=matsize(m)[1]; if((k==1 && m[1,2]%2==0) || (k==2 && m[1,2]%2==0 && m[2,2]%2==0), print(lift(P)," (mod ",p,")"); print(lift(m)," (mod ",p,")"); print("kronecker(",a,",",p,")=-1"); w=[p,lift(P),lift(m)] ), if(d==0, print(lift(P)," (mod ",p,")"); print("kronecker(",a,",",p,")=-1"); w=[p,lift(P),lift(m)] ) ) ); w } ssu(N)= { local(v,i); v=vector(10000); i=0; forstep(m=0,4*N,4, forstep(n=-2*N+1,2*N-1,2, if(gcd(m,n)==1 && R3(m,n), i=i+1; v[i]=[max(abs(m),abs(n)),[m,n]] ) ) ); v=v[1..i]; \\ print(v); v=vecsort(v,(x,y)->x[1]-y[1]); \\ print(v); for(k=1,i, v[k][1]=2*v[k][2][1]/v[k][2][2] ); v } ssu1(N)= { local(v,i); v=vector(10000); i=0; forstep(m=0,4*N,4, forstep(n=-2*N+1,2*N-1,2, if(gcd(m,n)==1 && R(m,n), i=i+1; v[i]=[max(abs(m),abs(n)),[m,n]] ) ) ); v=v[1..i]; \\ print(v); v=vecsort(v,(x,y)->x[1]-y[1]); \\ print(v); for(k=1,i, v[k][1]=2*v[k][2][1]/v[k][2][2] ); v } ppu(U)=vector(length(U),i,U[i][1]); \\ \\ P(x) is polynomial a*x^2+b*x+c \\ \\ check \\ y^2=P(x) (mod p) have no rational solutions for some p. \\ i.e. kronecker(a*x^2+b*x+c,p)==-1 \\ check(P,pmax)= { local(v,x,ff,a,b,c); v=vector(3,i,polcoeff(P,3-i,'x)); a=v[1]; b=v[2]; c=v[3]; forprime(p=17,pmax, if(p%8==1, ff=0; for(x=0,p-1, if(kronecker(a*x^2+b*x+c,p)>0, \\ print([x,p]); ff=1 ) ); if(ff==0, print("y^2=",P,"have no rational solutions with y!=0 (mod ",p,")"); break ) ) ); ff } check2(P,p1,pmax)= { local(v,x,ff,a,b,c); v=vector(3,i,polcoeff(P,3-i,'x)); a=v[1]; b=v[2]; c=v[3]; \\ print([a,b,c]); forprime(p=p1,pmax, ff=0; \\ print("y^2=",a*'x^2+b*'x+c,"(mod ",p,")"); for(x=0,p-1, if(kronecker(a*x^2+b*x+c,p)>0, \\ print([x]); ff=1 ) ); if(ff==0, print("y^2=",P," have no rational solutions with y!=0 (mod ",p,")"); break \\ ,print("y^2=",P," have some rational solutions with y!=0 (mod ",p,")"); ) ); ff } \\ \\ u0=m/k gcd(m,k)=1 0< m,k <=N \\ A(n,u)=n-u^2 \\ B(n,u)=n+u^2 \\ 2*u*Y^2=A(n,u)*X~2+2*B(n,u)*X+2*A(n,u) \\ 2*u*T^2=B(n,u)*X^2+2*A(n,u)*X+2*B(n,u) \\ aa(n,m,k)=n*m^2-k^2; bb(n,m,k)=n*m^2+k^2; pr(n,N1,N)= { local(v,dd,dd2,c,A,B,C,A2,B2); c=0; print("P2 := ProjectiveSpace(Rationals(), 2);"); for(m=1,N, for(k=1,N, if(max(m,k)>=N1 && gcd(m,k)==1, A=aa(n,m,k); B=bb(n,m,k); C=2*m*k; dd=B^2-2*A^2; if(dd!=0, c=c+1; print1("/* u0=",m/n," */"); print1(" C1 := Conic(P2, ",-C*y^2+A*x^2+2*B*x*z+2*A*z^2,");"); print(" A,m:=HasRationalPoint(C1);"); print1("If A then C1a := Conic(P2, ",-C*y^2+B*x^2+2*A*x*z+2*B*z^2,");"); print(" HasRationalPoint(C1a);"); print1(" C1b := Conic(P2, ",-C*y^2+B*x^2+2*A*x*z+2*B*z^2,");"); print(" HasRationalPoint(C1b);"); if(c%5==0, print("/* ------------------------------------------------ */"); c=0, print() ) ) ) ) ) } \\ \\ f=TT2(n,u0,xk(x)) \\ PP4(f)= { local(m,h,k,v,d1,d2,d3,d); m=factor(f); k=matsize(m)[1]; h=1; for(i=1,k, if(m[i,2]==-2, h=f*(m[i,1])^2; v=vector(5,i,polcoeff(h,5-i,'x)); d1=lcm(denominator(v[1]),denominator(v[5])); d2=lcm(denominator(v[2]),denominator(v[4])); d3=lcm(d1,d2); d=lcm(d3,denominator(v[3])); h=h*d*core(d); break ) ); h } uxy(u1,x1,y1)=u0=u1;x0=x1;y0=y1;xx(u0,x0,y0); uxt(n,u1,x1,t1)=u0=u1;x0=x1;t0=t1;xxt(n,u0,x0,t0); msq(x)=abs(x)*core(x);