We relate the connection between the sizes of circuits in suborbital graph for the normalizer of Γ0(m) in PSL(2,ℝ) and the congruence equations arising from related group action. We give a number theoretic result which says that all prime divisors of 3u2 ∓ 3u + 1 for any integer u must be congruent to 1 ( mod 3).
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