﻿ proposition 37
```/*****************************************************************************/
/*									     */
/*  FACTOR (a**p+b**p)/(a+b)						     */
/*  11/27/06 (dkc)							     */
/*									     */
/*  This C program finds a and b such that every prime factor of	     */
/*  (a**p+b**p)/(a+b) other than p is of the form p**2*k+1.  p**2 must	     */
/*  divide a, b, a+b or a-b.  Whether a (or b) is a pth power modulo p**2    */
/*  when p does not divide a (or b) is determined.  Whether a+b and a-b      */
/*  are pth powers modulo p**2 when p does not divide a+b or a-b is	     */
/*  determined.  (These are the "weak" Furtwangler conditions.)              */
/*									     */
/*  Note:  The least residues modulo p**2 table ("residue") is dependent on  */
/*  p.	Modify the look-up table accordingly.				     */
/*									     */
/*  The output is "(a<<16)|b".  If the "weak" Furtwangler conditions aren't  */
/*  satisfied, then an error is indicated ("error[1]" is set to a non-zero   */
/*  value).								     */
/*									     */
/*****************************************************************************/
#include <math.h>
#include <stdio.h>
#include "table11.h"
unsigned int lmbd(unsigned int mode, unsigned int a);
void bigprod(unsigned int a, unsigned int b, unsigned int c, unsigned int *p);
void bigbigd(unsigned int *a, unsigned int *b);
void differ(unsigned int *a, unsigned int *b);
void dummy(unsigned int a, unsigned int b, unsigned int c);
void bigbigs(unsigned int *addend, unsigned int *augend);
void hugeprod(unsigned int a, unsigned int b, unsigned int c, unsigned d,
unsigned int *e, unsigned int f);
void bigbigq(unsigned int a, unsigned int b, unsigned int c, unsigned int d,
unsigned int *e, unsigned int f, unsigned int g);
void bigresx(unsigned int a, unsigned int b, unsigned int c, unsigned int d,
unsigned int *e, unsigned int f);
void quotient(unsigned int *a, unsigned int *b, unsigned int c);
int main ()
{
//
// Note: The maximum "dbeg" value for p=3 is about 5000.
//	 The maximum "dbeg" value for p=5 is about 1000.
//	 The maximum "dbeg" value for p=7 is about 250;
//	 The maximum "dbeg" value for p=11 is about 50;
//
unsigned int p=3;	     // input prime
unsigned int dbeg=1000;	     // starting "a" value
unsigned int dend=1;	     // ending "a" value
//unsigned int stop=0x44;
unsigned int sumdif=1;	     // select [(a**p+b**p)/(a+b)] if "sumdif" is non-
// zero, or [(a**p-b**p)/(a-b)] otherwise
unsigned int correct=1;
// There is a small probability that (a**p+b**p)/(a+b) is not
// completely factored if "correct" is not set to 1.

extern unsigned short residue[];
extern unsigned short table[];
extern unsigned int tmptab[];
extern unsigned int output[];
extern unsigned int error[];
extern unsigned int tmpsav;
extern unsigned int count;
unsigned int maxsiz=15600;
unsigned int tsize=303;
unsigned int tmpsiz;
unsigned int outsiz=1999;
unsigned int save[16];	 // solutions array
unsigned int savsiz=15;  // size of solutions array minus one
unsigned int d,e,a,b,temp;
unsigned int i,j,k,l,m;
unsigned int flag,limit,fflag;
unsigned short rem;
unsigned int S[2],T[2],V[2],W[2],X[3],Y[4],Z[4];
unsigned int ps;
unsigned int n=0;
double sqrt2=1.4142135;
FILE *Outfp;
Outfp = fopen("out37.dat","w");
ps=p*p;
/*********************************/
/*  extend prime look-up table	 */
/*********************************/
error[0]=0;
tmpsiz=0;
for (i=0; i<tsize; i++) {
j = (int)(table[i]);
if (((j-1)/p)*p==(j-1)) {
tmptab[tmpsiz] = j;
tmpsiz=tmpsiz+1;
}
}
for (d=2007; d<4000000; d++) {
if (((d-1)/p)*p!=(d-1))
continue;
if(d==(d/2)*2) continue;
if(d==(d/3)*3) continue;
if(d==(d/5)*5) continue;
if(d==(d/7)*7) continue;
if(d==(d/11)*11) continue;
if(d==(d/13)*13) continue;
if(d==(d/17)*17) continue;
if(d==(d/19)*19) continue;
/************************************************/
/*  look for prime factors using look-up table	*/
/************************************************/
l = (int)(2.0 + sqrt((double)d));
k=0;
if (l>table[tsize-1]) {
error[0]=1;
goto bskip;
}
else {
for (i=0; i<tsize; i++) {
if (table[i] < l) k=i;
else break;
}
}
flag=1;
l=k;
for (i=0; i<=l; i++) {
k = table[i];
if ((d/k)*k == d) {
flag=0;
break;
}
}
if (flag==1) {
tmptab[tmpsiz]=d;
tmpsiz = tmpsiz + 1;
if (tmpsiz>=maxsiz)
break;
}
}
tmpsav=tmpsiz;
limit=(tmptab[tmpsiz-1])>>16;
limit=limit*limit;
/***********************************/
/*  factor (d**p + e**p)/(d + e)   */
/***********************************/
error[1]=0;
error[2]=0;
count=0;
for (d=dbeg; d>=dend; d--) {
for (e=d-1; e>0; e--) {
//    if (e!=stop) continue;
/*********************************************/
/*  check if p**2 divides d, e, d+e or d-e   */
/*********************************************/
if ((d/ps)*ps==d)
goto zskip;
if ((e/ps)*ps==e)
goto zskip;
if (((d+e)/ps)*ps==(d+e))
goto zskip;
if (((d-e)/ps)*ps!=(d-e))
continue;
/******************************************/
/*  check for common factors of d and e   */
/******************************************/
zskip:if((d==(d/2)*2)&&(e==(e/2)*2)) continue;
if((d==(d/3)*3)&&(e==(e/3)*3)) continue;
if((d==(d/5)*5)&&(e==(e/5)*5)) continue;
if((d==(d/7)*7)&&(e==(e/7)*7)) continue;
/***********************/
/*  Euclidean G.C.D.   */
/***********************/
a=d;
b=e;
if (b>a) {
temp=a;
a=b;
b=temp;
}
loop: temp = a - (a/b)*b;
a=b;
b=temp;
if (b!=0) goto loop;
if (a!=1) continue;
/***************************************/
/*  check for Furtwangler conditions   */
/***************************************/
fflag=1;
if ((d/ps)*ps!=d) {
flag=0;
rem=d-(d/ps)*ps;
for (l=0; l<p-1; l++) {
if (rem==residue[l])
flag=1;
}
if (flag==0)
fflag=0;
}
if ((e/ps)*ps!=e) {
flag=0;
rem=e-(e/ps)*ps;
for (l=0; l<p-1; l++) {
if (rem==residue[l])
flag=1;
}
if (flag==0)
fflag=0;
}
if ((((d+e)/ps)*ps!=(d+e))&&(((d-e)/ps)*ps!=(d-e))) {
flag=0;
rem=(d+e)-((d+e)/ps)*ps;
for (l=0; l<p-1; l++) {
if (rem==residue[l])
flag=1;
}
if (flag==0)
fflag=0;
flag=0;
rem=(d-e)-((d-e)/ps)*ps;
for (l=0; l<p-1; l++) {
if (rem==residue[l])
flag=1;
}
if (flag==0)
fflag=0;
}
/************************************/
/*  compute (d**p + e**p)/(d + e)   */
/************************************/
Y[0]=0;
Y[1]=0;
Y[2]=0;
Y[3]=d;
for (i=0; i<p-1; i++)
hugeprod(Y[0], Y[1], Y[2], Y[3], Y, d);
Z[0]=0;
Z[1]=0;
Z[2]=0;
Z[3]=e;
for (i=0; i<p-1; i++)
hugeprod(Z[0], Z[1], Z[2], Z[3], Z, e);
if (sumdif==1) {
bigbigs(Y, Z);
temp=d+e;
if (((d+e)/p)*p==(d+e))
temp=temp*p;
bigbigq(Z[0], Z[1], Z[2], Z[3], Y, 0, temp);
}
else {
bigbigd(Y, Z);
temp=d-e;
if (((d-e)/p)*p==(d-e))
temp=temp*p;
bigbigq(Z[0], Z[1], Z[2], Z[3], Y, 0, temp);
}
S[0]=Y[2];
S[1]=Y[3];
W[0]=S[0];
W[1]=S[1];
/************************************************/
/*  look for prime factors using look-up table	*/
/************************************************/
if (S[0]==0)
l = 32 - lmbd(1, S[1]);
else
l = 64 - lmbd(1, S[0]);
j=l-(l/2)*2;
l=l/2;
l = 1 << l;
if (j==1)
l=(int)(((double)(l))*sqrt2);
l=l+1;
flag=0;
if (l>tmptab[tmpsiz-1]) {
flag=1;
k=tmpsiz-1;
}
else {
k=0;
for (i=0; i<tmpsiz; i++) {
if (tmptab[i] < l) k=i;
else break;
}
}
m=0;
for (i=0; i<=k; i++) {
l = tmptab[i];
quotient(S, T, l);
V[0]=T[0];
V[1]=T[1];
bigprod(T[0], T[1], l, X);
if ((S[0]!=X[1]) || (S[1]!=X[2]))
continue;
if (((l-1)/ps)*ps!=(l-1))
aloop:	 S[0]=V[0];
S[1]=V[1];
save[m]=l;
if (m < savsiz) m=m+1;
else {
error[0]=3;
goto bskip;
}
quotient(S, T, l);
V[0]=T[0];
V[1]=T[1];
bigprod(T[0], T[1], l, X);
if ((S[0]==X[1]) && (S[1]==X[2])) goto aloop;
}
/***********************************************/
/*  output prime factors satisfying criterion  */
/***********************************************/
if ((S[0]!=0) || (S[1]!=1)) {
if ((flag==1) && (correct==1)) {
if (S[0]==0)
j = (32 - lmbd(1, S[1]));
else
j = (64 - lmbd(1, S[0]));
k=j-(j/2)*2;
j=j/2;
j = 1 << j;
if (k==1)
j=(int)(((double)(j))*sqrt2);
for (i=tmptab[tmpsiz-1]; i<j; i+=2*p) {
quotient(S, T, i);
bigprod(T[0], T[1], i, X);
if ((X[1]==S[0]) && (X[2]==S[1])) {
if (((i-1)/ps)*ps!=(i-1))
if (T[0]<=limit) {   // largest prime in table is 0x126f5f
S[0]=T[0];      // for p=7
S[1]=T[1];
save[m]=i;
if (m < savsiz) m=m+1;
else {
error[0]=3;
goto bskip;
}
goto cskip;
}
else {
error[0]=4;
goto bskip;
}
}
}
}
cskip:	 T[0]=0;
T[1]=1;
differ(S, T);
quotient(T, T, ps);
bigprod(T[0], T[1], ps, X);
T[0]=0;
T[1]=1;
differ(S, T);
if ((X[1]!=T[0]) || (X[2]!=T[1]))
if (n>outsiz) {
error[0]=6;
goto bskip;
}
output[n]=((int)(d) << 16) | (int)(e);
if (m>0)
printf("d=%d, e=%d, m=%d \n",d,e,m+1);
T[0]=S[0];
T[1]=S[1];
for (i=0; i<m; i++) {
bigprod(T[0], T[1], save[i], X);
T[0] = X[1];
T[1] = X[2];
}
if ((T[0]!=W[0]) || (T[1]!=W[1])) {
error[0]=7;
goto bskip;
}
if (m>0) {
n=n+1;
count=count+1;
}
if (fflag==0) {
error[1]=8;
goto bskip;
}
}
else {
if (n>outsiz) {
error[0]=6;
goto bskip;
}
output[n]=((int)(d) << 16) | (int)(e);
if (m>1)
printf("d=%d, e=%d, m=%d \n",d,e,m);
S[0]=0;
S[1]=1;
for (i=0; i<m; i++) {
bigprod(S[0], S[1], save[i], X);
S[0] = X[1];
S[1] = X[2];
}
if ((S[0]!=W[0]) || (S[1]!=W[1])) {
error[0]=7;
goto bskip;
}
if (m>1) {
n=n+1;
count=count+1;
}
if (fflag==0) {
error[1]=8;
goto bskip;
}
}
}
}
bskip:
output[n]=-1;
fprintf(Outfp," error0=%d error1=%d \n",error[0],error[1]);
fprintf(Outfp," count=%d \n",n-1);
if (n!=0) {
for (i=0; i<n-1; i++)
fprintf(Outfp," %#10x \n",output[i]);
}
if (error[1]!=0)
printf(" error \n");
return(0);
}
```