﻿ proposition 23
```/*****************************************************************************/
/*									     */
/*  FACTOR (a**p+b**p)/(a+b)						     */
/*  11/08/06 (dkc)							     */
/*									     */
/*  This C program determines if q is a pth power with respect to	     */
/*  (a**p + b**p)/(a+b).  q must divide a, b, a+b or a-b.  If p>3, whether   */
/*  p divides a when q divides a and q is not a pth power modulo p**2	     */
/*  is determined (simiarly for b).  Whether p divides a-b or a+b when q     */
/*  divides a-b or a+b when q is not a pth power modulo p**2 is determined.  */
/*  q should be a prime.						     */
/*									     */
/*  The output is "(a<<16)|b".  If p does not divide a when q divides a,     */
/*  then an error is indicated ("error[1]" is set to a non-zero value).      */
/*  b is treated similarly.  If p does not divide a-b or a+b when q divides  */
/*  a-b or a+b, then an error is indicated.				     */
/*									     */
/*  Note: The user must ascertain that "base" is not a pth power modulo p^2. */
/*									     */
/*****************************************************************************/
#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 shift16(unsigned int a, unsigned int b, unsigned int c, unsigned int d,
unsigned int *e);
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=7;	     // input prime
unsigned int dbeg=200;	     // starting "a" value
unsigned int dend=1;	     // ending "a" value
//unsigned int stop=0x3c1;
unsigned int base=29;	    // q value
// 1, 8 are pth powers modulus p**2 for p=3
// 1, 7, 18, 24 are pth powers modulus p**2 for p=5
// 1, 18, 19, 30, 31, 48 are pth powers modulus p**2 for p=7
// 1, 3, 9, 27, 40, 81, 94, 112, 118, 120 are pth powers
// modulus p**2 for p=11
unsigned int sumdif=1;	 // select [(a**p+b**p)/(a+b)] when "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.
unsigned int out=1;    // results copied to the output array when set

extern unsigned short table[];
extern unsigned int tmptab[];
extern unsigned int output[];
extern unsigned int error[];
extern unsigned int tmpsav;
extern unsigned int count;
extern unsigned int pcount;
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,bsave;
unsigned int S[2],T[2],U[2],X[3],Y[4],Z[4];
unsigned int BT[4],BV[4],BW[4],BX[4];
unsigned int n=0;
double sqrt2=1.4142135;
FILE *Outfp;
Outfp = fopen("out23a.dat","w");
/*********************************/
/*  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;
error[3]=0;
error[4]=0;
error[5]=0;
error[6]=0;
count=0;
pcount=0;
for (d=dbeg; d>=dend; d--) {
for (e=d-1; e>0; e--) {
//    if (e!=stop) continue;
/******************************************/
/*  check for common factors of d and e   */
/******************************************/
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 if q divides d, e, d+e or d-e   */
/******************************************/
bsave=0;
if (p!=3) {
if ((d/base)*base==d) {
bsave=d;
goto zskip;
}
if ((e/base)*base==e) {
bsave=e;
goto zskip;
}
}
if (((d+e)/base)*base==(d+e))
goto zskip;
if (((d-e)/base)*base!=(d-e))
continue;
/************************************/
/*  compute (d**p + e**p)/(d + e)   */
/************************************/
zskip: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);
}
BW[0]=Y[0];
BW[1]=Y[1];
BW[2]=Y[2];
BW[3]=Y[3];
/************************************************/
/*  look for prime factors using look-up table	*/
/************************************************/
if ((Y[0]==0)&&(Y[1]==0)) {
if (Y[2]==0)
l = 32 - lmbd(1, Y[3]);
else
l = 64 - lmbd(1, Y[2]);
j=l-(l/2)*2;
l=l/2;
l = 1 << l;
if (j==1)
l=(int)(((double)(l))*sqrt2);
l=l+1;
}
else
l=0x7fffffff;
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];
bigbigq(Y[0], Y[1], Y[2], Y[3], BT, 0, l);
BV[0]=BT[0];
BV[1]=BT[1];
BV[2]=BT[2];
BV[3]=BT[3];
if (l>65535) {
hugeprod(BT[0], BT[1], BT[2], BT[3], BX, l>>16);
shift16(BX[0], BX[1], BX[2], BX[3], BX);
hugeprod(BT[0], BT[1], BT[2], BT[3], BT, l&65535);
bigbigs(BX, BT);
}
else
hugeprod(BT[0], BT[1], BT[2], BT[3], BT, l);
if ((Y[0]!=BT[0])||(Y[1]!=BT[1])||(Y[2]!=BT[2])||(Y[3]!=BT[3])) continue;
bigresx(0, (l-1)/p, 0, l, U, base);
if ((U[0]!=0)||(U[1]!=1)) {
}
aloop:	 Y[0]=BV[0];
Y[1]=BV[1];
Y[2]=BV[2];
Y[3]=BV[3];
save[m]=l;
if (m < savsiz) m=m+1;
else {
error[0]=3;
goto bskip;
}
bigbigq(Y[0], Y[1], Y[2], Y[3], BT, 0, l);
BV[0]=BT[0];
BV[1]=BT[1];
BV[2]=BT[2];
BV[3]=BT[3];
if (l>65535) {
hugeprod(BT[0], BT[1], BT[2], BT[3], BX, l>>16);
shift16(BX[0], BX[1], BX[2], BX[3], BX);
hugeprod(BT[0], BT[1], BT[2], BT[3], BT, l&65535);
bigbigs(BX, BT);
}
else
hugeprod(BT[0], BT[1], BT[2], BT[3], BT, l);
if ((Y[0]==BT[0])&&(Y[1]==BT[1])&&(Y[2]==BT[2])&&(Y[3]==BT[3])) goto aloop;
}
/***********************************************/
/*  output prime factors satisfying criterion  */
/***********************************************/
if ((Y[0]!=0)||(Y[1]!=0))
continue;
if (Y[2]>0x3fffffff)
continue;
S[0]=Y[2];
S[1]=Y[3];
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])) {
bigresx(0, (i-1)/p, 0, i, U, base);
if ((U[0]!=0)||(U[1]!=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, p);
bigresx(T[0], T[1], S[0], S[1], U, base);
if ((U[0]==0)&&(U[1]==1)) {
if (n>outsiz) {
error[0]=6;
goto bskip;
}
if (m>0)
printf("d=%d, e=%d, m=%d \n",d,e,m+1);
output[n]=((int)(d) << 16) | (int)(e);
BT[0]=0;
BT[1]=0;
BT[2]=S[0];
BT[3]=S[1];
for (i=0; i<m; i++) {
l=save[i];
if (l>65535) {
hugeprod(BT[0], BT[1], BT[2], BT[3], BX, l>>16);
shift16(BX[0], BX[1], BX[2], BX[3], BX);
hugeprod(BT[0], BT[1], BT[2], BT[3], BT, l&65535);
bigbigs(BX, BT);
}
else
hugeprod(BT[0], BT[1], BT[2], BT[3], BT, l);
}
if ((BT[0]!=BW[0])||(BT[1]!=BW[1])||(BT[2]!=BW[2])||(BT[3]!=BW[3])) {
error[0]=7;
goto bskip;
}
if (bsave!=0) {
if ((bsave/p)*p!=bsave) {
error[1]=12;
error[2]=d;
error[3]=e;
goto bskip;
}
}
else {
if ((((d+e)/p)*p!=(d+e))&&(((d-e)/p)*p!=(d-e))) {
error[1]=12;
error[2]=d;
error[3]=e;
goto bskip;
}
}
if (out!=0) {
n=n+1;
count=count+1;
}
else {
if (m>0) {
n=n+1;
count=count+1;
}
}
if (m>0) {
error[2]=error[2]+1;
if (out!=0)
m=error[2];
else
m=1;
if (m<100) {
error[2*m+3]=d;
error[2*m+4]=e;
}
}
}
else {
}
}
else {
if (n>outsiz) {
error[0]=6;
goto bskip;
}
if (m>1)
printf("d=%d, e=%d, m=%d \n",d,e,m);
output[n]=((int)(d) << 16) | (int)(e);
BT[0]=0;
BT[1]=0;
BT[2]=0;
BT[3]=1;
for (i=0; i<m; i++) {
l=save[i];
if (l>65535) {
hugeprod(BT[0], BT[1], BT[2], BT[3], BX, l>>16);
shift16(BX[0], BX[1], BX[2], BX[3], BX);
hugeprod(BT[0], BT[1], BT[2], BT[3], BT, l&65535);
bigbigs(BX, BT);
}
else
hugeprod(BT[0], BT[1], BT[2], BT[3], BT, l);
}
if ((BT[0]!=BW[0])||(BT[1]!=BW[1])||(BT[2]!=BW[2])||(BT[3]!=BW[3])) {
error[0]=7;
goto bskip;
}
if (bsave!=0) {
if ((bsave/p)*p!=bsave) {
error[1]=12;
error[2]=d;
error[3]=e;
goto bskip;
}
}
else {
if ((((d+e)/p)*p!=(d+e))&&(((d-e)/p)*p!=(d-e))) {
error[1]=12;
error[2]=d;
error[3]=e;
goto bskip;
}
}
if (out!=0) {
n=n+1;
count=count+1;
}
else {
if (m>1) {
n=n+1;
count=count+1;
}
}
if (m>1) {
error[2]=error[2]+1;
if (out!=0)
m=error[2];
else
m=1;
if (m<100) {
error[2*m+3]=d;
error[2*m+4]=e;
}
}
}
}
}
bskip:
output[n]=0xffffffff;
fprintf(Outfp," error0=%d error1=%d count=%d asave=%d bsave=%d \n",error[0],
error[1],error[2],error[3],error[4]);
fprintf(Outfp," count=%d \n",n-1);
if (n!=0) {
for (i=0; i<n-1; i++)
fprintf(Outfp," %#10x \n",output[i]);
}
fclose(Outfp);
if (error[1]!=0)
printf(" error \n");
return(0);
}
```