﻿ proposition 39
```/*****************************************************************************/
/*									     */
/*  FACTOR (a**p+b**p)/(a+b)						     */
/*  11/27/06 (dkc)							     */
/*									     */
/*  This C program determines if a, b, and p(a+b) are pth powers w.r.t.      */
/*  (a**p+b**p)/(a+b).	p**2 must divide a, b, a-b, or a+b.		     */
/*									     */
/*  Note:  The least-residues modulo p**2 table ("residue") is dependent on  */
/*  p.	Modify the look-up table accordingly.				     */
/*									     */
/*  If a-b is not a pth power w.r.t. (a**p+b**p)/(a+b), then an error is     */
/*  indicated.								     */
/*									     */
/*  Note: Prime [(a^p+b^p)/(a+b)] (or [(a^p+b^p)/(a+b)] with small prime     */
/*  factors) are usually not checked.					     */
/*									     */
/*****************************************************************************/
#include <math.h>
#include <stdio.h>
#include "table12.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 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);
void newprod(unsigned int a, unsigned int b, unsigned int c, unsigned *d,
unsigned int e);
int main ()
{
//
// Note: The maximum "dbeg" value for p=3 is about 2^32
//	 The maximum "dbeg" value for p=5 is about 2^20
//	 The maximum "dbeg" value for p=7 is about 2^14
//	 The maximum "dbeg" value for p=11 is about 2^9
//
unsigned int p=3;	   // input prime
unsigned int dbeg=900000;	 // starting "a" value
unsigned int dend=1;	  // ending "a" value
//unsigned int stop=3627;
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=0;
// 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 residue[2]={1,8};					   // p=3
//unsigned int residue[4]={1,7,18,24};			     // p=5
//unsigned int residue[6]={1,18,19,30,31,48};			 // p=7
//unsigned int residue[10]={1,3,9,27,40,81,94,112,118,120};  // p=11

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 pcount;
unsigned int qcount;
unsigned int maxsiz=100000;
unsigned int tsize=1228;
unsigned int tmpsiz;
unsigned int outsiz=499;
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,tflag,aflag,bflag,limit;
unsigned int S[4],T[4],U[4],V[4],W[4],X[4],Y[4],Z[4];
unsigned short rem;
unsigned int ps,iters;
unsigned int n=0;
unsigned int wrap;
double sqrt2=1.4142135;
unsigned int histom[16],histop[16];
FILE *Outfp;
Outfp = fopen("out39b.dat","w");
iters=0;
for (i=0; i<16; i++) {
histom[i]=0;
histop[i]=0;
}
/*********************************/
/*  extend prime look-up table	 */
/*********************************/
ps=p*p;
tmpsiz=0;
for (i=0; i<tsize; i++) {
j = (int)(table[i]);
if (((j-1)/ps)*ps==(j-1)) {
tmptab[tmpsiz] = j;
tmpsiz=tmpsiz+1;
}
}
for (d=9975; d<(9971*9971); d++) {
if (((d-1)/ps)*ps!=(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;
}
}
printf("size=%d, prime=%d \n",tmpsiz,tmptab[tmpsiz-1]);
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;
count=0;
pcount=0;
qcount=0;
wrap=0;
for (d=dbeg; d>=dend; d--) {
if (p>5) {
if (wrap>8) {
printf("d=%d \n",d);
wrap=0;
}
else
wrap=wrap+1;
}
for (e=d-1; e>0; e--) {
//      if (e!=stop) continue;
/******************************************/
/* check if p^2 divides a, b, a+b, or a-b */
/******************************************/
if ((d/ps)*ps==d)
goto dskip;
if ((e/ps)*ps==e)
goto dskip;
if (((d-e)/ps)*ps==(d-e))
goto dskip;
if (((d+e)/ps)*ps!=(d+e))
continue;
/******************************************/
/*  check for common factors of d and e   */
/******************************************/
dskip: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   */
/***************************************/
tflag=1;
if ((d/p)*p!=d) {
flag=0;
rem=d-(d/ps)*ps;
for (l=0; l<p-1; l++) {
if (rem==residue[l])
flag=1;
}
if (flag==0)
tflag=0;
}
if ((e/p)*p!=e) {
flag=0;
rem=e-(e/ps)*ps;
for (l=0; l<p-1; l++) {
if (rem==residue[l])
flag=1;
}
if (flag==0)
tflag=0;
}
if ((((d-e)/p)*p!=(d-e))&&((d+e)/p)*p!=(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)
tflag=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)
tflag=0;
}
/************************************/
/*  compute (d**p + e**p)/(d + e)   */
/************************************/
for (i=0; i<16; i++)
save[i]=0;
Y[0]=0;
Y[1]=0;
Y[2]=0;
Y[3]=d;
for (i=0; i<p-1; i++)
newprod(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++)
newprod(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[0];
S[1]=Y[1];
S[2]=Y[2];
S[3]=Y[3];
W[0]=S[0];
W[1]=S[1];
W[2]=S[2];
W[3]=S[3];
/************************************************/
/*  look for prime factors using look-up table	*/
/************************************************/
if (S[0]!=0) {
printf("S value too big \n");
goto zskip;
}
if (S[1]!=0)
l=96-lmbd(1,S[1]);
else {
if (S[2]!=0)
l=64-lmbd(1,S[2]);
else
l=32-lmbd(1,S[3]);
}
j=l-(l/2)*2;
l=l/2;
if (l>28) {
flag=1;
k=tmpsiz-1;
goto ajump;
}
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;
}
}
ajump:m=0;
for (i=0; i<=k; i++) {
l = tmptab[i];
bigbigq(S[0],S[1],S[2],S[3],T,0,l);
V[0]=T[0];
V[1]=T[1];
V[2]=T[2];
V[3]=T[3];
if (T[0]!=0) {
printf("multiplier too large \n");
goto zskip;
}
newprod(T[1],T[2],T[3],X,l);
if ((S[0]!=X[0])||(S[1]!=X[1])||(S[2]!=X[2])||(S[3]!=X[3]))
continue;
aloop:	 S[0]=V[0];
S[1]=V[1];
S[2]=V[2];
S[3]=V[3];
save[m]=l;
if (m < savsiz) m=m+1;
else {
error[0]=3;
goto bskip;
}
bigbigq(S[0],S[1],S[2],S[3],T,0,l);
V[0]=T[0];
V[1]=T[1];
V[2]=T[2];
V[3]=T[3];
newprod(T[1],T[2],T[3],X,l);
if ((S[0]==X[0])&&(S[1]==X[1])&&(S[2]==X[2])&&(S[3]==X[3]))
goto aloop;
}
/***********************************************/
/*  output prime factors satisfying criterion  */
/***********************************************/
if ((S[0]!=0)||(S[1]!=0)||((S[2]&0xc0000000)!=0))
S[0]=S[2];
S[1]=S[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*ps) {
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, d);
if ((U[0]!=0)||(U[1]!=1))
if (T[0]<=limit) {   // largest prime in table is 71458759
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]))
quotient(T, T, p);
aflag=1;
bflag=1;
bigresx(T[0], T[1], S[0], S[1], U, d);
if ((U[0]==0)&&(U[1]==1)) {
bigresx(T[0], T[1], S[0], S[1], U, e);
if ((U[0]!=0)||(U[1]!=1))
if (sumdif!=0)
bigresx(T[0], T[1], S[0], S[1], U, p*(d+e));
else
bigresx(T[0], T[1], S[0], S[1], U, p*(d-e));
if ((U[0]!=0)||(U[1]!=1))
if (sumdif!=0)
bigresx(T[0], T[1], S[0], S[1], U, d-e);
else
bigresx(T[0], T[1], S[0], S[1], U, d+e);
if ((U[0]!=0)||(U[1]!=1))
aflag=0;
bigresx(T[0], T[1], S[0], S[1], U, p);
if ((U[0]!=0)||(U[1]!=1))
bflag=0;
for (i=0; i<m; i++) {
l=save[i];
bigresx(0, (l-1)/p, 0, l, U, d);
if ((U[0]!=0)||(U[1]!=1))
bigresx(0, (l-1)/p, 0, l, U, e);
if ((U[0]!=0)||(U[1]!=1))
if (sumdif!=0)
bigresx(0, (l-1)/p, 0, l, U, p*(d+e));
else
bigresx(0, (l-1)/p, 0, l, U, p*(d-e));
if ((U[0]!=0)||(U[1]!=1))
if (sumdif!=0)
bigresx(0, (l-1)/p, 0, l, U, d-e);
else
bigresx(0, (l-1)/p, 0, l, U, d+e);
if ((U[0]!=0)||(U[1]!=1))
aflag=0;
bigresx(0, (l-1)/p, 0, l, U, p);
if ((U[0]!=0)||(U[1]!=1))
bflag=0;
}
if (aflag==0) {
printf("not a-b: d=%d, e=%d, m=%d, flags=%d %d \n",d,e,m+1,aflag,bflag);
if ((p==3)||(p==5))
goto bskip;
}
if ((aflag==bflag)&&(m>0)) {
error[1]=11;
error[4]=error[4]+1;
//	       goto bskip;
}
histom[m+1]=histom[m+1]+1;
if ((m==1)&&(S[1]==save[0]))
histop[2]=histop[2]+1;
if ((m==2)&&(S[1]==save[0])&&(S[1]==save[1]))
histop[3]=histop[3]+1;
if ((m==3)&&(S[1]==save[0])&&(S[1]==save[1])&&(S[1]==save[2]))
histop[4]=histop[4]+1;
if (((m>0)&&(p!=3))||((m>1)&&(p==3))) {
printf("d=%d, e=%d, m=%d, flags=%d %d, f=%d, %d, %d, %d, %d, %d  %#010x %#010x \n",d,e,m+1,aflag,bflag,save[0],save[1],save[2],save[3],save[4],save[5],S[0],S[1]);
if (iters<5000) {
fprintf(Outfp,"d=%d, e=%d, m=%d, flags=%d %d, f=%d, %d, %d, %d, %d, %d  %#010x %#010x \n",d,e,m+1,aflag,bflag,save[0],save[1],save[2],save[3],save[4],save[5],S[0],S[1]);
iters=iters+1;
}
}
T[0]=0;
T[1]=0;
T[2]=S[0];
T[3]=S[1];
for (i=0; i<m; i++) {
l=save[i];
newprod(T[1],T[2],T[3],X,l);
T[0]=X[0];
T[1]=X[1];
T[2]=X[2];
T[3]=X[3];
}
if ((T[0]!=W[0])||(T[1]!=W[1])||(T[2]!=W[2])||(T[3]!=W[3])) {
error[0]=7;
goto bskip;
}
if (tflag==0) {
error[1]=7;
printf("error: d=%d, e=%d, flag=%d \n",d,e,error[1]);
goto bskip;
}
}
else {
}
}
else {
if ((S[0]!=0)||(S[1]!=0))
S[0]=S[2];
S[1]=S[3];
aflag=1;
bflag=1;
for (i=0; i<m; i++) {
l=save[i];
bigresx(0, (l-1)/p, 0, l, U, d);
if ((U[0]!=0)||(U[1]!=1))
bigresx(0, (l-1)/p, 0, l, U, e);
if ((U[0]!=0)||(U[1]!=1))
if (sumdif!=0)
bigresx(0, (l-1)/p, 0, l, U, p*(d+e));
else
bigresx(0, (l-1)/p, 0, l, U, p*(d-e));
if ((U[0]!=0)||(U[1]!=1))
if (sumdif!=0)
bigresx(0, (l-1)/p, 0, l, U, d-e);
else
bigresx(0, (l-1)/p, 0, l, U, d+e);
if ((U[0]!=0)||(U[1]!=1))
aflag=0;
bigresx(0, (l-1)/p, 0, l, U, p);
if ((U[0]!=0)||(U[1]!=1))
bflag=0;
}
if (aflag==0) {
printf("not a-b: d=%d, e=%d, m=%d, flags=%d %d \n",d,e,m,aflag,bflag);
if ((p==3)||(p==5))
goto bskip;
}
if ((aflag==bflag)&&(m>1)) {
error[1]=11;
error[4]=error[4]+1;
//	    goto bskip;
}
histom[m]=histom[m]+1;
if ((m==2)&&(save[0]==save[1]))
histop[2]=histop[2]+1;
if ((m==3)&&(save[0]==save[1])&&(save[0]==save[2]))
histop[3]=histop[3]+1;
if ((m==4)&&(save[0]==save[1])&&(save[0]==save[2])&&(save[0]==save[3]))
histop[4]=histop[4]+1;
if (((m>1)&&(p!=3))||((m>2)&&(p==3))) {
printf("d=%d, e=%d, m=%d, flags=%d %d, f=%d, %d, %d, %d, %d, %d \n",d,e,m,aflag,bflag,save[0],save[1],save[2],save[3],save[4],save[5]);
if (iters<2000) {
fprintf(Outfp,"d=%d, e=%d, m=%d, flags=%d %d, f=%d, %d, %d, %d, %d, %d \n",d,e,m,aflag,bflag,save[0],save[1],save[2],save[3],save[4],save[5]);
iters=iters+1;
}
}
S[0]=0;
S[1]=0;
S[2]=0;
S[3]=1;
for (i=0; i<m; i++) {
l=save[i];
newprod(S[1],S[2],S[3],X,l);
S[0]=X[0];
S[1]=X[1];
S[2]=X[2];
S[3]=X[3];
}
if ((S[0]!=W[0])||(S[1]!=W[1])||(S[2]!=W[2])||(S[3]!=W[3])) {
error[0]=7;
goto bskip;
}
if (tflag==0) {
error[1]=7;
printf("error: d=%d, e=%d, flag=%d \n",d,e,error[1]);
goto bskip;
}
}
}
}
bskip:
printf("\n");
fprintf(Outfp,"\n");
fprintf(Outfp," error0=%d error1=%d count=%d asave=%d bsave=%d \n",error[0],
error[1],error[2],error[3],error[4]);
printf("\n");
fprintf(Outfp,"\n");
for (i=0; i<16; i++)
fprintf(Outfp,"i=%d h=%d, %d, \n",i,histom[i],histop[i]);
zskip:
return(0);
}
```