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arith1.c

/* $Id: arith1.c,v 1.39.2.6 2002/09/05 16:49:25 cvsadmin Exp $

Copyright (C) 2000  The PARI group.

This file is part of the PARI/GP package.

PARI/GP is free software; you can redistribute it and/or modify it under the
terms of the GNU General Public License as published by the Free Software
Foundation. It is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY WHATSOEVER.

Check the License for details. You should have received a copy of it, along
with the package; see the file 'COPYING'. If not, write to the Free Software
Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. */

/*********************************************************************/
/**                                                                 **/
/**                     ARITHMETIC FUNCTIONS                        **/
/**                         (first part)                            **/
/**                                                                 **/
/*********************************************************************/
#include "pari.h"

/*********************************************************************/
/**                                                                 **/
/**                  ARITHMETIC FUNCTION PROTOTYPES                 **/
/**                                                                 **/
/*********************************************************************/
GEN
garith_proto(GEN f(GEN), GEN x, int do_error)
{
  long tx=typ(x),lx,i;
  GEN y;

  if (is_matvec_t(tx))
  {
    lx=lg(x); y=cgetg(lx,tx);
    for (i=1; i<lx; i++) y[i] = (long) garith_proto(f,(GEN) x[i], do_error);
    return y;
  }
  if (tx != t_INT && do_error) err(arither1);
  return f(x);
}

GEN
arith_proto(long f(GEN), GEN x, int do_error)
{
  long tx=typ(x),lx,i;
  GEN y;

  if (is_matvec_t(tx))
  {
    lx=lg(x); y=cgetg(lx,tx);
    for (i=1; i<lx; i++) y[i] = (long) arith_proto(f,(GEN) x[i], do_error);
    return y;
  }
  if (tx != t_INT && do_error) err(arither1);
  return stoi(f(x));
}

GEN
arith_proto2(long f(GEN,GEN), GEN x, GEN n)
{
  long l,i,tx = typ(x);
  GEN y;

  if (is_matvec_t(tx))
  {
    l=lg(x); y=cgetg(l,tx);
    for (i=1; i<l; i++) y[i] = (long) arith_proto2(f,(GEN) x[i],n);
    return y;
  }
  if (tx != t_INT) err(arither1);
  tx=typ(n);
  if (is_matvec_t(tx))
  {
    l=lg(n); y=cgetg(l,tx);
    for (i=1; i<l; i++) y[i] = (long) arith_proto2(f,x,(GEN) n[i]);
    return y;
  }
  if (tx != t_INT) err(arither1);
  return stoi(f(x,n));
}

static GEN
arith_proto2gs(long f(GEN,long), GEN x, long y)
{
  long l,i,tx = typ(x);
  GEN t;

  if (is_matvec_t(tx))
  {
    l=lg(x); t=cgetg(l,tx);
    for (i=1; i<l; i++) t[i]= (long) arith_proto2gs(f,(GEN) x[i],y);
    return t;
  }
  if (tx != t_INT) err(arither1);
  return stoi(f(x,y));
}

GEN
garith_proto2gs(GEN f(GEN,long), GEN x, long y)
{
  long l,i,tx = typ(x);
  GEN t;

  if (is_matvec_t(tx))
  {
    l=lg(x); t=cgetg(l,tx);
    for (i=1; i<l; i++) t[i]= (long) garith_proto2gs(f,(GEN) x[i],y);
    return t;
  }
  if (tx != t_INT) err(arither1);
  return f(x,y);
}

/*********************************************************************/
/**                                                                 **/
/**               ORDER of INTEGERMOD x  in  (Z/nZ)*                **/
/**                                                                 **/
/*********************************************************************/

GEN
order(GEN x)
{
  long av=avma,av1,i,e;
  GEN o,m,p;

  if (typ(x) != t_INTMOD || !gcmp1(mppgcd((GEN) x[1],(GEN) x[2])))
    err(talker,"not an element of (Z/nZ)* in order");
  o=phi((GEN) x[1]); m=decomp(o);
  for (i = lg(m[1])-1; i; i--)
  {
    p=gcoeff(m,i,1); e=itos(gcoeff(m,i,2));
    do
    {
      GEN o1=divii(o,p), y=powgi(x,o1);
      if (!gcmp1((GEN)y[2])) break;
      e--; o = o1;
    }
    while (e);
  }
  av1=avma; return gerepile(av,av1,icopy(o));
}

/******************************************************************/
/*                                                                */
/*                 GENERATOR of (Z/mZ)*                           */
/*                                                                */
/******************************************************************/

GEN
ggener(GEN m)
{
  return garith_proto(gener,m,1);
}

GEN
gener(GEN m)
{
  long av=avma,av1,k,i,e;
  GEN x,t,q,p;

  if (typ(m) != t_INT) err(arither1);
  e = signe(m);
  if (!e) err(talker,"zero modulus in znprimroot");
  if (is_pm1(m)) { avma=av; return gmodulss(0,1); }
  if (e<0) m = absi(m);

  e = mod4(m);
  if (e == 0) /* m = 0 mod 4 */
  {
    if (cmpis(m,4)) err(generer); /* m != 4, non cyclic */
    return gmodulsg(3,m);
  }
  if (e == 2) /* m = 0 mod 2 */
  {
    q=shifti(m,-1); x = (GEN) gener(q)[2];
    if (!mod2(x)) x = addii(x,q);
    av1=avma; return gerepile(av,av1,gmodulcp(x,m));
  }

  t=decomp(m); if (lg(t[1]) != 2) err(generer);
  p=gcoeff(t,1,1); e=itos(gcoeff(t,1,2)); q=subis(p,1);
  if (e>=2)
  {
    x = (GEN) gener(p)[2];
    if (gcmp1(powmodulo(x,q, sqri(p)))) x = addii(x,p);
    av1=avma; return gerepile(av,av1,gmodulcp(x,m));
  }

  p=(GEN)decomp(q)[1]; k=lg(p)-1; x=stoi(1);
  for(;;)
  {
    x[2]++;
    if (gcmp1(mppgcd(m,x)))
    {
      for (i=k; i; i--)
      if (gcmp1(powmodulo(x, divii(q,(GEN)p[i]), m))) break;
      if (!i) break;
    }
  }
  av1=avma; return gerepile(av,av1,gmodulcp(x,m));
}

GEN
znstar(GEN n)
{
  GEN p1,z,q,u,v,d,list,ep,h,gen,moduli,p,a;
  long i,j,nbp,sizeh,av,tetpil;

  if (typ(n) != t_INT) err(arither1);
  if (!signe(n))
  {
    z=cgetg(4,t_VEC);
    z[1]=deux; p1=cgetg(2,t_VEC);
    z[2]=(long)p1; p1[1]=deux; p1=cgetg(2,t_VEC);
    z[3]=(long)p1; p1[1]=lneg(gun);
    return z;
  }
  av=avma; n=absi(n);
  if (cmpis(n,2)<=0)
  {
    avma=av; z=cgetg(4,t_VEC);
    z[1]=un;
    z[2]=lgetg(1,t_VEC);
    z[3]=lgetg(1,t_VEC);
    return z;
  }
  list=factor(n); ep=(GEN)list[2]; list=(GEN)list[1]; nbp=lg(list)-1;
  h      = cgetg(nbp+2,t_VEC);
  gen    = cgetg(nbp+2,t_VEC);
  moduli = cgetg(nbp+2,t_VEC);
  switch(mod8(n))
  {
    case 0:
      h[1] = lmul2n(gun,itos((GEN)ep[1])-2); h[2] = deux;
      gen[1] = lstoi(5); gen[2] = laddis(gmul2n((GEN)h[1],1),-1);
      moduli[1] = moduli[2] = lmul2n(gun,itos((GEN)ep[1]));
      sizeh = nbp+1; i=3; j=2; break;
    case 4:
      h[1] = deux;
      gen[1] = lstoi(3);
      moduli[1] = lmul2n(gun,itos((GEN)ep[1]));
      sizeh = nbp; i=j=2; break;
    case 2: case 6:
      sizeh = nbp-1; i=1; j=2; break;
    default: /* 1, 3, 5, 7 */
      sizeh = nbp; i=j=1;
  }
  for ( ; j<=nbp; i++,j++)
  {
    p = (GEN)list[j]; q = gpuigs(p, itos((GEN)ep[j])-1);
    h[i] = lmulii(addis(p,-1),q); p1 = mulii(p,q);
    gen[i] = gener(p1)[2];
    moduli[i] = (long)p1;
  }
#if 0
  if (nbp == 1 && is_pm1(ep[1]))
    gen[1] = lmodulcp((GEN)gen[1],n);
  else
#endif
    for (i=1; i<=sizeh; i++)
    {
      q = (GEN)moduli[i]; a = (GEN)gen[i];
      u = mpinvmod(q,divii(n,q));
      gen[i] = lmodulcp(addii(a,mulii(mulii(subsi(1,a),u),q)), n);
    }
  
  for (i=sizeh; i>=2; i--)
    for (j=i-1; j>=1; j--)
      if (!divise((GEN)h[j],(GEN)h[i]))
      {
      d=bezout((GEN)h[i],(GEN)h[j],&u,&v);
        q=divii((GEN)h[j],d);
      h[j]=(long)mulii((GEN)h[i],q); h[i]=(long)d;
      gen[j]=ldiv((GEN)gen[j], (GEN)gen[i]);
      gen[i]=lmul((GEN)gen[i], powgi((GEN)gen[j], mulii(v,q)));
      }
  q=gun; for (i=1; i<=sizeh && !gcmp1((GEN)h[i]); i++) q=mulii(q,(GEN)h[i]);
  setlg(h,i); setlg(gen,i); z=cgetg(4,t_VEC);
  z[1]=(long)q; z[2]=(long)h; z[3]=(long)gen;
  tetpil=avma; return gerepile(av,tetpil,gcopy(z));
}

/*********************************************************************/
/**                                                                 **/
/**                     INTEGRAL SQUARE ROOT                        **/
/**                                                                 **/
/*********************************************************************/
extern ulong mpsqrtl(GEN a);

GEN
gracine(GEN a)
{
  return garith_proto(racine,a,1); /* hm. --GN */
}

/* Use l as lgefint(a) [a may have more digits] */
static GEN
racine_r(GEN a, long l)
{
  long av,s;
  GEN x,y,z;

  if (l <= 4) return utoi(mpsqrtl(a));
  av = avma;
  s = 2 + ((l-1) >> 1);
  setlgefint(a, s);
  x = addis(racine_r(a, s), 1); setlgefint(a, l);
  /* x = good approx (from above) of sqrt(a): about l/2 correct bits */
  x = shifti(x, (l - s)*(BITS_IN_LONG/2));
  do
  { /* one or two iterations should do the trick */
    z = shifti(addii(x,divii(a,x)), -1);
    y = x; x = z;
  }
  while (cmpii(x,y) < 0);
  avma = (long)y;
  return gerepileuptoint(av,y);
}

GEN
racine_i(GEN a, int roundup)
{
  long k,m,l = lgefint(a), av = avma;
  GEN x = racine_r(a, l);
  if (roundup && signe(x))
  {
    m = x[lgefint(x)-1];
    k = (m * m != a[l-1] || !egalii(sqri(x),a));
    avma = (long)x;
    if (k) x = gerepileuptoint(av, addis(x,1));
  }
  return x;
}

GEN
racine(GEN a)
{
  if (typ(a) != t_INT) err(arither1);
  switch (signe(a))
  {
    case 1: return racine_i(a,0);
    case 0: return gzero;
    case -1:
    {
      GEN x = cgetg(3,t_COMPLEX);
      x[1] = zero;
      x[2] = (long)racine_i(a,0); return x;
    }
  }
  return NULL; /* not reached */
}

/*********************************************************************/
/**                                                                 **/
/**                      PERFECT SQUARE                             **/
/**                                                                 **/
/*********************************************************************/
extern ulong usqrtsafe(ulong a);

static int
carremod(ulong A)
{
  static int carresmod64[]={
    1,1,0,0,1,0,0,0,0,1, 0,0,0,0,0,0,1,1,0,0, 0,0,0,0,0,1,0,0,0,0,
    0,0,0,1,0,0,1,0,0,0, 0,1,0,0,0,0,0,0,0,1, 0,0,0,0,0,0,0,1,0,0, 0,0,0,0};
  static int carresmod63[]={
    1,1,0,0,1,0,0,1,0,1, 0,0,0,0,0,0,1,0,1,0, 0,0,1,0,0,1,0,0,1,0,
    0,0,0,0,0,0,1,1,0,0, 0,0,0,1,0,0,1,0,0,1, 0,0,0,0,0,0,0,0,1,0, 0,0,0};
  static int carresmod65[]={
    1,1,0,0,1,0,0,0,0,1, 1,0,0,0,1,0,1,0,0,0, 0,0,0,0,0,1,1,0,0,1,
    1,0,0,0,0,1,1,0,0,1, 1,0,0,0,0,0,0,0,0,1, 0,1,0,0,0,1,1,0,0,0, 0,1,0,0,1};
  static int carresmod11[]={1,1,0,1,1,1,0,0,0,1, 0};
  return (carresmod64[A & 0x3fUL]
    && carresmod63[A % 63UL]
    && carresmod65[A % 65UL]
    && carresmod11[A % 11UL]);
}

/* emulate carrecomplet on single-word positive integers */
ulong
ucarrecomplet(ulong A)
{
  if (carremod(A))
  {
    ulong a = usqrtsafe(A);
    if (a * a == A) return a;
  }
  return 0;
}

long
carrecomplet(GEN x, GEN *pt)
{
  long av;
  GEN y;

  switch(signe(x))
  {
    case -1: return 0;
    case 0: if (pt) *pt=gzero; return 1;
  }
  if (lgefint(x) == 3)
  {
    long a = ucarrecomplet((ulong)x[2]);
    if (!a) return 0;
    if (pt) *pt = stoi(a);
    return 1;
  }
  if (!carremod((ulong)smodis(x, 64*63*65*11))) return 0;
  av=avma; y = racine(x);
  if (!egalii(sqri(y),x)) { avma=av; return 0; }
  if (pt) { *pt = y; avma=(long)y; } else avma=av; 
  return 1;
}

static GEN
polcarrecomplet(GEN x, GEN *pt)
{
  long i,l,av,av2;
  GEN y,a,b;

  if (!signe(x)) return gun;
  l=lgef(x); if ((l&1) == 0) return gzero; /* odd degree */
  i=2; while (isexactzero((GEN)x[i])) i++;
  if (i&1) return gzero;
  av2 = avma; a = (GEN)x[i];
  switch (typ(a))
  {
    case t_POL: case t_INT:
      y = gcarrecomplet(a,&b); break;
    default:
      y = gcarreparfait(a); b = NULL; break;
  }
  if (y == gzero) { avma = av2; return gzero; }
  av = avma; x = gdiv(x,a);
  y = gtrunc(gsqrt(greffe(x,l,1),0)); av2 = avma;
  if (!gegal(gsqr(y), x)) { avma=av; return gzero; }
  if (pt)
  { 
    avma = av2; 
    if (!gcmp1(a))
    {
      if (!b) b = gsqrt(a,DEFAULTPREC);
      y = gmul(b,y);
    }
    *pt = gerepileupto(av,y);
  }
  else avma = av;
  return gun;
}

GEN
gcarrecomplet(GEN x, GEN *pt)
{
  long tx = typ(x),l,i;
  GEN z,y,p,t;

  if (!pt) return gcarreparfait(x);
  if (is_matvec_t(tx))
  {
    l=lg(x); y=cgetg(l,tx); z=cgetg(l,tx);
    for (i=1; i<l; i++)
    {
      t=gcarrecomplet((GEN)x[i],&p);
      y[i] = (long)t;
      z[i] = gcmp0(t)? zero: (long)p;
    }
    *pt=z; return y;
  }
  if (tx == t_POL) return polcarrecomplet(x,pt);
  if (tx != t_INT) err(arither1);
  return stoi(carrecomplet(x,pt));
}

GEN
gcarreparfait(GEN x)
{
  GEN p1,p2,p3,p4;
  long tx=typ(x),l,i,av,v;

  switch(tx)
  {
    case t_INT:
      return carreparfait(x)? gun: gzero;

    case t_REAL:
      return (signe(x)>=0)? gun: gzero;

    case t_INTMOD:
      if (!signe(x[2])) return gun;
      av=avma; p1=factor(absi((GEN)x[1]));
      p2=(GEN)p1[1]; l=lg(p2); p3=(GEN)p1[2];
      for (i=1; i<l; i++)
      {
      v = pvaluation((GEN)x[2],(GEN)p2[i],&p4);
      if (v < itos((GEN)p3[i]))
      {
        if (v&1) break;
        if (!egalii((GEN)p2[i], gdeux))
            { if (kronecker(p4,(GEN)p2[i]) == -1) break; }
          else
        {
          v = itos((GEN)p3[i]) - v;
          if ((v>=3 && mod8(p4) != 1 ) ||
              (v==2 && mod4(p4) != 1)) break;
        }
      }
      }
      avma=av; return i<l? gzero: gun;

    case t_FRAC: case t_FRACN:
      av=avma; l=carreparfait(mulii((GEN)x[1],(GEN)x[2]));
      avma=av; return l? gun: gzero;

    case t_COMPLEX:
      return gun;

    case t_PADIC:
      p4=(GEN)x[4]; if (!signe(p4)) return gun;
      if (valp(x)&1) return gzero;
      if (cmpis((GEN)x[2],2))
        return (kronecker((GEN)x[4],(GEN)x[2])== -1)? gzero: gun;

      v=precp(x); /* here p=2 */
      if ((v>=3 && mod8(p4) != 1 ) ||
          (v==2 && mod4(p4) != 1)) return gzero;
      return gun;

    case t_POL:
      return polcarrecomplet(x,NULL);

    case t_SER:
      if (!signe(x)) return gun;
      if (valp(x)&1) return gzero;
      return gcarreparfait((GEN)x[2]);

    case t_RFRAC: case t_RFRACN:
      av=avma;
      l=itos(gcarreparfait(gmul((GEN)x[1],(GEN)x[2])));
      avma=av; return stoi(l);

    case t_QFR: case t_QFI:
      return gcarreparfait((GEN)x[1]);

    case t_VEC: case t_COL: case t_MAT:
      l=lg(x); p1=cgetg(l,tx);
      for (i=1; i<l; i++) p1[i]=(long)gcarreparfait((GEN)x[i]);
      return p1;
  }
  err(typeer,"issquare");
  return NULL; /* not reached */
}

/*********************************************************************/
/**                                                                 **/
/**                        KRONECKER SYMBOL                         **/
/**                                                                 **/
/*********************************************************************/
#define ome(t) (labs(mod8(t)-4) == 1)

GEN
gkronecker(GEN x, GEN y)
{
  return arith_proto2(kronecker,x,y);
}

long
kronecker(GEN x, GEN y)
{
  GEN z;
  long av,r,s=1;

  av=avma;
  switch (signe(y))
  {
    case -1: y=negi(y); if (signe(x)<0) s= -1; break;
    case 0: return is_pm1(x);
  }
  r=vali(y);
  if (r)
  {
    if (mpodd(x))
    {
      if (odd(r) && ome(x)) s= -s;
      y=shifti(y,-r);
    }
    else { avma=av; return 0; }
  }
  x=modii(x,y);
  while (signe(x))
  {
    r=vali(x);
    if (r)
    {
      if (odd(r) && ome(y)) s= -s;
      x=shifti(x,-r);
    }
    /* x=3 mod 4 && y=3 mod 4 ? (both are odd here) */
    if (y[lgefint(y)-1]&2 && x[lgefint(x)-1]&2) s = -s;
    z=resii(y,x); y=x; x=z;
  }
  avma=av; return is_pm1(y)? s: 0;
}

GEN
gkrogs(GEN x, long y)
{
  return arith_proto2gs(krogs,x,y);
}

long
krogs(GEN x, long y)
{
  long av,r,s=1,x1,z;

  av=avma;
  if (y<=0)
  {
    if (y) { y = -y; if (signe(x)<0) s = -1; }
    else  return is_pm1(x);
  }
  r=vals(y);
  if (r)
  {
    if (mpodd(x))
    {
      if (odd(r) && ome(x)) s= -s;
      y>>=r;
    }
    else return 0;
  }
  x1=smodis(x,y);
  while (x1)
  {
    r=vals(x1);
    if (r)
    {
      if (odd(r) && (labs((y&7)-4)==1)) s= -s;
      x1>>=r;
    }
    if (y&2 && x1&2) s= -s;
    z=y%x1; y=x1; x1=z;
  }
  avma=av; return (y==1)? s: 0;
}

long
krosg(long s, GEN x)
{
  long av = avma, y = kronecker(stoi(s),x);
  avma = av; return y;
}

long
kross(long x, long y)
{
  long r,s=1,x1,z;

  if (y<=0)
  {
    if (y) { y= -y; if (x<0) s = -1; }
    else  return (labs(x)==1);
  }
  r=vals(y);
  if (r)
  {
    if (odd(x))
    {
      if (odd(r) && labs((x&7)-4) == 1) s = -s;
      y>>=r;
    }
    else return 0;
  }
  x1=x%y; if (x1<0) x1+=y;
  while (x1)
  {
    r=vals(x1);
    if (r)
    {
      if (odd(r) && labs((y&7)-4) == 1) s= -s;
      x1>>=r;
    }
    if (y&2 && x1&2) s= -s;
    z=y%x1; y=x1; x1=z;
  }
  return (y==1)? s: 0;
}

long
hil0(GEN x, GEN y, GEN p)
{
  return p? hil(x,y,p): hil(x,y,gzero);
}

#define eps(t) (((signe(t)*(t[lgefint(t)-1]))&3)==3)
long
hil(GEN x, GEN y, GEN p)
{
  long a,b,av,tx,ty,z;
  GEN p1,p2,u,v;

  if (gcmp0(x) || gcmp0(y)) return 0;
  av=avma; tx=typ(x); ty=typ(y);
  if (tx>ty) { p1=x; x=y; y=p1; a=tx,tx=ty; ty=a; }
  switch(tx)
  {
    case t_INT:
      switch(ty)
      {
      case t_INT:
        if (signe(p)<=0)
          return (signe(x)<0 && signe(y)<0)? -1: 1;
        a = odd(pvaluation(x,p,&u));
          b = odd(pvaluation(y,p,&v));
        if (egalii(p,gdeux))
          {
          z = (eps(u) && eps(v))? -1: 1;
          if (a && ome(v)) z= -z;
          if (b && ome(u)) z= -z;
          }
          else
        {
          z = (a && b && eps(p))? -1: 1;
          if (a && kronecker(v,p)<0) z= -z;
          if (b && kronecker(u,p)<0) z= -z;
        }
        avma=av; return z;
      case t_REAL:
        return (signe(x)<0 && signe(y)<0)? -1: 1;
      case t_INTMOD:
          p = (GEN)y[1];
        if (egalii(gdeux,p)) err(hiler1);
        return hil(x,(GEN)y[2],p);
      case t_FRAC: case t_FRACN:
        p1=mulii((GEN)y[1],(GEN)y[2]); z=hil(x,p1,p);
        avma=av; return z;
      case t_PADIC:
          p = (GEN)y[2];
        if (egalii(gdeux,p) && precp(y) <= 1) err(hiler1);
        p1 = odd(valp(y))? mulii(p,(GEN)y[4]): (GEN)y[4];
        z=hil(x,p1,p); avma=av; return z;
      }
      break;

    case t_REAL:
      if (! is_frac_t(ty)) break;
      if (signe(x) > 0) return 1;
      return signe(y[1])*signe(y[2]);

    case t_INTMOD:
      p = (GEN)x[1];
      if (egalii(gdeux,p)) err(hiler1);
      switch(ty)
      {
        case t_INTMOD:
          if (!egalii(p, (GEN)y[1])) break;
          return hil((GEN)x[2],(GEN)y[2],p);
        case t_FRAC: case t_FRACN:
        return hil((GEN)x[2],y,p);
        case t_PADIC:
          if (!egalii(p, (GEN)y[2])) break;
          return hil((GEN)x[2],y,p);
      }
      break;

    case t_FRAC: case t_FRACN:
      p1=mulii((GEN)x[1],(GEN)x[2]);
      switch(ty)
      {
      case t_FRAC: case t_FRACN:
        p2=mulii((GEN)y[1],(GEN)y[2]);
        z=hil(p1,p2,p); avma=av; return z;
      case t_PADIC:
        z=hil(p1,y,NULL); avma=av; return z;
      }
      break;

    case t_PADIC:
      p = (GEN)x[2];
      if (ty != t_PADIC || !egalii(p,(GEN)y[2])) break;
      if (egalii(p, gdeux) && (precp(x) <= 1 || precp(y) <= 1)) err(hiler1);
      p1 = odd(valp(x))? mulii(p,(GEN)x[4]): (GEN)x[4];
      p2 = odd(valp(y))? mulii(p,(GEN)y[4]): (GEN)y[4];
      z=hil(p1,p2,p); avma=av; return z;
  }
  err(talker,"forbidden or incompatible types in hil");
  return 0; /* not reached */
}
#undef eps
#undef ome

/*******************************************************************/
/*                                                                 */
/*                       SQUARE ROOT MODULO p                      */
/*                                                                 */
/*******************************************************************/

#define sqrmod(x,p) (resii(sqri(x),p))

/* Assume p is prime. return NULL if (a,p) = -1 */
GEN
mpsqrtmod(GEN a, GEN p)
{
  long av = avma, av1,lim,i,k,e;
  GEN p1,q,v,y,w,m;

  if (typ(a) != t_INT || typ(p) != t_INT) err(arither1);
  if (signe(p) <= 0 || is_pm1(p)) err(talker,"not a prime in mpsqrtmod");
  p1 = addsi(-1,p); e = vali(p1);
  if (e == 0) /* p = 2 */
  {
    avma = av;
    if (!egalii(p, gdeux))
      err(talker,"composite modulus in mpsqrtmod: %Z",p);
    if (!signe(a) || !mod2(a)) return gzero;
    return gun;
  }
  q = shifti(p1,-e); /* q = (p-1)/2^oo is odd */
  if (e == 1) y = p1;
  else /* look for an odd power of a primitive root */
    for (k=2; ; k++)
    { /* loop terminates for k < p (even if p composite) */
  
      i = krosg(k,p);
      if (i >= 0)
      {
        if (i) continue;
        err(talker,"composite modulus in mpsqrtmod: %Z",p);
      }
      av1 = avma;
      y = m = powmodulo(stoi(k),q,p);
      for (i=1; i<e; i++)
      if (gcmp1(m = sqrmod(m,p))) break;
      if (i == e) break; /* success */
      avma = av1;
    }

  p1 = powmodulo(a, shifti(q,-1), p); /* a ^ [(q-1)/2] */
  if (!signe(p1)) { avma=av; return gzero; }
  v = modii(mulii(a, p1), p);
  w = modii(mulii(v, p1), p);
  lim = stack_lim(av,1);
  while (!gcmp1(w))
  { /* a*w = v^2, y primitive 2^e-th root of 1
       a square --> w even power of y, hence w^(2^(e-1)) = 1 */
    p1 = sqrmod(w,p);
    for (k=1; !gcmp1(p1) && k < e; k++) p1 = sqrmod(p1,p);
    if (k == e) { avma=av; return NULL; } /* p composite or (a/p) != 1 */
    /* w ^ (2^k) = 1 --> w = y ^ (u * 2^(e-k)), u odd */
    p1 = y;
    for (i=1; i < e-k; i++) p1 = sqrmod(p1,p);
    y = sqrmod(p1, p); e = k;
    w = modii(mulii(y, w), p);
    v = modii(mulii(v, p1), p);
    if (low_stack(lim, stack_lim(av,1)))
    {
      GEN *gptr[3]; gptr[0]=&y; gptr[1]=&w; gptr[2]=&v;
      if(DEBUGMEM>1) err(warnmem,"mpsqrtmod");
      gerepilemany(av,gptr,3);
    }
  }
  av1 = avma;
  p1 = subii(p,v); if (cmpii(v,p1) > 0) v = p1; else avma = av1;
  return gerepileuptoint(av, v);
}

/*******************************************************************/
/*                                                                 */
/*                       n-th ROOT MODULO p                        */
/*                                                                 */
/*******************************************************************/

/* UNCLEAN
 * p-1=l^e*r
 * l doit etre premier
 * q=p-1
 * q=(l^e)*r
 * e>=1
 * pgcd(r,l)=1
 * return a non l-power residue and set zeta to a primitive l root of unity.
 */

static GEN
mplgenmod(GEN l, long e, GEN r,GEN p,GEN *zeta)
{
  ulong av1;
  GEN m,m1;
  long k,i; 
  av1 = avma;
  for (k=1; ; k++)
  {
    m1 = m = powmodulo(stoi(k+1),r,p);
    if (gcmp1(m)) {avma=av1; continue;}
    for (i=1; i<e; i++)
      if (gcmp1(m=powmodulo(m,l,p))) break;
    if (i==e) break;
    avma = av1;
  }
  *zeta=m;
  return m1;
}
/* resoud x^l=a mod (p)
 * l doit etre premier
 * q=p-1
 * q=(l^e)*r
 * e>=1
 * pgcd(r,l)=1
 * m=y^(q/l)
 * y n'est pas une puissance l-ieme
 * m!=1
 * ouf!
 * FIXME: use baby-step giant-step.
 */
static GEN
mpsqrtlmod(GEN a, GEN l, GEN p, GEN q,long e, GEN r, GEN y, GEN m)
{
  ulong av = avma, tetpil,lim;
  long i,k;
  GEN p1,p2,u1,u2,v,w,z;
  /* y contient un generateur de (Z/pZ)^* eleve a la puis (p-1)/(l^e) */
  bezout(r,l,&u1,&u2);
  v=powmodulo(a,u2,p);
  w=powmodulo(a,modii(mulii(negi(u1),r),q),p);
  lim = stack_lim(av,1);
  while (!gcmp1(w))
  {
    /* if p is not prime, next loop will not end */
    k=0;
    p1=w;
    do
    {
      z=p1;
      p1=powmodulo(p1,l,p);
      k++;
    }while(!gcmp1(p1));
    if (k==e) { avma=av; return NULL; }
    p2 = modii(mulii(z,m),p);
    for(i=1; !gcmp1(p2); i++) p2 = modii(mulii(p2,m),p);/*should be a baby step
                                            giant step instead*/
    p1=powmodulo(y,modii(mulsi(i,gpowgs(l,e-k-1)),q),p);
    m=powmodulo(m,stoi(i),p);
    e = k;
    v = modii(mulii(p1,v),p);
    y = powmodulo(p1,l,p);
    w = modii(mulii(y,w),p);
    if (low_stack(lim, stack_lim(av,1)))
    {
      GEN *gptr[4];
      if(DEBUGMEM>1) err(warnmem,"mpsqrtlmod");
      gptr[0]=&y; gptr[1]=&v; gptr[2]=&w; gptr[3]=&m;
      gerepilemany(av,gptr,4);
    }
  }
  tetpil=avma; return gerepile(av,tetpil,icopy(v));
}
/* a and n are integers, p is prime

return a solution of 

x^n=a mod p 

1)If there is no solution return NULL and if zetan is not NULL set zetan to gzero.

2) If there is solution there are exactly  m=gcd(p-1,n) of them.

If zetan is not NULL, zetan is set to a primitive mth root of unity so that
the set of solutions is {x*zetan^k;k=0 to m-1}

If a=0 ,return 0 and if zetan is not NULL zetan is set to gun
*/
GEN 
mpsqrtnmod(GEN a, GEN n, GEN p, GEN *zetan)
{
  ulong ltop=avma,lbot=0,av1,lim;
  GEN m,u1,u2;
  GEN q,z;
  GEN *gptr[2];
  if (typ(a) != t_INT || typ(n) != t_INT || typ(p)!=t_INT)
    err(typeer,"mpsqrtnmod");
  if(!signe(n))
    err(talker,"1/0 exponent in mpsqrtnmod");
  if(gcmp1(n)) {if (zetan) *zetan=gun;return gcopy(a);}
  a=modii(a,p);
  if(gcmp0(a)) {if (zetan) *zetan=gun;avma=ltop;return gzero;}
  q=addsi(-1,p);
  m=bezout(n,q,&u1,&u2);
  if (zetan) z=gun;
  lim=stack_lim(ltop,1);
  if (!gcmp1(m))
  {
    GEN F=decomp(m);
    long i,j,e; 
    GEN r,zeta,y,l;
    av1=avma;
    for (i = lg(F[1])-1; i; i--)
    {
      l=gcoeff(F,i,1); j=itos(gcoeff(F,i,2));
      e=pvaluation(q,l,&r);
      y=mplgenmod(l,e,r,p,&zeta);
      if (zetan) z=modii(mulii(z,powmodulo(y,gpowgs(l,e-j),p)),p);
      do
      {
      lbot=avma;
      if (!is_pm1(a) || signe(a)<0)
        a=mpsqrtlmod(a,l,p,q,e,r,y,zeta);
      else
        a=icopy(a);
      if (!a){avma=ltop;if (zetan)  *zetan=gzero;return NULL;}
      j--;
      }while (j);
      if (low_stack(lim, stack_lim(ltop,1)))
      /* n can have lots of prime factors*/
      {
      if(DEBUGMEM>1) err(warnmem,"mpsqrtnmod");
      if (zetan)
      {
        z=gcopy(z);
        gptr[0]=&a;gptr[1]=&z;
        gerepilemanysp(av1,lbot,gptr,2);
      }
      else
        a=gerepile(av1,lbot,a);
      lbot=av1;
      }
    }
  }
  if (cmpii(m,n))
  {
    GEN b=modii(u1,q);
    lbot=avma;
    a=powmodulo(a,b,p);
  }
  if (zetan)
  {
    *zetan=gcopy(z);
    gptr[0]=&a;gptr[1]=zetan;
    gerepilemanysp(ltop,lbot,gptr,2);
  }
  else
    a=gerepile(ltop,lbot,a);
  return a;
}

/*********************************************************************/
/**                                                                 **/
/**                        GCD & BEZOUT                             **/
/**                                                                 **/
/*********************************************************************/

/* Ultra-fast private ulong gcd for trial divisions.  Called with y odd;
   x can be arbitrary (but will most of the time be smaller than y).
   Will also be used from inside ifactor2.c, so it's `semi-private' really.
   --GN */

/* Gotos are Harmful, and Programming is a Science.  E.W.Dijkstra. */
ulong
ugcd(ulong x, ulong y)         /* assume y&1==1, y > 1 */
{
  if (!x) return y;
  /* fix up x */
  while (!(x&1)) x>>=1;
  if (x==1) return 1;
  if (x==y) return y;
  else if (x>y) goto xislarger;/* will be rare, given how we'll use this */
  /* loop invariants: x,y odd and distinct. */
 yislarger:
  if ((x^y)&2)                 /* ...01, ...11 or vice versa */
    y=(x>>2)+(y>>2)+1;         /* ==(x+y)>>2 except it can't overflow */
  else                         /* ...01,...01 or ...11,...11 */
    y=(y-x)>>2;                /* now y!=0 in either case */
  while (!(y&1)) y>>=1;        /* kill any windfall-gained powers of 2 */
  if (y==1) return 1;          /* comparand == return value... */
  if (x==y) return y;          /* this and the next is just one comparison */
  else if (x<y) goto yislarger;/* else fall through to xislarger */

 xislarger:                    /* same as above, seen through a mirror */
  if ((x^y)&2)
    x=(x>>2)+(y>>2)+1;
  else
    x=(x-y)>>2;                /* x!=0 */
  while (!(x&1)) x>>=1;
  if (x==1) return 1;
  if (x==y) return y;
  else if (x>y) goto xislarger;/* again, a decent [g]cc should notice that
                                  it can re-use the comparison */
  goto yislarger;
}
/* Gotos are useful, and Programming is an Art.  D.E.Knuth. */
/* PS: Of course written with Dijkstra's lessons firmly in mind... --GN */

/* modified right shift binary algorithm with at most one division */
long
cgcd(long a,long b)
{
  long v;
  a=labs(a); if (!b) return a;
  b=labs(b); if (!a) return b;
  if (a>b) { a %= b; if (!a) return b; }
  else { b %= a; if (!b) return a; }
  v=vals(a|b); a>>=v; b>>=v;
  if (a==1 || b==1) return 1L<<v;
  if (b&1)
    return ((long)ugcd((ulong)a, (ulong)b)) << v;
  else
    return ((long)ugcd((ulong)b, (ulong)a)) << v;
}
long
clcm(long a,long b)
{
  long d;
  if(!a) return 0;
  d=cgcd(a,b);
  if(d!=1) return a*(b/d);
  return a*b;
}

/* assume a>b>0, return gcd(a,b) as a GEN (for mppgcd) */
static GEN
mppgcd_cgcd(ulong a, ulong b)
{
  GEN r = cgeti(3);
  long v;

  r[1] = evalsigne(1)|evallgefint(3);
  a %= b; if (!a) { r[2]=(long)b; return r; }
  v=vals(a|b); a>>=v; b>>=v;
  if (a==1 || b==1) { r[2]=(long)(1UL<<v); return r; }
  if (b&1)
    r[2] = (long)(ugcd((ulong)a, (ulong)b) << v);
  else
    r[2] = (long)(ugcd((ulong)b, (ulong)a) << v);
  return r;
}

void mppgcd_plus_minus(GEN x, GEN y, GEN t);
ulong mppgcd_resiu(GEN y, ulong x);

/* uses modified right-shift binary algorithm now --GN 1998Jul23 */
GEN
mppgcd(GEN a, GEN b)
{
  long av,v,w;
  GEN t, p1;

  if (typ(a) != t_INT || typ(b) != t_INT) err(arither1);
  switch (absi_cmp(a,b))
  {
    case 0: return absi(a);
    case -1: t=b; b=a; a=t;
  }
  if (!signe(b)) return absi(a);
  /* here |a|>|b|>0. Try single precision first */
  if (lgefint(a)==3)
    return mppgcd_cgcd((ulong)a[2], (ulong)b[2]);
  if (lgefint(b)==3)
  {
    ulong u = mppgcd_resiu(a,(ulong)b[2]);
    if (!u) return absi(b);
    return mppgcd_cgcd((ulong)b[2], u);
  }

  /* larger than gcd: "avma=av" gerepile (erasing t) is valid */
  av = avma; new_chunk(lgefint(b));
  t = resii(a,b);
  if (!signe(t)) { avma=av; return absi(b); }

  a = b; b = t;
  v = vali(a); a = shifti(a,-v); setsigne(a,1);
  w = vali(b); b = shifti(b,-w); setsigne(b,1);
  if (w < v) v = w;
  switch(absi_cmp(a,b))
  {
    case  0: avma=av; a=shifti(a,v); return a;
    case -1: p1=b; b=a; a=p1;
  }
  if (is_pm1(b)) { avma=av; return shifti(gun,v); }

  /* we have three consecutive memory locations: a,b,t.
   * All computations are done in place */

  /* a and b are odd now, and a>b>1 */
  while (lgefint(a) > 3)
  {
    /* if a=b mod 4 set t=a-b, otherwise t=a+b, then strip powers of 2 */
    /* so that t <= (a+b)/4 < a/2 */
    mppgcd_plus_minus(a,b, t);
    if (is_pm1(t)) { avma=av; return shifti(gun,v); }
    switch(absi_cmp(t,b))
    {
      case -1: p1 = a; a = b; b = t; t = p1; break;
      case  1: p1 = a; a = t; t = p1; break;
      case  0: avma = av; b=shifti(b,v); return b;
    }
  }
  {
    long r[] = {evaltyp(t_INT)|m_evallg(3), evalsigne(1)|evallgefint(3), 0};
    r[2] = (long) ugcd((ulong)b[2], (ulong)a[2]);
    avma = av; return shifti(r,v);
  }
}
GEN
mpppcm(GEN x, GEN y)
{
  ulong av;
  GEN p1,p2;
  if (typ(x) != t_INT || typ(y) != t_INT) err(arither1);
  if (!signe(x)) return gzero;
  av = avma;
  p1 = mppgcd(x,y); if (!is_pm1(p1)) y = divii(y,p1);
  p2 = mulii(x,y);
  if (signe(p2) < 0) setsigne(p2,1);
  return gerepileuptoint(av, p2);
}
/* Extended bezout. Return d=pgcd(a,b) and &u,&v */
long
cbezout(long a,long b,long *uu,long *vv)
{
  long av=avma,v,q,u,t,s, d = labs(a), r = labs(b);
  GEN p1;

  if (!b)
  {
    *vv=0;
    if (!a) { *uu=1; return 1; }
    if (a<0)
      { *uu=-1; return -a; }
    else
      { *uu= 1; return  a; }
  }
  u=1; t=0;
  for(;;)
  {
    if (!r)
    {
      if (a<0) u = -u;
      p1 = mulss(a,u); s = signe(p1);
      if (!s) v = d / b;
      else if (!is_bigint(p1))
      {
        ulong ul;
        long sb = (b<0);
        b = labs(b);
        if (s < 0)
        {
          ul = (p1[2]+d);
          ul /= b; v = sb? -(long)ul: (long)ul;
        }
        else
        {
          ul = p1[2]-d;
          ul /= b; v = sb? (long)ul: -(long)ul;
        }
      }
      else v = - itos(divis(addsi(-d,p1), b));
      avma=av; *uu = u; *vv = v; return d;
    }
    v=d; d=r; q = v/r; r = v - q*r;
    v=t; t = u - q*t; u=v;
  }
}

GEN
bezout(GEN a, GEN b, GEN *ptu, GEN *ptv)
{
  GEN u,v,t,x,y,q,r, *tmp;
  long av, av2, lim;

  if (typ(a) != t_INT || typ(b) != t_INT) err(arither1);
  if (absi_cmp(a,b) < 0)
  {
    t=b; b=a; a=t;
    tmp=ptu; ptu=ptv; ptv=tmp;
  }
  /* now |a| >= |b| */
  if (!signe(b))
  {
    *ptv = gzero;
    switch(signe(a))
    {
      case  0: *ptu = gun; return gzero;
      case  1: *ptu = gun; return icopy(a);
      case -1: *ptu = negi(gun); return negi(a);
    }
  }
  if (!is_bigint(a))
  {
    long uu,vv, dd = cbezout(itos(a),itos(b),&uu,&vv);
    *ptu = stoi(uu); *ptv = stoi(vv); return stoi(dd);
  }
  av = avma;
  (void)new_chunk(lgefint(b) + (lgefint(a)<<1)); /* room for x, u and v */
  x = a; y = b; u = gun; v = gzero;
  av2 = avma;
  lim = stack_lim(av2,3);
  do
  {
    q = dvmdii(x,y,&r);
    v = subii(v, mulii(q,u));
    t=v; v=u; u=t;
    x=y; y=r;
    if (low_stack(lim, stack_lim(av2,3)))
    {
      GEN *gptr[2]; gptr[0]=&x; gptr[1]=&y;
      if(DEBUGMEM>1) err(warnmem,"bezout");
      gerepilemany(av2,gptr,2);
    }
  } while (signe(y));
  u = divii(subii(x, mulii(b,v)), a);
  avma = av; x = icopy(x); v = icopy(v); u = icopy(u);
  if (signe(x) < 0)
  {
    setsigne(x,1);
    setsigne(u,-signe(u));
    setsigne(v,-signe(v));
  }
  *ptu = u; *ptv = v; return x;
}

/*********************************************************************/
/**                                                                 **/
/**                      CHINESE REMAINDERS                         **/
/**                                                                 **/
/*********************************************************************/

/*  P.M. & M.H.
 *
 *  Chinese Remainder Theorem.  x and y must have the same type (integermod,
 *  polymod, or polynomial/vector/matrix recursively constructed with these
 *  as coefficients). Creates (with the same type) a z in the same residue
 *  class as x and the same residue class as y, if it is possible.
 *
 *  We also allow (during recursion) two identical objects even if they are
 *  not integermod or polymod. For example, if
 *
 *    x = [1. mod(5, 11), mod(X + mod(2, 7), X^2 + 1)]
 *    y = [1, mod(7, 17), mod(X + mod(0, 3), X^2 + 1)],
 *
 *  then chinois(x, y) returns
 *
 *    [1, mod(16, 187), mod(X + mod(9, 21), X^2 + 1)]
 *
 *  Someone else may want to allow power series, complex numbers, and
 *  quadratic numbers.
 */
GEN
chinois(GEN x, GEN y)
{
  long i,lx,vx,av,tetpil, tx = typ(x);
  GEN z,p1,p2,d,u,v;

  if (gegal(x,y)) return gcopy(x);
  if (tx == typ(y)) switch(tx)
  {
    case t_POLMOD:
      if (gegal((GEN)x[1],(GEN)y[1]))  /* same modulus */
      {
      z=cgetg(3,tx);
      z[1]=lcopy((GEN)x[1]);
      z[2]=(long)chinois((GEN)x[2],(GEN)y[2]);
        return z;
      }  /* fall through */
    case t_INTMOD:
      z=cgetg(3,tx); av=avma;
      d=gbezout((GEN)x[1],(GEN)y[1],&u,&v);
      if (!gegal(gmod((GEN)x[2],d), gmod((GEN)y[2],d))) break;
      p1 = gdiv((GEN)x[1],d);
      p2 = gadd((GEN)x[2], gmul(gmul(u,p1), gadd((GEN)y[2],gneg((GEN)x[2]))));

      tetpil=avma; z[1]=lmul(p1,(GEN)y[1]); z[2]=lmod(p2,(GEN)z[1]);
      gerepilemanyvec(av,tetpil,z+1,2); return z;

    case t_POL:
      lx=lgef(x); vx=varn(x); z=cgetg(lx,tx);
      if (lx!=lgef(y) || vx!=varn(y)) break;
      z[1]=evalsigne(1)|evallgef(lx)|evalvarn(vx);
      for (i=2; i<lx; i++) z[i]=(long)chinois((GEN)x[i],(GEN)y[i]);
      return z;

    case t_VEC: case t_COL: case t_MAT:
      lx=lg(x); z=cgetg(lx,tx); if (lx!=lg(y)) break;
      for (i=1; i<lx; i++) z[i]=(long)chinois((GEN)x[i],(GEN)y[i]);
      return z;
  }
  err(talker,"incompatible arguments in chinois");
  return NULL; /* not reached */
}

/* return lift(chinois(x2 mod x1, y2 mod y1))
 * assume(x1,y1)=1, xi,yi integers and z1 = x1*y1
 */
GEN
chinois_int_coprime(GEN x2, GEN y2, GEN x1, GEN y1, GEN z1)
{
  long av = avma;
  GEN ax,p1;
  (void)new_chunk((lgefint(z1)<<1)+lgefint(x1)+lgefint(y1));
  ax = mulii(mpinvmod(x1,y1), x1);
  p1 = addii(x2, mulii(ax, subii(y2,x2)));
  avma = av; return modii(p1,z1);
}

/*********************************************************************/
/**                                                                 **/
/**                      INVERSE MODULO b                           **/
/**                                                                 **/
/*********************************************************************/

GEN
mpinvmod(GEN a, GEN m)
{
  GEN res;
  if (! invmod(a,m,&res))
    err(talker,"impossible inverse modulo: %Z",gmodulcp(res,m));
  return res;
}

/*********************************************************************/
/**                                                                 **/
/**                   POWERING MODULO (A^N mod M)                   **/
/**                                                                 **/
/*********************************************************************/
GEN resiimul(GEN x, GEN y);
GEN resmod2n(GEN x, long y);
GEN _resii(GEN x, GEN y) { return resii(x,y); }

GEN
init_remainder(GEN M)
{
  GEN sM = cgetg(3, t_VEC);
  GEN Mr = cgetr(lgefint(M) + 1);
  affir(M, Mr);
  sM[1] = (long)M;
  sM[2] = (long)linv(Mr);
  return sM;
}

/* optimal on UltraSPARC */
static long RESIIMUL_LIMIT = 150;

void
setresiilimit(long n) { RESIIMUL_LIMIT = n; }

GEN
powmodulo(GEN A, GEN N, GEN M)
{
  GEN y;
  long av = avma,av1,lim,man,k,nb,s, *p;
  GEN (*mul)(GEN,GEN) = mulii;
  GEN (*res)(GEN,GEN) = _resii;

  if (typ(A) != t_INT || typ(N) != t_INT || typ(M) != t_INT) err(arither1);
  s = signe(N);
  if (!s)
  {
    k = signe(resii(A,M)); avma=av;
    return k? gun: gzero;
  }
  if (s < 0) { A = mpinvmod(A,M); N = absi(N); }
  else
  {
    A = modii(A,M);
    if (!signe(A)) { avma=av; return gzero; }
  }
  y = A;
  if (lgefint(y)==3) switch(y[2])
  {
    case 1: /* y = 1 */
      avma=av; return gun;
    case 2: /* y = 2, use shifti not mulii */
    mul = (GEN (*)(GEN,GEN))shifti; A = (GEN)1L;
  }

  /* TODO: Move this out of here and use for general modular computations */
  if ((k = vali(M)) && k == expi(M))
  { /* M is a power of 2 */
    M = (GEN)k;
    res = (GEN(*)(GEN,GEN))resmod2n;
  }
  else if (lgefint(M) > RESIIMUL_LIMIT && (lgefint(N) > 3 || N[2] > 4))
  { /* compute x % M using multiplication by 1./M */
    M = init_remainder(M);
    res = (GEN(*)(GEN,GEN))resiimul;
  }

  p = N+2; man = *p; /* see puissii */
  k = 1+bfffo(man); man<<=k; k = BITS_IN_LONG-k;
  av1=avma; lim=stack_lim(av1,1);
  for (nb=lgefint(N)-2;;)
  {
    for (; k; man<<=1,k--)
    {
      y = res(sqri(y), M);
      if (man < 0) y = res(mul(y,A), M);
      if (low_stack(lim, stack_lim(av1,1)))
      {
        if (DEBUGMEM>1) err(warnmem,"powmodulo");
        y = gerepileuptoint(av1,y);
      }
    }
    if (--nb == 0) break;
    man = *++p, k = BITS_IN_LONG;
  }
  return gerepileupto(av,y);
}

/*********************************************************************/
/**                                                                 **/
/**                NEXT / PRECEDING (PSEUDO) PRIME                  **/
/**                                                                 **/
/*********************************************************************/
GEN
gnextprime(GEN n)
{
  return garith_proto(nextprime,n,0);
}

GEN
gprecprime(GEN n)
{
  return garith_proto(precprime,n,0);
}

GEN
gisprime(GEN x, long flag)
{
  switch (flag)
  {
  case 0:
    return arith_proto(isprime,x,1);
  case 1:
    return garith_proto2gs(plisprime,x,0);
  case 2:
    return garith_proto2gs(plisprime,x,1);
  }
  err(flagerr,"gisprime");
  return 0;
}

long
isprime(GEN x)
{
  return millerrabin(x,10);
}

GEN
gispsp(GEN x)
{
  return arith_proto(ispsp,x,1);
}

long
ispsp(GEN x)
{
  return millerrabin(x,1);
}

GEN
gmillerrabin(GEN n, long k)
{
  return arith_proto2gs(millerrabin,n,k);
}

/*********************************************************************/
/**                                                                 **/
/**                    FUNDAMENTAL DISCRIMINANTS                    **/
/**                                                                 **/
/*********************************************************************/

/* gissquarefree, issquarefree moved into arith2.c with the other
   basic arithmetic functions (issquarefree is the square of the
   Moebius mu function, after all...) --GN */

GEN
gisfundamental(GEN x)
{
  return arith_proto(isfundamental,x,1);
}

long
isfundamental(GEN x)
{
  long av,r;
  GEN p1;

  if (gcmp0(x)) return 0;
  r=mod4(x);
  if (!r)
  {
    av=avma; p1=shifti(x,-2);
    r=mod4(p1); if (!r) return 0;
    if (signe(x)<0) r=4-r;
    r = (r==1)? 0: issquarefree(p1);
    avma=av; return r;
  }
  if (signe(x)<0) r=4-r;
  return (r==1) ? issquarefree(x) : 0;
}

GEN
quaddisc(GEN x)
{
  long av=avma,tetpil=avma,i,r,tx=typ(x);
  GEN p1,p2,f,s;

  if (tx != t_INT && ! is_frac_t(tx)) err(arither1);
  f=factor(x); p1=(GEN)f[1]; p2=(GEN)f[2];
  s=gun;
  for (i=1; i<lg(p1); i++)
    if ( odd(mael(p2,i,2)) ) { tetpil=avma; s=gmul(s,(GEN)p1[i]); }
  r=mod4(s); if (gsigne(x)<0) r=4-r;
  if (r>1) { tetpil=avma; s=shifti(s,2); }
  return gerepile(av,tetpil,s);
}

/*********************************************************************/
/**                                                                 **/
/**                              FACTORIAL                          **/
/**                                                                 **/
/*********************************************************************/
GEN muluu(ulong x, ulong y);

GEN
mpfact(long n)
{
  long lx, av = avma;
  ulong k,l;
  GEN x;

  if (n<2)
  {
    if (n<0) err(facter);
    return gun;
  }
  if (n < 60)
  {
    x = gdeux;
    for (k=3; k<=n; k++) x = mulsi(k,x);
    return gerepileuptoint(av,x);
  }
  lx = 1; x = cgetg(1 + n/2, t_VEC);
  for (k=2;; k++)
  {
    l = n+2-k; if (l <= k) break;
    x[lx++] = (long)muluu(k,l);
  }
  if (l == k) x[lx++] = lstoi(k);
  setlg(x, lx);
  return gerepileuptoint(av, divide_conquer_prod(x, mulii));
}

GEN
mpfactr(long n, long prec)
{
  long av = avma, lim,k;
  GEN f = cgetr(prec);

  affsr(1,f);
  if (n<2)
  {
    if (n<0) err(facter);
    return f;
  }
  lim = stack_lim(av,1);
  for (k=2; k<=n; k++)
  {
    f = mulsr(k,f);
    if (low_stack(lim, stack_lim(av,1)))
    {
      if(DEBUGMEM>1) err(warnmem,"mpfactr k=%ld",k);
      f = gerepileuptoleaf(av,f);
    }
  }
  return gerepileuptoleaf(av,f);
}

/*******************************************************************/
/**                                                               **/
/**                      LUCAS & FIBONACCI                        **/
/**                                                               **/
/*******************************************************************/

void
lucas(long n, GEN *ln, GEN *ln1)
{
  long taille,av;
  GEN z,t;

  if (!n) { *ln=stoi(2); *ln1=stoi(1); return; }

  taille=(long)(pariC3*(1+labs(n))+3);
  *ln=cgeti(taille); *ln1=cgeti(taille);
  av=avma; lucas(n/2,&z,&t);
  switch(n % 4)
  {
    case -3:
      addsiz(2,sqri(z),*ln1);
      subiiz(addsi(1,mulii(z,t)),*ln1,*ln); break;
    case -2:
      addsiz(2,sqri(z),*ln); addsiz(1,mulii(z,t),*ln1); break;
    case -1:
      subisz(sqri(z),2,*ln1);
      subiiz(subis(mulii(z,t),1),*ln1,*ln); break;
    case  0: subisz(sqri(z),2,*ln); subisz(mulii(z,t),1,*ln1); break;
    case  1: subisz(mulii(z,t),1,*ln); addsiz(2,sqri(t),*ln1); break;
    case  2: addsiz(2,sqri(z),*ln); addsiz(1,mulii(z,t),*ln1); break;
    case  3: addsiz(1,mulii(z,t),*ln); subisz(sqri(t),2,*ln1);
  }
  avma=av;
}

GEN
fibo(long n)
{
  long av = avma;
  GEN ln,ln1;

  lucas(n-1,&ln,&ln1);
  return gerepileupto(av, divis(addii(shifti(ln,1),ln1), 5));
}

/*******************************************************************/
/*                                                                 */
/*                      CONTINUED FRACTIONS                        */
/*                                                                 */
/*******************************************************************/
static GEN sfcont2(GEN b, GEN x, long k);

GEN
gcf(GEN x)
{
  return sfcont(x,x,0);
}

GEN
gcf2(GEN b, GEN x)
{
  return contfrac0(x,b,0);
}

GEN
gboundcf(GEN x, long k)
{
  return sfcont(x,x,k);
}

GEN
contfrac0(GEN x, GEN b, long flag)
{
  long lb,tb,i;
  GEN y;

  if (!b || gcmp0(b)) return sfcont(x,x,flag);
  tb = typ(b);
  if (tb == t_INT) return sfcont(x,x,itos(b));
  if (! is_matvec_t(tb)) err(typeer,"contfrac0");

  lb=lg(b); if (lb==1) return cgetg(1,t_VEC);
  if (tb != t_MAT) return sfcont2(b,x,flag);
  if (lg(b[1])==1) return sfcont(x,x,flag);
  y = (GEN) gpmalloc(lb*sizeof(long));
  for (i=1; i<lb; i++) y[i]=coeff(b,1,i);
  x = sfcont2(y,x,flag); free(y); return x;
}

GEN
sfcont(GEN x, GEN x1, long k)
{
  long  av,lx=lg(x),tx=typ(x),e,i,j,l,l1,lx1,tetpil,f;
  GEN   y,p1,p2,p3,p4,yp;

  if (is_scalar_t(tx))
  {
    if (gcmp0(x)) { y=cgetg(2,t_VEC); y[1]=zero; return y; }
    switch(tx)
    {
      case t_INT:
        y=cgetg(2,t_VEC); y[1]=lcopy(x); return y;
      case t_REAL:
        l=avma; p1=cgetg(3,t_FRACN);
      p2 = rcopy(x); settyp(p2,t_INT); setlgefint(p2,lx);
      p1[1] = (long) p2; e = bit_accuracy(lx)-1-expo(x);
      if (e<0) err(talker,"integral part not significant in scfont");

      l1 = (e>>TWOPOTBITS_IN_LONG)+3; p2=cgeti(l1);
      p2[1]= evalsigne(1)|evallgefint(l1);
      p2[2]= (1L<<(e&(BITS_IN_LONG-1)));
      for (i=3; i<l1; i++) p2[i]=0;
      p1[2]=(long) p2; p3=cgetg(3,t_FRACN);
      p3[2]=lcopy(p2);
      p3[1]=laddsi(signe(x),(GEN)p1[1]);
      p1=sfcont(p1,p1,k); tetpil=avma;
      return gerepile(l,tetpil,sfcont(p3,p1,k));

      case t_FRAC: case t_FRACN:
        l=avma; lx1=lgefint(x[2]);
      l1 = (long) ((double) BYTES_IN_LONG/4.0*46.093443*(lx1-2)+3);
        if (k>0 && ++k > 0 && l1 > k) l1 = k; /* beware overflow */
      if (l1>LGBITS) l1=LGBITS;
      if (lgefint(x[1]) >= lx1) p1=icopy((GEN)x[1]);
      else affii((GEN)x[1],p1=cgeti(lx1));
      p2=icopy((GEN)x[2]); lx1=lg(x1);
      y=cgetg(l1,t_VEC); f=(x!=x1); i=0;
      while (!gcmp0(p2) && i<=l1-2)
      {
        i++; y[i]=ldvmdii(p1,p2,&p3);
        if (signe(p3)>=0) affii(p3,p1);
          else
        {
          p4=addii(p3,p2); affii(p4,p1); cgiv(p4);
          y[1]=laddsi(-1,(GEN)y[1]);
        }
        cgiv(p3); p4=p1; p1=p2; p2=p4;
        if (f)
          {
          if (i>=lx1) { i--; break; }
            if (!egalii((GEN)y[i],(GEN)x1[i]))
            {
              av=avma;
              if (gcmp1(absi(subii((GEN)x1[i],(GEN)y[i]))))
              {
                if (i>=lx1-1 || !gcmp1((GEN)x1[i+1]))
                  affii((GEN)x1[i],(GEN)y[i]);
              }
              else i--;
              avma=av; break;
            }
          }
      }
      if (i>=2 && gcmp1((GEN)y[i]))
      {
        cgiv((GEN)y[i]); --i; cgiv((GEN)y[i]);
        y[i]=laddsi(1,(GEN)y[i]);
      }
      setlg(y,i+1); return gerepileupto(l, y);
    }
    err(typeer,"sfcont");
  }

  switch(tx)
  {
    case t_POL:
      y=cgetg(2,t_VEC); y[1]=lcopy(x); break;
    case t_SER:
      av=avma; p1=gtrunc(x); tetpil=avma;
      y=gerepile(av,tetpil,sfcont(p1,p1,k)); break;
    case t_RFRAC:
    case t_RFRACN:
      av=avma; l1=lgef(x[1]); if (lgef(x[2])>l1) l1=lgef(x[2]);
      if (k>0 && l1>k+1) l1=k+1;
      yp=cgetg(l1,t_VEC); p1=(GEN)x[1]; i=0; p2=(GEN)x[2];
      while (!gcmp0(p2) && i<=l1-2)
      {
      i++; yp[i]=ldivres(p1,p2,&p3);
      p1=p2; p2=p3;
      }
      tetpil=avma; y=cgetg(i+1,t_VEC);
      for (j=1; j<=i; j++) y[j]=lcopy((GEN)yp[j]);
      y=gerepile(av,tetpil,y); break;
    default: err(typeer,"sfcont");
      return NULL; /* not reached */
  }
  return y;
}

static GEN
sfcont2(GEN b, GEN x, long k)
{
  long l1 = lg(b), tx = typ(x), i,tetpil, av = avma;
  GEN y,p1;

  if (k)
  {
    if (k>=l1) err(typeer,"sfcont2");
    l1 = k+1;
  }
  y=cgetg(l1,t_VEC);
  if (l1==1) return y;
  if (is_scalar_t(tx))
  {
    if (!is_intreal_t(tx) && !is_frac_t(tx)) err(typeer,"sfcont2");
  }
  else if (tx == t_SER) x = gtrunc(x);

  if (!gcmp1((GEN)b[1])) x = gmul((GEN)b[1],x);
  i = 2; y[1] = lfloor(x); p1 = gsub(x,(GEN)y[1]);
  for (  ; i<l1 && !gcmp0(p1); i++)
  {
    x = gdiv((GEN)b[i],p1);
    if (tx == t_REAL)
    {
      long e = expo(x);
      if (e>0 && (e>>TWOPOTBITS_IN_LONG)+3 > lg(x)) break;
    }
    y[i] = lfloor(x);
    p1 = gsub(x,(GEN)y[i]);
  }
  setlg(y,i); tetpil=avma;
  return gerepile(av,tetpil,gcopy(y));
}

GEN
pnqn(GEN x)
{
  long av=avma,tetpil,lx,ly,tx=typ(x),i;
  GEN y,p0,p1,q0,q1,a,b,p2,q2;

  if (! is_matvec_t(tx)) err(typeer,"pnqn");
  lx=lg(x); if (lx==1) return idmat(2);
  p0=gun; q0=gzero;
  if (tx != t_MAT)
  {
    p1=(GEN)x[1]; q1=gun;
    for (i=2; i<lx; i++)
    {
      a=(GEN)x[i];
      p2=gadd(gmul(a,p1),p0); p0=p1; p1=p2;
      q2=gadd(gmul(a,q1),q0); q0=q1; q1=q2;
    }
  }
  else
  {
    ly=lg(x[1]);
    if (ly==2)
    {
      p1=cgetg(lx,t_VEC); for (i=1; i<lx; i++) p1[i]=mael(x,i,1);
      tetpil=avma; return gerepile(av,tetpil,pnqn(p1));
    }
    if (ly!=3) err(talker,"incorrect size in pnqn");
    p1=gcoeff(x,2,1); q1=gcoeff(x,1,1);
    for (i=2; i<lx; i++)
    {
      a=gcoeff(x,2,i); b=gcoeff(x,1,i);
      p2=gadd(gmul(a,p1),gmul(b,p0)); p0=p1; p1=p2;
      q2=gadd(gmul(a,q1),gmul(b,q0)); q0=q1; q1=q2;
    }
  }
  tetpil=avma; y=cgetg(3,t_MAT);
  p2=cgetg(3,t_COL); y[1]=(long)p2; p2[1]=lcopy(p1); p2[2]=lcopy(q1);
  p2=cgetg(3,t_COL); y[2]=(long)p2; p2[1]=lcopy(p0); p2[2]=lcopy(q0);
  return gerepile(av,tetpil,y);
}

GEN
bestappr(GEN x, GEN k)
{
  long av=avma,tetpil,tx,tk=typ(k),lx,i,e;
  GEN p0,p1,p,q0,q1,q,a,y;

  if (tk != t_INT)
  {
    if (tk != t_REAL && !is_frac_t(tk))
      err(talker,"incorrect bound type in bestappr");
    k = gcvtoi(k,&e);
  }
  if (cmpis(k,1) < 0) k=gun;
  tx=typ(x); if (tx == t_FRACN) x = gred(x);
  switch(tx)
  {
    case t_INT:
      if (av==avma) return icopy(x);
      tetpil=avma; return gerepile(av,tetpil,icopy(x));

    case t_FRAC:
      if (cmpii((GEN)x[2],k) <= 0)
      {
        if (av==avma) return gcopy(x);
        tetpil=avma; return gerepile(av,tetpil,gcopy(x));
      }

    case t_REAL:
      p1=gun; p0=gfloor(x); q1=gzero; q0=gun; a=p0;
      while (gcmp(q0,k)<=0)
      {
      x = gsub(x,a);
      if (gcmp0(x)) { p1=p0; q1=q0; break; }

      x = ginv(x); a = (gcmp(x,k)<0)? gfloor(x): k;
      p = addii(mulii(a,p0),p1); p1=p0; p0=p;
        q = addii(mulii(a,q0),q1); q1=q0; q0=q;
      }
      tetpil=avma; return gerepile(av,tetpil,gdiv(p1,q1));

   case t_COMPLEX: case t_POL: case t_SER: case t_RFRAC:
   case t_RFRACN: case t_VEC: case t_COL: case t_MAT:
      lx = (tx==t_POL)? lgef(x): lg(x); y=cgetg(lx,tx);
      for (i=1; i<lontyp[tx]; i++) y[i]=x[i];
      for (   ; i<lx; i++) y[i]=(long)bestappr((GEN)x[i],k);
      return y;
  }
  err(typeer,"bestappr");
  return NULL; /* not reached */
}

/***********************************************************************/
/**                                                                   **/
/**         FUNDAMENTAL UNIT AND REGULATOR (QUADRATIC FIELDS)         **/
/**                                                                   **/
/***********************************************************************/

GEN
gfundunit(GEN x)
{
  return garith_proto(fundunit,x,1);
}

static GEN
get_quad(GEN f, GEN pol, long r)
{
  GEN y, c=(GEN)f[2], p1=(GEN)c[1], q1=(GEN)c[2];
  y=cgetg(4,t_QUAD); y[1]=(long)pol;
  y[2]=r? lsubii(p1,q1): (long)p1;
  y[3]=(long)q1; return y;
}

/* replace f by f * [a,1; 1,0] */
static void
update_f(GEN f, GEN a)
{
  GEN p1;
  p1=gcoeff(f,1,1);
  coeff(f,1,1)=laddii(mulii(a,p1), gcoeff(f,1,2));
  coeff(f,1,2)=(long)p1;

  p1=gcoeff(f,2,1);
  coeff(f,2,1)=laddii(mulii(a,p1), gcoeff(f,2,2));
  coeff(f,2,2)=(long)p1;
}

GEN
fundunit(GEN x)
{
  long av = avma, av2,tetpil,lim,r,flp,flq;
  GEN pol,y,a,u,v,u1,v1,sqd,f;

  if (typ(x) != t_INT) err(arither1);
  if (signe(x)<=0) err(arither2);
  r=mod4(x); if (r==2 || r==3) err(funder2,"fundunit");

  sqd=racine(x); av2=avma; lim=stack_lim(av2,2);
  a = shifti(addsi(r,sqd),-1);
  f=cgetg(3,t_MAT); f[1]=lgetg(3,t_COL); f[2]=lgetg(3,t_COL);
  coeff(f,1,1)=(long)a; coeff(f,1,2)=un;
  coeff(f,2,1)=un;      coeff(f,2,2)=zero;
  v = gdeux; u = stoi(r);
  for(;;)
  {
    u1=subii(mulii(a,v),u);       flp=egalii(u,u1); u=u1;
    v1=divii(subii(x,sqri(u)),v); flq=egalii(v,v1); v=v1;
    if (flq) break; a = divii(addii(sqd,u),v);
    if (flp) break; update_f(f,a);
    if (low_stack(lim, stack_lim(av2,2)))
    {
      GEN *gptr[4]; gptr[0]=&a; gptr[1]=&f; gptr[2]=&u; gptr[3]=&v;
      if(DEBUGMEM>1) err(warnmem,"fundunit");
      gerepilemany(av2,gptr,4);
    }
  }
  pol = quadpoly(x); y = get_quad(f,pol,r);
  if (!flq) v1 = y; else { update_f(f,a); v1 = get_quad(f,pol,r); }

  u1=gconj(y); tetpil=avma; y=gdiv(v1,u1);
  if (signe(y[3]) < 0) { tetpil=avma; y=gneg(y); }
  return gerepile(av,tetpil,y);
}

GEN
gregula(GEN x, long prec)
{
  return garith_proto2gs(regula,x,prec);
}

GEN
regula(GEN x, long prec)
{
  long av = avma,av2,lim,r,fl,rexp;
  GEN reg,rsqd,y,u,v,u1,v1, sqd = racine(x);

  if (signe(x)<=0) err(arither2);
  r=mod4(x); if (r==2 || r==3) err(funder2,"regula");

  rsqd = gsqrt(x,prec);
  if (egalii(sqri(sqd),x)) err(talker,"square argument in regula");

  rexp=0; reg=cgetr(prec); affsr(2,reg);
  av2=avma; lim = stack_lim(av2,2);
  u = stoi(r); v = gdeux;
  for(;;)
  {
    u1 = subii(mulii(divii(addii(u,sqd),v), v), u);
    v1 = divii(subii(x,sqri(u1)),v); fl = egalii(v,v1);
    if (fl || egalii(u,u1)) break;
    reg = mulrr(reg, divri(addir(u1,rsqd),v));
    rexp += expo(reg); setexpo(reg,0);
    u = u1; v = v1;
    if (rexp & ~EXPOBITS) err(muler4);
    if (low_stack(lim, stack_lim(av2,2)))
    {
      GEN *gptr[3]; gptr[0]=&reg; gptr[1]=&u; gptr[2]=&v;
      if(DEBUGMEM>1) err(warnmem,"regula");
      gerepilemany(av2,gptr,3);
    }
  }
  reg = gsqr(reg); setexpo(reg,expo(reg)-1);
  if (fl) reg = mulrr(reg, divri(addir(u1,rsqd),v));
  y = mplog(divri(reg,v));
  if (rexp)
  {
    u1 = mulsr(rexp, glog(gdeux, prec));
    setexpo(u1, expo(u1)+1);
    y = addrr(y,u1);
  }
  return gerepileupto(av, y);
}

/*************************************************************************/
/**                                                                     **/
/**                            CLASS NUMBER                             **/
/**                                                                     **/
/*************************************************************************/

static GEN
gclassno(GEN x)
{
  return garith_proto(classno,x,1);
}

static GEN
gclassno2(GEN x)
{
  return garith_proto(classno2,x,1);
}

GEN
qfbclassno0(GEN x,long flag)
{
  switch(flag)
  {
    case 0: return gclassno(x);
    case 1: return gclassno2(x);
    default: err(flagerr,"qfbclassno");
  }
  return NULL; /* not reached */
}

/* f^h = 1, return order(f) */
static GEN
find_order(GEN f, GEN h)
{
  GEN fh, p,e;
  long i,j,lim;

  p = decomp(h);
  e =(GEN)p[2];
  p =(GEN)p[1];
  for (i=1; i<lg(p); i++)
  {
    lim = itos((GEN)e[i]);
    for (j=1; j<=lim; j++)
    {
      GEN p1 = divii(h,(GEN)p[i]);
      fh = powgi(f,p1);
      if (!is_pm1(fh[1])) break;
      h = p1;
    }
  }
  return h;
}

static GEN
end_classno(GEN h, GEN hin, GEN forms, long lform)
{
  long i,com;
  GEN a,b,p1,q,fh,fg, f = (GEN)forms[0];

  h = find_order(f,h); /* H = <f> */
  q = ground(gdiv(hin,h)); /* approximate order of G/H */
  for (i=1; i < lform; i++)
  {
    long av = avma; 
    fg = powgi((GEN)forms[i], h);
    fh = powgi(fg, q);
    a = (GEN)fh[1];
    if (is_pm1(a)) continue;
    b = (GEN)fh[2]; p1 = fg;
    for (com=1; ; com++)
    {
      if (egalii((GEN)p1[1], a) && absi_equal((GEN)p1[2], b)) break;
      p1 = gmul(p1,fg);
    }
    if (signe(p1[2]) == signe(b)) com = -com;
    /* f_i ^ h(q+com) = 1 */
    avma = av; q = addsi(com,q);
  }
  return mulii(q,h);
}

static GEN
to_form(GEN x, long f)
{
  return redimag(primeform(x, stoi(f), 0));
}

static GEN
conductor_part(GEN x, GEN *ptD, GEN *ptreg, GEN *ptfa)
{
  long n,i,k,s=signe(x),fl2;
  GEN e,p,H,d,D,fa,reg;

  fa = auxdecomp(absi(x),1);
  e = (GEN)fa[2]; fa = (GEN)fa[1];
  n = lg(fa); d = gun;
  for (i=1; i<n; i++)
    if (mod2((GEN)e[i])) d = mulii(d,(GEN)fa[i]);
  fl2 = (mod4(x)==0); /* 4 | x */
  if (mod4(d) == 2-s) fl2 = 0;
  else
  {
    if (!fl2) err(funder2,"classno2");
    d = shifti(d,2);
  }
  H = gun; D = (s<0)? negi(d): d; /* d = abs(D) */
  /* f \prod_{p|f}  [ 1 - (D/p) p^-1 ] */
  for (i=1; i<n; i++)
  {
    k = itos((GEN)e[i]); p = (GEN)fa[i];
    if (fl2 && i==1) k -= 2; /* p = 2 */
    if (k >= 2)
    {
      H = mulii(H, subis(p, kronecker(D,p)));
      if (k>=4) H = mulii(H,gpowgs(p,(k>>1)-1));
    }
  }
  
  /* divide by [ O^* : O_K^* ] */
  if (s < 0)
  {
    reg = NULL;
    if (!is_bigint(d))
      switch(itos(d))
      {
        case 4: H = divis(H,2); break;
        case 3: H = divis(H,3); break;
      }
  } else {
    reg = regula(D,DEFAULTPREC);
    if (!egalii(x,D))
      H = divii(H, ground(gdiv(regula(x,DEFAULTPREC), reg)));
  }
  *ptfa = fa; *ptD = D; if (ptreg) *ptreg = reg;
  return H;
}


static long
two_rank(GEN x)
{
  GEN p = (GEN)decomp(absi(x))[1];
  long l = lg(p)-1;
#if 0 /* positive disc not needed */
  if (signe(x) > 0)
  {
    long i;
    for (i=1; i<=l; i++)
      if (mod4((GEN)p[i]) == 3) { l--; break; }
  }
#endif
  return l-1;
}

#define MAXFORM 11
#if 0 /* gcc-ism */
#  define _low(x) ({GEN __x=(GEN)x; modBIL(__x);})
#else
#  define _low(x) (__x = (GEN)(x), __x[lgefint(__x)-1])
#endif

/* h(x) for x<0 using Baby Step/Giant Step.
 * Assumes G is not too far from being cyclic.
 * 
 * Compute G^2 instead of G so as to kill most of the non-cyclicity */
GEN
classno(GEN x)
{
  long av = avma, av2,c,lforms,k,l,i,j,j1,j2,lim,com,s, forms[MAXFORM];
  long r2;
  GEN a,b,count,index,tabla,tablb,hash,p1,p2,hin,h,f,fh,fg,ftest;
  GEN Hf,D,fa;
  byteptr p = diffptr;
  GEN __x;

  if (typ(x) != t_INT) err(arither1);
  s=signe(x); if (s>=0) return classno2(x);

  k=mod4(x); if (k==1 || k==2) err(funder2,"classno");
  if (cmpis(x,-12) >= 0) return gun;

  Hf = conductor_part(x,&D,NULL,&fa);
  if (cmpis(D,-12) >= 0) return gerepileuptoint(av, icopy(Hf));

  p2 = gsqrt(absi(D),DEFAULTPREC);
  p1 = divrr(p2,mppi(DEFAULTPREC));
  p2 = gtrunc(shiftr(gsqrt(p2,DEFAULTPREC),1));
  s = 1000;
  if (signe(p2))
  {
    if (is_bigint(p2)) err(talker,"discriminant too big in classno");
    s = itos(p2); if (s < 1000) s = 1000;
  }
  r2 = two_rank(D);
  c=lforms=0;
  while (c <= s && *p)
  {
    c += *p++; k = krogs(D,c);
    if (!k) continue;

    av2 = avma;
    if (k > 0)
    {
      divrsz(mulsr(c,p1),c-1, p1);
      if (lforms < MAXFORM && c > 2) forms[lforms++] = c; 
    }
    else divrsz(mulsr(c,p1),c+1, p1);
    avma = av2; 
  }
  h = hin = shifti(ground(p1), -r2);
  s = 2*itos(gceil(gpui(p1,ginv(stoi(4)),DEFAULTPREC)));
  if (s>=10000) s=10000;

  count = new_chunk(256); for (i=0; i<=255; i++) count[i]=0;
  index = new_chunk(257);
  tabla = new_chunk(10000);
  tablb = new_chunk(10000);
  hash  = new_chunk(10000);
  f = gsqr(to_form(D, forms[0]));
  p1 = fh = powgi(f, h);
  for (i=0; i<s; i++)
  {
    tabla[i] = _low(p1[1]);
    tablb[i] = _low(p1[2]);
    count[tabla[i]&255]++; p1=compimag(p1,f);
  }
  index[0]=0; for (i=0; i<=254; i++) index[i+1] = index[i]+count[i];
  for (i=0; i<s; i++) hash[index[tabla[i]&255]++]=i;
  index[0]=0; for (i=0; i<=255; i++) index[i+1] = index[i]+count[i];

  fg=gpuigs(f,s); av2=avma; lim=stack_lim(av2,2);
  ftest=gpuigs(p1,0);
  for (com=0; ; com++)
  {
    a = (GEN)ftest[1]; k = _low(a);
    b = (GEN)ftest[2]; l = _low(b); j = k&255;
    for (j1=index[j]; j1 < index[j+1]; j1++)
    {
      j2 = hash[j1];
      if (tabla[j2] == k && tablb[j2] == l)
      {
        p1 = gmul(gpuigs(f,j2),fh);
        if (egalii((GEN)p1[1], a) && absi_equal((GEN)p1[2], b))
        { /* p1 = ftest or ftest^(-1), we are done */
          if (signe(p1[2]) == signe(b)) com = -com;
          h = addii(addis(h,j2), mulss(s,com));
          forms[0] = (long)f;
          for (i=1; i<lforms; i++)
            forms[i] = (long)gsqr(to_form(D, forms[i]));
          h = end_classno(h,hin,forms,lforms);
          h = mulii(h,Hf);
          return gerepileuptoint(av, shifti(h, r2));
        }
      }
    }
    ftest = gmul(ftest,fg);
    if (is_pm1(ftest[1])) err(impl,"classno with too small order");
    if (low_stack(lim, stack_lim(av2,2))) ftest = gerepileupto(av2,ftest);
  }
}

/* use Euler products */
GEN
classno2(GEN x)
{
  ulong av = avma;
  long n,i,k,s = signe(x);
  GEN p1,p2,p3,p4,p5,p7,Hf,Pi,reg,logd,d,D;

  if (typ(x) != t_INT) err(arither1);
  if (!s) err(arither2);
  if (s < 0 && cmpis(x,-12) >= 0) return gun;

  Hf = conductor_part(x, &D, &reg, &p1);
  if (s < 0 && cmpis(D,-12) >= 0)
    return gerepileuptoint(av, icopy(Hf)); /* |D| < 12*/

  Pi = mppi(DEFAULTPREC);
  d = absi(D); logd = glog(d,DEFAULTPREC);
  p1 = mpsqrt(gdiv(gmul(d,logd), gmul2n(Pi,1)));
  if (s > 0)
  {
    p2 = subsr(1, gmul2n(divrr(mplog(reg),logd),1));
    if (gcmp(gsqr(p2), divsr(2,logd)) >= 0) p1 = gmul(p2,p1);
  }
  p1 = gtrunc(p1);
  if (is_bigint(p1)) err(talker,"discriminant too large in classno");
  n = itos(p1);

  p4 = divri(Pi,d); p7 = ginv(mpsqrt(Pi));
  p1 = gsqrt(d,DEFAULTPREC); p3 = gzero;
  if (s > 0)
  {
    for (i=1; i<=n; i++)
    {
      k = krogs(D,i); if (!k) continue;
      p2 = mulir(mulss(i,i),p4);
      p5 = subsr(1,mulrr(p7,incgam3(ghalf,p2,DEFAULTPREC)));
      p5 = addrr(divrs(mulrr(p1,p5),i), eint1(p2,DEFAULTPREC));
      p3 = (k>0)? addrr(p3,p5): subrr(p3,p5);
    }
    p3 = shiftr(divrr(p3,reg),-1);
  }
  else
  {
    p1 = gdiv(p1,Pi);
    for (i=1; i<=n; i++)
    {
      k = krogs(D,i); if (!k) continue;
      p2 = mulir(mulss(i,i),p4);
      p5 = subsr(1,mulrr(p7,incgam3(ghalf,p2,DEFAULTPREC)));
      p5 = addrr(p5, divrr(divrs(p1,i),mpexp(p2)));
      p3 = (k>0)? addrr(p3,p5): subrr(p3,p5);
    }
  }
  return gerepileuptoint(av, mulii(Hf,ground(p3)));
}

GEN
hclassno(GEN x)
{
  long d,a,b,h,b2,f;

  d= -itos(x); if (d>0 || (d & 3) > 1) return gzero;
  if (!d) return gdivgs(gun,-12);
  if (-d > (VERYBIGINT>>1))
    err(talker,"discriminant too big in hclassno. Use quadclassunit");
  h=0; b=d&1; b2=(b-d)>>2; f=0;
  if (!b)
  {
    for (a=1; a*a<b2; a++)
      if (b2%a == 0) h++;
    f = (a*a==b2); b=2; b2=(4-d)>>2;
  }
  while (b2*3+d<0)
  {
    if (b2%b == 0) h++;
    for (a=b+1; a*a<b2; a++)
      if (b2%a == 0) h+=2;
    if (a*a==b2) h++;
    b+=2; b2=(b*b-d)>>2;
  }
  if (b2*3+d==0)
  {
    GEN y = cgetg(3,t_FRAC);
    y[1]=lstoi(3*h+1); y[2]=lstoi(3); return y;
  }
  if (f) return gaddsg(h,ghalf);
  return stoi(h);
}

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