﻿ proposition 19
```/*****************************************************************************/
/*									     */
/*  FACTOR (a**p+b**p)/(a+b)						     */
/*  11/07/06 (dkc)							     */
/*									     */
/*  This C program determines if (a**p+b**p)/(a+b) is prime or p times a     */
/*  prime.  p must divide a, b, a-b, or a+b if p>3.  p must divide a, b,     */
/*  a-b or p**2 must divide a+b if p=3.  If p divides a, then q must divide  */
/*  a.	If p divides b, then q must divide b.  If p divides a-b or a+b,      */
/*  then q must divide a-b or a+b.  Whether q is a pth power modulus	     */
/*  [(a**p+b**p)/(a+b)] is determined.	p should not divide q.		     */
/*									     */
/*  The output is "(a<<16)|b".  If q is not a pth power modulus [(a**p+b**p)/*/
/*  (a+b)], then an error is indicated ("error[1]" is set to a non-zero      */
/*  value).								     */
/*									     */
/*****************************************************************************/
#include <math.h>
#include <stdio.h>
#include "table9.h"
void hugeprod(unsigned int a0, unsigned int a2, unsigned int a4,
unsigned int a6, unsigned int *product, unsigned int m0);
void dummy(unsigned int a, unsigned int b, unsigned int c);
unsigned int lmbd(unsigned int mode, unsigned int a);
void sum(unsigned int *addend, unsigned int *augend);
void differ(unsigned int *minuend, unsigned int *subtrahend);
void bigprod(unsigned int a, unsigned int b, unsigned int c, unsigned int *p);
void quotient(unsigned int *a, unsigned int *b, unsigned int);
void bigresx(unsigned int a, unsigned int b, unsigned int c, unsigned int d,
unsigned int *e, unsigned int f);
void bigbigs(unsigned int *a, unsigned int *b);
void bigbigd(unsigned int *a, unsigned int *b);
void bigbigq(unsigned int a, unsigned int b, unsigned int c, unsigned int d,
unsigned int *e, unsigned int f, unsigned int g);
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=5;	// input prime
unsigned int dbeg=500;	// starting "a" value
unsigned int dend=1;	// ending "a" value
//unsigned int stop=1;
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 base=51;	// q value

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 tsize=172;
unsigned int tmpsiz;
unsigned int outsiz=1999;
unsigned int d,e,a,b,temp;
unsigned int i,j,k,l,m;
unsigned int flag;
unsigned int S[2],T[2],U[2],X[3],Y[4],Z[4];
unsigned int ps,pb;
unsigned int n=0;
double sqrt2=1.4142135;
FILE *Outfp;
Outfp = fopen("out19.dat","w");
/*********************************/
/*  extend prime look-up table	 */
/*********************************/
for (i=0; i<tsize; i++) tmptab[i] = (int)(table[i]);
tmpsiz=tsize;
for (d=1033; d<170000; d++) {
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)(100.0 + 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;
}
}
tmpsav=tmpsiz;
/***********************************/
/*  factor (d**p + e**p)/(d + e)   */
/***********************************/
pb=p*base;
ps=p*p;
error[0]=0;	// clear error array
error[1]=0;
error[2]=0;
error[3]=0;
error[4]=0;
error[5]=0;
count=0;
for (d=dbeg; d>=dend; d--) {
for (e=d-1; e>0; e--) {
//    if (e!=stop) continue;
/******************************************/
/*  check if p divides d, e, d+e or d-e   */
/******************************************/
if ((d/pb)*pb==d)
goto zskip;
if ((e/pb)*pb==e)
goto zskip;
if (p==3) {
if (sumdif==1) {
if (((d+e)/p)*p==(d+e)) {
if (((d+e)/ps)*ps!=(d+e))
continue;
}
}
else {
if (((d-e)/p)*p==(d-e)) {
if (((d-e)/ps)*ps!=(d-e))
continue;
}
}
}
if (((d+e)/p)*p==(d+e)) {
if (((d+e)/base)*base==(d+e))
goto zskip;
if (((d-e)/base)*base==(d-e))
goto zskip;
else
continue;
}
if (((d-e)/p)*p==(d-e)) {
if (((d-e)/base)*base==(d-e))
goto zskip;
if (((d+e)/base)*base==(d+e))
goto zskip;
else
continue;
}
else
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;
/************************************/
/*  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];
/************************************************/
/*  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;
}
}
for (i=0; i<=k; i++) {
m = tmptab[i];
quotient(S, T, m);
bigprod(T[0], T[1], m, X);
if ((S[0]==X[1]) && (S[1]==X[2]))
}
/***********************************************/
/*  output prime factors satisfying criterion  */
/***********************************************/
if (flag==1) {
for (i=tmptab[tmpsiz-1]; i<=l; i+=2) {
quotient(S, T, i);
bigprod(T[0], T[1], i, X);
if ((X[1]==S[0]) && (X[2]==S[1]))
}
}
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)) {
error[1]+=1;
}
if (n>outsiz) {
error[0]=6;
n=n-1;
}
output[n]=((int)(d) << 16) | (int)(e);
n=n+1;
count=count+1;
}
}
bskip:
output[n]=-1;
fprintf(Outfp," error0=%d error1=%d \n",
error[0],error[1]);
fprintf(Outfp," count=%d \n",n);
if (n!=0) {
for (i=0; i<n-1; i++)
fprintf(Outfp," %#10x \n",output[i]);
}
if (error[1]!=0)
printf(" error \n");
fclose(Outfp);
return(0);
}
```