﻿ Vandiver's theorem
```/*****************************************************************************/
/*									     */
/*  FACTOR (a**p+b**p)/(a+b)						     */
/*  11/27/06 (dkc)							     */
/*									     */
/*  This C program finds a and b such that (a**p + b**p)/(a + b) is a cube   */
/*  or p times a cube.	The prime factors of [(a**p+b**p)/(a+b)] must be of  */
/*  the form p**2*k+1.	p is set to 3.	Only a and b where (a**p+b**p)/(a+b) */
/*  has one distinct prime factor are output.				     */
/*									     */
/*  The output is "a, b, split".  If 2p does not divide a, b, a-b, or a+b,   */
/*  then "split" is set to 1.  If a is even, p divides a, and p**2 does not  */
/*  divide a, then an error is indicated ("error[1]" is set to 7).  b and    */
/*  a-b are treated similarly.	If a+b is even, p divides a+b, and p**3      */
/*  does not divide a+b, then an error is indicated ("error[1]" is set to    */
/*  7).  If a+b is odd, p divides a+b, and p**2 divides a+b, then an error   */
/*  is indicated ("error[1]" is set to 7).                                   */
/*									     */
/*  If a is even, p does not divide a, and (a/2)**(p-1) is not congruent to  */
/*  1 modulo p**3, then an error is indicated ("error[3]" is set to 9).      */
/*  b is treated similarly.						     */
/*									     */
/*  If a is odd, p divides a+b, and a**(p-1) is not congruent to 1 modulo    */
/*  p**3, then an error is indicated ("error[3]" is set to 9).  b and a-b    */
/*  are treated similarly.						     */
/*									     */
/*  If p**3 divides a, b, a-b or p**4 divides a+b, 2 does not divide a, and  */
/*  a**(p-1) is not congruent to 1 modulo p**3, then an error is indicated   */
/*  ("error[2]" is set to 8). b and a-b are treated similarly.  If p**3      */
/*  divides a, b, or a-b or p**4 divides a+b, 2 does not divide a+b, p does  */
/*  not divide a+b, and (a+b)**(p-1) is not congruent to 1 modulo p**3,      */
/*  then an error is indicated.  If p**3 divides a, b, or a-b or p**4	     */
/*  divides a+b, 2 does not divide a+b, p divides a+b, and [(a+b)/p]**(p-1)  */
/*  is not congruent to 1 modulo p**3, then an error is indicated.	     */
/*									     */
/*  Proposition (28) can be verified by examining the output and determining */
/*  if p**3 divides a', b', or a'-b' or p**4 divides a'+b' for some          */
/*  representation [((a')**p+(b')**p)/(a'+b')] of T**p.  Use "prop28c.c"     */
/*  to group the representations of T**p and factor a, b, a-b, and a+b (the  */
/*  output of this program is input to "prop28c.c").                         */
/*									     */
/*  Note:  If p**4 divides a+b and [(a**p+b**p)/(a+b)] has only one distinct */
/*  factor, 2 does not divide a already implies that a**(p-1) is congruent   */
/*  to 1 modulo p**3 (similarly for b and a-b).  If p**3 divides a, b, or    */
/*  a-b and [(a**p+b**p)/(a+b)] has only one distinct prime factor, then     */
/*  p**3 divides a, b, or a-b for the other two representations (although    */
/*  this is probably not true for p>3).  The stipulation that p**3 divide    */
/*  a', b', or a'-b' or p**4 divide a'+b' is then really only useful when    */
/*  p**4 divides a'+b' (which is enough).  If p*T**p has two distinct prime  */
/*  factors, then [(a+b)/p]**(p-1) is not congruent to 1 modulo p**3 when    */
/*  p**4 divides a'+b' for the other group of three representations.         */
/*									     */
/*  Note:  If 2 does not divide a, the condition that p**3 divide a, b, or   */
/*  a-b or p**4 divide a+b is needed in case 2*p divides b or a-b	     */
/*  (similarly for b and a-b).						     */
/*									     */
/*****************************************************************************/
#include <math.h>
#include <stdio.h>
#include "table0b.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=10000;   // starting "a" value
unsigned int dend=1;	   // ending "a" value
//unsigned int stop=0x2e831;
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 int table3[];
extern unsigned int output[];
extern unsigned int error[];
unsigned int t3size=848;
unsigned int outsiz=1999;
unsigned int n=0;
unsigned int d,e,a,b,temp;
unsigned int i,j,k,l,lp,m,wrap;
unsigned int ps,pc,pf,qflag,flag,split;
unsigned int S[2],T[2],V[2],X[3];
double croot2,croot4,halfcr4;
FILE *Outfp;
Outfp = fopen("out28a.dat","w");
croot2=1.259921;
croot4=1.587401;
halfcr4=croot4*((double)(0.5));
/***********************************/
/*  factor (d**p + e**p)/(d + e)   */
/***********************************/
ps=p*p;
pc=ps*p;
pf=pc*p;
error[0]=0;	// clear error array
error[1]=0;
error[2]=0;
error[3]=0;
error[4]=0;
wrap=1;
for (d=dbeg; d>=dend; d--) {
if (wrap>99) {
printf("a=%d \n",d);
wrap=1;
}
else
wrap=wrap+1;
for (e=d-1; e>0; e--) {
dummy(d,e,2);
//    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;
/*******************************/
/* check if 2 and p are split  */
/********************************/
split=1;
if ((d/2)*2==d)
if ((d/p)*p==d)
split=0;
if ((e/2)*2==e)
if ((e/p)*p==e)
split=0;
if (((d-e)/2)*2==(d-e))
if (((d-e)/p)*p==(d-e))
split=0;
if (((d+e)/2)*2==(d+e))
if (((d+e)/p)*p==(d+e))
split=0;
/**********************************************************/
/* check if p**3 divides a, b, or a-b or p**4 divides a+b */
/**********************************************************/
qflag=0;
if ((d/pc)*pc==d)
qflag=1;
if ((e/pc)*pc==e)
qflag=1;
if (sumdif==1) {
if (((d+e)/pf)*pf==(d+e))
qflag=1;
if (((d-e)/pc)*pc==(d-e))
qflag=1;
}
if (sumdif==0) {
if (((d-e)/pf)*pf==(d-e))
qflag=1;
if (((d+e)/pc)*pc==(d+e))
qflag=1;
}
/************************************/
/*  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/3)*3;
l=l/3;
l = 1 << l;
if (j==0)
lp=(int)(((double)(l))*halfcr4);
if (j==1) {
lp=l;
l=(int)(((double)(l))*croot2);
}
if (j==2){
lp=(int)(((double)(l))*croot2);
l=(int)(((double)(l))*croot4);
}
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/3)*3!=m)
else
break;
}
if ((m/3)*3!=m) continue;
if ((S[0]!=0) || (S[1]!=1)) continue;
//
// p**2
//
if ((d/2)*2==d) {
if ((d/p)*p==d) {
if ((d/ps)*ps!=d) {
error[1]=7;
goto bskip;
}
}
}
if ((e/2)*2==e) {
if ((e/p)*p==e) {
if ((e/ps)*ps!=e) {
error[1]=7;
goto bskip;
}
}
}
if (sumdif!=0) {
if (((d-e)/2)*2==(d-e)) {
if (((d-e)/p)*p==(d-e)) {
if (((d-e)/ps)*ps!=(d-e)) {
error[1]=7;
goto bskip;
}
}
}
}
else {
if (((d+e)/2)*2==(d+e)) {
if (((d+e)/p)*p==(d+e)) {
if (((d+e)/ps)*ps!=(d+e)) {
error[1]=7;
goto bskip;
}
}
}
}
if (sumdif!=0) {
if (((d+e)/2)*2==(d+e)) {
if (((d+e)/p)*p==(d+e)) {
if (((d+e)/pc)*pc!=(d+e)) {
error[1]=7;
goto bskip;
}
}
}
if (((d+e)/2)*2!=(d+e)) {
if (((d+e)/p)*p==(d+e)) {
if (((d+e)/ps)*ps==(d+e)) {
error[1]=7;
goto bskip;
}
}
}
}
else {
if (((d-e)/2)*2==(d-e)) {
if (((d-e)/p)*p==(d-e)) {
if (((d-e)/pc)*pc!=(d-e)) {
error[1]=7;
goto bskip;
}
}
}
if (((d-e)/2)*2!=(d-e)) {
if (((d-e)/p)*p==(d-e)) {
if (((d-e)/ps)*ps==(d-e)) {
error[1]=7;
goto bskip;
}
}
}
}
//
// p**3
//
if (qflag!=0) {
if ((d/2)*2!=d) {
flag=0;
if (((d-1)/pc)*pc==(d-1))
flag=1;
if (((d+1)/pc)*pc==(d+1))
flag=1;
if (flag==0)
error[2]=8;
}
if ((e/2)*2!=e) {
flag=0;
if (((e-1)/pc)*pc==(e-1))
flag=1;
if (((e+1)/pc)*pc==(e+1))
flag=1;
if (flag==0)
error[2]=8;
}
if (sumdif!=0) {
if (((d-e)/2)*2!=(d-e)) {
flag=0;
if ((((d-e)-1)/pc)*pc==((d-e)-1))
flag=1;
if ((((d-e)+1)/pc)*pc==((d-e)+1))
flag=1;
if (flag==0)
error[2]=8;
}
if (((d+e)/2)*2!=(d+e)) {
if (((d+e)/p)*p==(d+e)) {
flag=0;
if (((((d+e)/p)-1)/pc)*pc==(((d+e)/p)-1))
flag=1;
if (((((d+e)/p)+1)/pc)*pc==(((d+e)/p)+1))
flag=1;
if (flag==0)
error[2]=8;
}
else {
flag=0;
if ((((d+e)-1)/pc)*pc==((d+e)-1))
flag=1;
if ((((d+e)+1)/pc)*pc==((d+e)+1))
flag=1;
if (flag==0)
error[2]=8;
}
}
}
else {
if (((d+e)/2)*2!=(d+e)) {
flag=0;
if ((((d+e)-1)/pc)*pc==((d+e)-1))
flag=1;
if ((((d+e)+1)/pc)*pc==((d+e)+1))
flag=1;
if (flag==0)
error[2]=8;
}
if (((d-e)/2)*2!=(d-e)) {
if (((d-e)/p)*p==(d-e)) {
flag=0;
if (((((d-e)/p)-1)/pc)*pc==(((d-e)/p)-1))
flag=1;
if (((((d-e)/p)+1)/pc)*pc==(((d-e)/p)+1))
flag=1;
if (flag==0)
error[2]=8;
}
else {
flag=0;
if ((((d-e)-1)/pc)*pc==((d-e)-1))
flag=1;
if ((((d-e)+1)/pc)*pc==((d-e)+1))
flag=1;
if (flag==0)
error[2]=8;
}
}
}
}
//
// "split" tests
//
if ((d/2)*2==d) {
if ((d/p)*p!=d) {
flag=0;
if ((((d/2)-1)/pc)*pc==((d/2)-1))
flag=1;
if ((((d/2)+1)/pc)*pc==((d/2)+1))
flag=1;
if (flag==0)
error[3]=9;
}
}
else {
if (sumdif==1) {
if (((d+e)/p)*p==(d+e)) {
flag=0;
if (((d-1)/pc)*pc==(d-1))
flag=1;
if (((d+1)/pc)*pc==(d+1))
flag=1;
if (flag==0)
error[3]=9;
}
}
else {
if (((d-e)/p)*p==(d-e)) {
flag=0;
if (((d-1)/pc)*pc==(d-1))
flag=1;
if (((d+1)/pc)*pc==(d+1))
flag=1;
if (flag==0)
error[3]=9;
}
}
}
if ((e/2)*2==e) {
if ((e/p)*p!=e) {
flag=0;
if ((((e/2)-1)/pc)*pc==((e/2)-1))
flag=1;
if ((((e/2)+1)/pc)*pc==((e/2)+1))
flag=1;
if (flag==0)
error[3]=9;
}
}
else {
if (sumdif==1) {
if (((d+e)/p)*p==(d+e)) {
flag=0;
if (((e-1)/pc)*pc==(e-1))
flag=1;
if (((e+1)/pc)*pc==(e+1))
flag=1;
if (flag==0)
error[3]=9;
}
}
else {
if (((d-e)/p)*p==(d-e)) {
flag=0;
if (((e-1)/pc)*pc==(e-1))
flag=1;
if (((e+1)/pc)*pc==(e+1))
flag=1;
if (flag==0)
error[3]=9;
}
}
}
if (sumdif==1) {
if (((d-e)/2)*2!=(d-e)) {
if (((d+e)/p)*p==(d+e)) {
flag=0;
if ((((d-e)-1)/pc)*pc==((d-e)-1))
flag=1;
if ((((d-e)+1)/pc)*pc==((d-e)+1))
flag=1;
if (flag==0)
error[3]=9;
}
}
}
else {
if (((d+e)/2)*2!=(d+e)) {
if (((d-e)/p)*p==(d-e)) {
flag=0;
if ((((d+e)-1)/pc)*pc==((d+e)-1))
flag=1;
if ((((d+e)+1)/pc)*pc==((d+e)+1))
flag=1;
if (flag==0)
error[3]=9;
}
}
}
if (n+2>outsiz) {
error[0]=6;
goto bskip;
}
output[n]=d;
output[n+1]=e;
output[n+2]=l;
output[n+3]=split;
n=n+4;
}
}
bskip:
output[n]=0xffffffff;
fprintf(Outfp," error0=%d error1=%d error2=%d error3=%d \n",error[0],
error[1],error[2],error[3]);
fprintf(Outfp," count=%d \n",(n+1)/4);
for (i=0; i<(n+1)/4; i++)
fprintf(Outfp," %#10x, %#10x, %#10x, %d, \n",output[4*i],output[4*i+1],
output[4*i+2],output[4*i+3]);
fclose(Outfp);
if ((error[1]!=0)||(error[2]!=0)||(error[3]!=0))
printf(" error \n");
return(0);
}
```