Equilateral pentagon and no sides intersecting others. A common example of this type of pentagon is the regular pentagon. Equilateral pentagon In geometry an equilateral pentagon is a polygon with five sides of equal length. Its five internal angles, in turn, can take a range of sets of values, thus permitting it to form a family of pentagons. The requirement is that all angles must add up to 540 degrees and must be between 0 and 360 degrees but not equal to 180 degrees. In contrast, the regular pentagon is unique, because it is equilateral and moreover it is equiangular (its five
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: