Equilateral 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 angles are equal; the measure is 108 degrees). Four intersecting equal circles arranged in a closed chain
Divergence of the sum of the reciprocals of the primes induction: The first partial sum is , which has the form . If the th partial sum (for ) has the form , then the st sum is as the st prime is odd; since this sum also has an form, this partial sum cannot be an integer (because 2 divides the denominator but not the numerator), and the induction continues. Another proof rewrites the expression for the sum of the first reciprocals of primes (or indeed the sum of the reciprocals of "any" set of primes) in terms of the least common denominator, which is the product of all