Equilateral pentagon of δ can be seen from the figure: Simplifying, δ is obtained as function of α and β: The remaining angles of the pentagon can be found geometrically: The remaining angles of the orange and blue triangles are readily found by noting that two angles of an isosceles triangle are equal while all three angles sum to 180°. Then formula_7 and the two remaining angles of the green triangle can be found from four equations stating that the sum of the angles of the pentagon is 540°, the sum of the angles of the green triangle is 180°, the angle
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: