Euclid’s Elements, Book IX, Proposition 20. Theorem : Prime numbers are more than any assigned multitude of prime numbers. That is, there are infinitely many primes. Proof : Suppose that there are primes : Let [more…]
In The Basel Problem we saw that Euler answered the question by showing that the sum of the reciprocals of the square numbers is equal to . That is The partial sums slowly converge to [more…]
Euler gave us this jewel which is regarded by many as the most beautiful equation in mathematics; it has both elegance and simplicity. It is comprised of five fundamental numbers and the operations of addition, [more…]
One sometimes gets very interesting results when one turns divergent sums upside down. The Basel problem was asked by Pietro Mengoli in . He asked what is the exact summation of the reciprocals of the [more…]
Euler’s totient function counts the number of positive integers up to that are relatively prime to , where is considered to be relatively prime to all . These numbers are called the totatives of . [more…]
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