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 these primes. Then each of these primes divides all but one of the numerator terms and hence does not divide the numerator itself; but each prime "does" divide the denominator. Thus the expression is irreducible and is non-integer. 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