﻿ proposition 22
```/*****************************************************************************/
/*									     */
/*  FACTOR (a**p+b**p)/(a+b)						     */
/*  11/03/06 (dkc)							     */
/*									     */
/*  This C program finds a and b such that (a**p+b**p)/(a+b) is a square or  */
/*  p times a square.  p is set to 3.					     */
/*									     */
/*  The output is "a, b".                                                    */
/*									     */
/*****************************************************************************/
#include <math.h>
#include <stdio.h>
#include "table0a.h"
unsigned int lmbd(unsigned int mode, unsigned int a);
void dummy(unsigned int a, unsigned int b, unsigned int c);
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);
int main ()
{
unsigned int p=3;	 // input prime
unsigned int dbeg=5000;  // 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

extern unsigned short table3[];
extern unsigned int output[];
extern unsigned int error[];
extern unsigned int count;
unsigned int t3size=2556;
unsigned int outsiz=1999;
unsigned int n=0;
unsigned int d,e,a,b,temp,flag;
unsigned int i,j,k,l,lp,m;
unsigned int S[2],T[2],V[2],X[3];
double sqrt2,halfsr2;
FILE *Outfp;
Outfp = fopen("out22a.dat","w");
sqrt2=1.4142135;
halfsr2=sqrt2*((double)(0.5));
/***********************************/
/*  factor (d**p + e**p)/(d + e)   */
/***********************************/
error[0]=0;	// clear error array
count=0;
for (d=dbeg; d>=dend; d--) {
for (e=d-1; e>0; e--) {
dummy(d,e,3);
//    if (e!=stop) continue;
/*******************************/
/*  check for common factors   */
/*******************************/
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)   */
/************************************/
S[0]=0;
S[1]=d;
for (i=0; i<p-1; i++) {
bigprod(S[0], S[1], d, X);
S[0]=X[1];
S[1]=X[2];
}
T[0]=0;
T[1]=e;
for (i=0; i<p-1; i++) {
bigprod(T[0], T[1], e, X);
T[0]=X[1];
T[1]=X[2];
}
if (sumdif==1) {
sum(S, T);
temp=d+e;
if (((d+e)/p)*p==(d+e))
temp=temp*p;
quotient(T, S, temp);
}
else {
differ(S, T);
temp=d-e;
if (((d-e)/p)*p==(d-e))
temp=temp*p;
quotient(T, S, temp);
}
/************************************************/
/*  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==0)
lp=(int)(((double)(l))*halfsr2);
if (j==1) {
lp=l;
l=(int)(((double)(l))*sqrt2);
}
lp=lp-1;
l=l+1;
if (l>table3[t3size-1]) {
error[0]=5;
goto bskip;
}
else {
j=0;
for (i=0; i<t3size; i++) {
if (table3[i] < lp) j=i;
else break;
}
k=j;
for (i=j; i<t3size; i++) {
if (table3[i] < l) k=i;
else break;
}
}
for (i=j; i<=k; i++) {
m=0;
l = table3[i];
quotient(S, V, l);
bigprod(V[0], V[1], l, X);
if ((S[0]!=X[1]) || (S[1]!=X[2])) continue;
aloop:	 S[0]=V[0];
S[1]=V[1];
m=m+1;
quotient(S, V, l);
bigprod(V[0], V[1], l, X);
if ((S[0]==X[1]) && (S[1]==X[2])) goto aloop;
if (m!=2)
else
break;
}
if (m!=2) continue;
if ((S[0]!=0) || (S[1]!=1)) continue;
if (n+1>outsiz) {
error[0]=6;
goto bskip;
}
output[n]=d;
output[n+1]=e;
n=n+2;
count=count+1;