Pentagon Pentagon In geometry, a pentagon (from the Greek πέντε "pente" and γωνία "gonia", meaning "five" and "angle") is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting. A self-intersecting "regular pentagon" (or "star pentagon") is called a pentagram. A "regular pentagon" has Schläfli symbol {5} and interior angles are 108°. A "regular pentagon" has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its
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: