Sum of angles of a triangle to zero. For a spherical triangle, the sum of the angles is greater than 180° and can be up to 540°. Specifically, the sum of the angles is where "f" is the fraction of the sphere's area which is enclosed by the triangle. Note that spherical geometry does not satisfy several of Euclid's axioms (including the parallel postulate.) Angles between adjacent sides of a triangle are referred to as "interior" angles in Euclidean and other geometries. "Exterior" angles can be also defined, and the Euclidean triangle postulate can be formulated as the exterior angle theorem. One can also consider the
Divergence of the sum of the reciprocals of the primes large , these bounds will turn out to be contradictory. Upper estimate: Lower estimate: This produces a contradiction: when , the estimates (2) and (3) cannot both hold, because . Here is another proof that actually gives a lower estimate for the partial sums; in particular, it shows that these sums grow at least as fast as . The proof is an adaptation of the product expansion idea of Euler. In the following, a sum or product taken over always represents a sum or product taken over a specified set of primes. The proof rests upon the following four inequalities: