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 Divergence of the sum of the reciprocals of the primes The sum of the reciprocals of all prime numbers diverges; that is: This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers. There are a variety of proofs of Euler's result, including a lower bound for the partial sums stating that for all natural numbers . The double natural logarithm (ln ln) indicates that the divergence might be very slow, which is indeed the case. See Meissel–Mertens constant. First, we describe how Euler originally discovered the result. He was considering