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Pell type reduction...

vv 13932
September 30 2024
4 0

The answer for your previous problem works here too.

restart;
eq:=x^2+y^2-N*(1+x*y):  # [x=0, y=sqrt(N)] and [x=sqrt(N),y=0] are solutions!
XY:=[X=x, Y=2*y-x*N]: xy:=solve(XY,[x,y])[]:
EQ:=simplify(eval(eq,xy)):
for N in [9, 49 , 729] do
  SOL:=isolve(EQ):
  sol[N]:=map(u -> simplify(eval(xy, u)), [SOL]);
  num:={seq}(simplify(sol[N])[],_Z1=0..2);
  print('N'=N, select(type,num, [anything=integer,anything=integer]));
od:

N = 9, {[x = -1497363, y = -13307787], 
  [x = -1497363, y = -168480], [x = -168480, y = -1497363], 
  [x = -168480, y = -18957], [x = -18957, y = -168480], 
  [x = -18957, y = -2133], [x = -2133, y = -18957], 
  [x = -2133, y = -240], [x = -240, y = -2133], 
  [x = -240, y = -27], [x = -27, y = -240], [x = -27, y = -3], 
  [x = -3, y = -27], [x = -3, y = 0], [x = 0, y = -3], 
  [x = 0, y = 3], [x = 3, y = 0], [x = 3, y = 27], 
  [x = 27, y = 3], [x = 27, y = 240], [x = 240, y = 27], 
  [x = 240, y = 2133], [x = 2133, y = 240], 
  [x = 2133, y = 18957], [x = 18957, y = 2133], 
  [x = 18957, y = 168480], [x = 168480, y = 18957], 
  [x = 168480, y = 1497363], [x = 1497363, y = 168480], 
  [x = 1497363, y = 13307787]}

N = 49, {[x = -96687343207, y = -4735705783543], 
  [x = -96687343207, y = -1974033600], 
  [x = -1974033600, y = -96687343207], 
  [x = -1974033600, y = -40303193], 
  [x = -40303193, y = -1974033600], [x = -40303193, y = -822857], 
  [x = -822857, y = -40303193], [x = -822857, y = -16800], 
  [x = -16800, y = -822857], [x = -16800, y = -343], 
  [x = -343, y = -16800], [x = -343, y = -7], [x = -7, y = -343], 
  [x = -7, y = 0], [x = 0, y = -7], [x = 0, y = 7], 
  [x = 7, y = 0], [x = 7, y = 343], [x = 343, y = 7], 
  [x = 343, y = 16800], [x = 16800, y = 343], 
  [x = 16800, y = 822857], [x = 822857, y = 16800], 
  [x = 822857, y = 40303193], [x = 40303193, y = 822857], 
  [x = 40303193, y = 1974033600], [x = 1974033600, y = 40303193], 
  [x = 1974033600, y = 96687343207], 
  [x = 96687343207, y = 1974033600], 
  [x = 96687343207, y = 4735705783543]}

N = 729, {[x = -4052517025117644747, y = -2954279352292037818803], 
  [x = -4052517025117644747, y = -5559018725201760], 
  [x = -5559018725201760, y = -4052517025117644747], 
  [x = -5559018725201760, y = -7625554438293], 
  [x = -7625554438293, y = -5559018725201760], 
  [x = -7625554438293, y = -10460313837], 
  [x = -10460313837, y = -7625554438293], 
  [x = -10460313837, y = -14348880], 
  [x = -14348880, y = -10460313837], [x = -14348880, y = -19683], 
  [x = -19683, y = -14348880], [x = -19683, y = -27], 
  [x = -27, y = -19683], [x = -27, y = 0], [x = 0, y = -27], 
  [x = 0, y = 27], [x = 27, y = 0], [x = 27, y = 19683], 
  [x = 19683, y = 27], [x = 19683, y = 14348880], 
  [x = 14348880, y = 19683], [x = 14348880, y = 10460313837], 
  [x = 10460313837, y = 14348880], 
  [x = 10460313837, y = 7625554438293], 
  [x = 7625554438293, y = 10460313837], 
  [x = 7625554438293, y = 5559018725201760], 
  [x = 5559018725201760, y = 7625554438293], 
  [x = 5559018725201760, y = 4052517025117644747], 
  [x = 4052517025117644747, y = 5559018725201760], 
  [x = 4052517025117644747, y = 2954279352292037818803]}

With sol[N] you may inspect the general solution depending on _Z1.


 

Solution...

vv 13932
September 30 2024
5 1

 

isolve is old and should be updated.

Actually isolve knows to solve generalized Pell equations. It only needs a little help to convert the diophantine quadratic to generalized Pell form.

At the end we must filter the obtained numeric solutions in order to eliminate the non-integer ones due to the form of xy, see below.

 

> 

restart;
> 

eq:=x^2 - 12*x*y + 6*y^2 + 4*x + 12*y - 3:
> 

XY:=[X = 2*x-12*y+4, Y = -5*y+3]; # via complete squares

[X = 2*x-12*y+4, Y = -5*y+3]

 (1)
> 

xy:=solve(XY,[x,y])[];

[x = (1/2)*X+8/5-(6/5)*Y, y = -(1/5)*Y+3/5]

 (2)
> 

EQ:=simplify(eval(eq,xy));

(1/4)*X^2+19/5-(6/5)*Y^2

 (3)
> 

SOL:=isolve( EQ ):
> 

sol:=map(u -> simplify(eval(xy, u)), [SOL]);

[[x = (1/10)*(30^(1/2)+7)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(-30^(1/2)+7)*(11+2*30^(1/2))^_Z1, y = (1/60)*(-30^(1/2)+12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(30^(1/2)+12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(3*30^(1/2)-17)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(-3*30^(1/2)-17)*(11+2*30^(1/2))^_Z1, y = (1/60)*(30^(1/2)-12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(-30^(1/2)-12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(-3*30^(1/2)-17)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(3*30^(1/2)-17)*(11+2*30^(1/2))^_Z1, y = (1/60)*(-30^(1/2)-12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(30^(1/2)-12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(-30^(1/2)+7)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(30^(1/2)+7)*(11+2*30^(1/2))^_Z1, y = (1/60)*(30^(1/2)+12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(-30^(1/2)+12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(30^(1/2)-7)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(-30^(1/2)-7)*(11+2*30^(1/2))^_Z1, y = (1/60)*(-30^(1/2)-12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(30^(1/2)-12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(3*30^(1/2)+17)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(-3*30^(1/2)+17)*(11+2*30^(1/2))^_Z1, y = (1/60)*(30^(1/2)+12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(-30^(1/2)+12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(-3*30^(1/2)+17)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(3*30^(1/2)+17)*(11+2*30^(1/2))^_Z1, y = (1/60)*(-30^(1/2)+12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(30^(1/2)+12)*(11+2*30^(1/2))^_Z1], [x = (1/10)*(-30^(1/2)-7)*(11-2*30^(1/2))^_Z1+8/5+(1/10)*(30^(1/2)-7)*(11+2*30^(1/2))^_Z1, y = (1/60)*(30^(1/2)-12)*(11-2*30^(1/2))^_Z1+3/5+(1/60)*(-30^(1/2)-12)*(11+2*30^(1/2))^_Z1]]

 (4)
> 

nops(sol)

8

 (5)
> 

num:={seq}(simplify(sol)[],_Z1=0..20): # some numeric solutions
> 

numsols:=select(type,num,  [anything=integer,anything=integer]);

{[x = 3, y = 1], [x = 3, y = 3], [x = 5, y = 1], [x = 5, y = 7], [x = 29, y = 3], [x = 29, y = 53], [x = 75, y = 7], [x = 75, y = 141], [x = 603, y = 53], [x = 603, y = 1151], [x = 1613, y = 141], [x = 1613, y = 3083], [x = 13205, y = 1151], [x = 13205, y = 25257], [x = 35379, y = 3083], [x = 35379, y = 67673], [x = 289875, y = 25257], [x = 289875, y = 554491], [x = 776693, y = 67673], [x = 776693, y = 1485711], [x = 6364013, y = 554491], [x = 6364013, y = 12173533], [x = 17051835, y = 1485711], [x = 17051835, y = 32617957], [x = 139718379, y = 12173533], [x = 139718379, y = 267263223], [x = 374363645, y = 32617957], [x = 374363645, y = 716109331], [x = 3067440293, y = 267263223], [x = 3067440293, y = 5867617361], [x = 8218948323, y = 716109331], [x = 8218948323, y = 15721787313], [x = 67343968035, y = 5867617361], [x = 67343968035, y = 128820318707], [x = 180442499429, y = 15721787313], [x = 180442499429, y = 345163211543], [x = 1478499856445, y = 128820318707], [x = 1478499856445, y = 2828179394181], [x = 3961516039083, y = 345163211543], [x = 3961516039083, y = 7577868866621], [x = 32459652873723, y = 2828179394181], [x = 32459652873723, y = 62091126353263], [x = 86972910360365, y = 7577868866621], [x = 86972910360365, y = 166367951854107], [x = 712633863365429, y = 62091126353263], [x = 712633863365429, y = 1363176600377593], [x = 1909442511888915, y = 166367951854107], [x = 1909442511888915, y = 3652517071923721], [x = 15645485341165683, y = 1363176600377593], [x = 15645485341165683, y = 29927794081953771], [x = 41920762351195733, y = 3652517071923721], [x = 41920762351195733, y = 80189007630467743], [x = 343488043642279565, y = 29927794081953771], [x = 343488043642279565, y = 657048293202605357], [x = 920347329214417179, y = 80189007630467743], [x = 920347329214417179, y = 1760505650798366613], [x = 7541091474788984715, y = 657048293202605357], [x = 7541091474788984715, y = 14425134656375364071], [x = 20205720480365982173, y = 1760505650798366613], [x = 20205720480365982173, y = 38650935309933597731], [x = 165560524401715384133, y = 14425134656375364071], [x = 165560524401715384133, y = 316695914147055404193], [x = 443605503238837190595, y = 38650935309933597731], [x = 443605503238837190595, y = 848560071167740783457], [x = 3634790445362949466179, y = 316695914147055404193], [x = 3634790445362949466179, y = 6952884976578843528163], [x = 9739115350774052210885, y = 848560071167740783457], [x = 9739115350774052210885, y = 18629670630380363638311], [x = 79799829273583172871773, y = 6952884976578843528163], [x = 79799829273583172871773, y = 152646773570587502215381], [x = 213816932213790311448843, y = 18629670630380363638311], [x = 213816932213790311448843, y = 409004193797200259259373], [x = 1751961453573466853712795, y = 152646773570587502215381], [x = 1751961453573466853712795, y = 3351276133576346205210207], [x = 4694233393352612799663629, y = 409004193797200259259373], [x = 4694233393352612799663629, y = 8979462592908025340067883], [x = 38463352149342687608809685, y = 3351276133576346205210207], [x = 38463352149342687608809685, y = 73575428165109029012409161], [x = 103059317721543691281150963, y = 8979462592908025340067883], [x = 103059317721543691281150963, y = 197139172850179357222234041], [x = 844441785831965660540100243, y = 73575428165109029012409161], [x = 2262610756480608595385657525, y = 197139172850179357222234041]}

 (6)
> 

nops(numsols);

82

 (7)
> 

{seq}( eval(eq, s), s=numsols); #check

{0}

 (8)
> 

 

 

Download dioph-sols-vv.mw

andmap...

vv 13932
September 26 2024
3 0

 

select(u -> nops(u)=m and andmap(isprime,u), L);

 

DirectSearch...

vv 13932
September 24 2024
3 1

 

> 

restart;
> 

dis:= (A,B) -> sqrt( (A[1]-B[1])^2 + (A[2]-B[2])^2 ):
P:= unapply(dis(A,B)+dis(B,C)+dis(C,A), [A,B,C]):
> 

A0:=[0,0]:
A1:=[u1,0]:
A2:=[u2,u3]:
A3:=v1*~A1:
A4:=A1+v2*~(A2-A1):
A5:=A1+v3*~(A2-A1):
A6:=A0+v4*~(A2-A0):
A7:=A0+v5*~(A2-A0):
A8:=A6+v6*~(A4-A6):
> 

eps:=0.05:
> 

perim:=
P(A0,A3,A6)=10, P(A6,A3,A4)=15, P(A3,A1,A4)=11,
P(A6,A8,A7)=9,  P(A8,A5,A7)=13, P(A8,A4,A5)=12,
P(A7,A5,A2)=20:
> 

vars := u1=1..50, u2=1..50, u3=1..50,
v1=eps ..1-eps, v2=eps ..1-eps, v3=eps ..1-eps, v4=eps ..1-eps, v5=eps ..1-eps, v6=eps ..1-eps:
> 

inisol:=[u1 = 8, u2 = 3, u3 = 12, v1 = 0.5, v2 = 0.3, v3 = 0.5, v4 = 0.09, v5 = 0.3, v6 = 0.5]:
> 

sol:=DirectSearch:-SolveEquations([perim], [vars], initialpoint=inisol );

[7.928429359693041*10^(-17), Vector(7, {(1) = 0.1798504456e-8, (2) = -0.1393335225e-8, (3) = -0.7304805649e-9, (4) = 0.7992083439e-8, (5) = -0.1359651947e-8, (6) = 0.1542570516e-8, (7) = 0.2339469063e-8}), [u1 = 8.211758237046912, u2 = 1.8442345202064532, u3 = 12.033442054587029, v1 = .5526589984562774, v2 = .26313930832995064, v3 = .5414796127553622, v4 = 0.7943775626427665e-1, v5 = .2941739734925204, v6 = .485843681048224], 5405]

 (1)
> 

plots:-display(plot(eval([A0,A1,A2,A0],sol[3])), plot(eval([A3,A4,A6,A3],sol[3])), plot(eval([A5,A7,A8,A5],sol[3])), scaling=constrained);

 
> 

eval([perim], sol[3]);

[HFloat(10.000000001798504) = 10, HFloat(14.999999998606665) = 15, HFloat(10.99999999926952) = 11, HFloat(9.000000007992083) = 9, HFloat(12.999999998640348) = 13, HFloat(12.00000000154257) = 12, HFloat(20.00000000233947) = 20]

 (2)
> 

eval(seq("A"||i=A||i,i=0..8), sol[3])

"A0" = [0, 0], "A1" = [HFloat(8.211758237046912), 0], "A2" = [HFloat(1.8442345202064532), HFloat(12.033442054587029)], "A3" = [HFloat(4.538302082851432), 0], "A4" = [HFloat(6.536212450422957), HFloat(3.166471619072571)], "A5" = [HFloat(4.763873960641556), HFloat(6.515863543831874)], "A6" = [HFloat(0.14650185231032542), HFloat(0.9559096369525809)], "A7" = [HFloat(0.5425257968612042), HFloat(3.539925463989865)], "A8" = [HFloat(3.2509023701302153), HFloat(2.029897207531015)]

 (3)
> 

 


(Edited - a typo).

Download perims-vv.mw

infinite...

vv 13932
September 20 2024
3 1

The curve length is infinite. The integrand has a singularity at t=ln(Pi/2):

restart;
f:= t -> sqrt((sin(t/cos(exp(t))) + t*(1/cos(exp(t)) + t*exp(t)*sin(exp(t))/cos(exp(t))^2)*cos(t/cos(exp(t))))^2 + (cos(t) - t*sin(t))^2 + 2*t):
a:=ln(Pi/2): evalf(a);
#                          0.4515827054

Digits:=15:
:-int(f, a..a+1/10, numeric, epsilon=1e-5);
#                     3.33001574946940* 10^13

D...

vv 13932
September 19 2024
0 3 
Optimization:-Minimize(AD, {CD=AB-2, BC=AB+2, AD=AB+BC+CD, AB>=1, BC>=1, CD>=1}, assume=integer);
#        [9, [AB = 3, AD = 9, BC = 5, CD = 1]]

 

D...

vv 13932
September 18 2024
2 0 
eval(E-A, solve({C-A + D-B + E-C = a, D-B = b}, {A,B,C,D,E}));
#                             -b + a

 

A more precise version and a simple proo...

vv 13932
September 17 2024
4 3

 

A more precise version and a simple proof

 

The cited result is true only when the curbe is not "too flat" and in those cases the result of the limit is different.

 

To simplify things we shall consider the Ox axes as one of the tangents to the curve.

So, we shall take  f(0) = 0, f'(0) = 0.

 

 

The tangent at the point P(a, f(a)) intersects the Ox axis for the abscissa  c = a - f(a)/f'(a);

 

So, the ratio r of the two areas is

 

> 

restart;
> 

r := (a*f(a)/2 - int(f(t),t=0..a))/ (1/2 * (a-f(a)/D(f)(a))* f(a));

2*((1/2)*a*f(a)-(int(f(t), t = 0 .. a)))/((a-f(a)/(D(f))(a))*f(a))

 (1)
> 

# f(0) = 0, f'(0)=0

0

 (2)
> 

eval( r,  f = (u -> u^2*g(u)) ):
series(%, a);

series(2/3+(2*(-(1/30)*((D@@2)(g))(0)+(1/12)*(D(g))(0)^2/g(0))/g(0))*a^2+O(a^3),a,3)

 (3)

Hence  limit(r, a=0)  = 2/3   indeed.

 

What happens if f is more "flat".  E.g. f(0) = 0, f'(0)=0, f''(0)=0

 

> 

eval( r,  f = (u -> u^3*g(u)) ):
series(%, a);

series(3/4+((1/40)*(D(g))(0)/g(0))*a+O(a^2),a,2)

 (4)

And so on; if the first nonzero derivative (at 0) is the 10th we obtain:

 

> 

eval( r,  f = (u -> u^10*g(u)) ):
series(%, a, 14);

series(10/11+((2/297)*(D(g))(0)/g(0))*a+(2*((7/2574)*((D@@2)(g))(0)-(31/10692)*(D(g))(0)^2/g(0))/g(0))*a^2+O(a^3),a,3)

 (5)

So, the limit will be  k/(k+1)  it  the first nonzero derivative of f at 0 is the k-th.


 

Download voller-vv.mw

NLPSolve...

vv 13932
September 07 2024
2 9

Optimization:-NLPSolve works very well and fast without the need of any simplification.

restart;
Digits:=15:
e:=(x,y) -> x^2 + 2*y^2 - 1:
h:=(x,y) -> (x-sin(x))^2 + (y-sin(y))^2 - 1:
constr:=
e(x1,y1)=0, h(x2,y2)=0, h(x3,y3)=0, r>=0,
D[1](e)(x1,y1)*(y1-y0) - D[2](e)(x1,y1)*(x1-x0) = 0,
D[1](h)(x2,y2)*(y2-y0) - D[2](h)(x2,y2)*(x2-x0) = 0,
D[1](h)(x3,y3)*(y3-y0) - D[2](h)(x3,y3)*(x3-x0) = 0,
(x1-x0)^2 + (y1-y0)^2 = r^2,
(x2-x0)^2 + (y2-y0)^2 = r^2,
(x3-x0)^2 + (y3-y0)^2 = r^2,
x3 >= x2 + 1/10:

sol:=Optimization:-Maximize(r, {constr},   x0=1..1.4, y0=1..1.4, x1=0..1, y1=0..1, x2=1..2, y2=1..2, x3=1..2, y3=1..2);

sol := [0.769863723979874459, [r = 0.769863723979874, x0 = 1.13805882552303, x1 = 0.699729170630851, x2 = 1.27446720511737, x3 = 1.89574141571275, y0 = 1.13805882552303, y1 = 0.505162888466810, y2 = 1.89574141571275, y3 = 1.27446720511737]]

p1:=plots:-implicitplot(e, -2..2, -2..2): p2:=plots:-implicitplot(h, -2..2, -2..2):
p3:=plottools:-circle(eval([x0,y0],sol[2]), eval(r,sol[2]), color=red):
p4:=plots:-pointplot(eval([[x0,y0],[x1,y1],[x2,y2],[x3,y3]],sol[2]), symbolsize=8, color=blue):
p5:=plots:-textplot(eval({seq}([x||i,y||i,P__||i], i=0..3), sol[2]), align={below, left}, color=blue):
plots:-display(p1,p2,p3,p4,p5);

row...

vv 13932
September 04 2024
2 0

 

data_new := DataFrame(data_2, rows=[1,2]);

 

Bezout...

vv 13932
August 28 2024
1 0

For a system of m polynomial equations of (total) degrees d_1, ..., d_m in n unknowns over C (or other algebraically closed field), the standard result is given by Bezout' theorem:

For m=n, the number of solutions is either infinite or <= d_1* d_2 * ...* d_m.

For details see: Bézout's theorem - Wikipedia

 

COLOR...

vv 13932
August 28 2024
2 0 
restart;
# f := floor;
f := x -> x^2:
X := [seq(x, x=-3..3,0.02)]: n:=nops(X):
Y := map(f, X):
C := Array(1..n, i -> X[i], datatype=float[8]):
plot(X,Y, color=COLOR(HUE, C), thickness=6);

dsolve...

vv 13932
August 23 2024
1 4

The best method is probably using dsolve + fsolve, because fsolve + int(..., numeric) is too slow.

restart:
Digits:=15:
#eqList := ...
FS:=proc(a,f,t) 
    local u, ds, F;
    ds:=dsolve({diff(u(t),t) = f, u(-4)=0}, numeric, output=operator);
    F:=eval(u,ds);
    fsolve(F-a)
end proc: 
G:=e -> FS(op(1,e), op([2,1],e), op([2,2,1], e)):
map(G, eqList);

 [-4.00000000000000, -3.13995448471848, -2.11370056919619, 
  -1.52883183230736, -1.04840878317075, -0.706638188721254,
   -0.479577206139816, -0.315169354360080, -0.182432574734532, 
   -0.0625487819911634, 0.0891458203730475, 0.368690494498621, 
   1.50469145760957, 2.11552551065259, 2.46119669196488, 
   2.72088546517742, 2.96213514705916, 3.21307491249018, 
   3.59812548343646, 5.16349647450855, 6.56002211630892, 
   6.81413891022544, 6.66825093298453, 6.40577992538593, 
   6.19352683224088, 6.02201310955201, 5.95433562651451, 
   6.06761314123142, 6.32677648880848]

nops(eqList)=nops(%);
                            29 = 29

select...

vv 13932
August 19 2024
1 0 
restart
L := [3*a0 + 2*a1, -a1/3 + a0/2 + 12*a3]:
x:=a3:

solve(select(has, L, x), x);

 

Answer...

vv 13932
August 18 2024
3 1

 

ex := x^4 + 3*(1+x^2)*f(x) + (x*x^2+x+1)*((D)(f)(x))^2 + (3*x+3)*(D@D)(f)(x):
eval( ex, f = (x -> (a1+a2*x+c*x^2+O(x^3)) ) ): 
series(%, x);
 #                   a2^2  + 3 a1 + 6 c + O(x)

Next time please post text, not pictures.

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