/* This C program finds all projective planes P(a,b,c) that satisfy the conditions of 
Theorem 1.5 in "Some non-finitely generated Cox rings" by J. Gonzalez and K. Karu */

#include<stdio.h>
#include<stdlib.h>
#include<math.h>

#define N 30

int gcd(int a, int b);

void main() {
  int a, b, c, e, f, g, ab,  A, B, D, del, S1, S2;
  
  for (a=1; a<=N; a++) 		/* run over a,b,c = 1,...,N with no common divisor */
  for (b=1; b<=N; b++) {
      ab = gcd(a,b);
      for (c=1; c<=N; c++)
	if (gcd(ab,c)==1) {	/* look for a relation  (e,f,-g) */
	  B = (int) floor(sqrt(a*b/c));		/* B is a bound for g */
	  for (g=0,D=-1,f=0; g<=B&&D<=0; )	/* run g=1,...,B and f=1,2,... */
	     for(g+=ab, D=g*c-b, f=1; D>0 && D%a; D-=b, f++); /* stop when D=gc-fb is positive and divisible by a */
	  if (D>0 && g*g*c<=a*b) {	/* we found a relation with w<=1 */
	      e=D/a;
				/* find the smallest del>0 (\delta in the paper), where
				   B=b+e*del, A=a-f*del, are both  divisible by g */
	      for (B=b, A=a, del=0; del<g && (B%g || A%g); B+=e, A-=f, del++);
	      S1 = (a-A%(f*g))/(f*g)+1;	/* S1, S2 are number of lattice points in the 2nd column from the right,left */
	      S2 = (b-B%(e*g))/(e*g)+1;
	      if (S1<2) 
		printf("P(%d, %d, %d), relation=(%d, %d, %d), w=%d/%d, S1=%d, S2=%d\n",a,b,c,e,f,-g, c*g*g, a*b,S1,S2);
	      else if (S1==2 && S2==2&& 2*del != g && a<=b)
		printf("P(%d, %d, %d), relation=(%d, %d, %d), w=%d/%d, S1=%d, S2=%d\n",a,b,c,e,f,-g, c*g*g, a*b,S1,S2);
	      else if (S1==2 && S2>2 && (A%(f*g))*(S2-1)<f*g && del*(S2-1)<g && del*S2 !=g)
		 printf("P(%d, %d, %d), relation=(%d, %d, %d), w=%d/%d, S1=%d, S2=%d\n",a,b,c,e,f,-g, c*g*g, a*b,S1,S2);
	  }
	}
    }
}
 
int gcd(int a, int b) {
  if (a) 
    return gcd(b%a,a);
  else
    return b;
}