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Partial sums of arithmetical functions

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In this post I would like to present some interesting generalized formulas for computing the partial sums of some arithmetical functions.

Continued fraction factorization method

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Prime factorization of composite integers has many applications in number theory,  especially in computing many important arithmetic functions for large non-trivial inputs.

One such function is Euler's totient function `φ(n)`, which it's practically impossible to compute it for a "hard" large random composite `n`, if the prime factorization of `n` is not known.

This led to the creation of public-key cryptography systems (such as the RSA algorithm), which are systems responsible for secure communication online and rely on the assumption that it's very hard to factorize a large integer `n` that is the product of two large random prime numbers.

Interesting exercises and identities in number theory

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In this post I would like to present some interesting exercises in number theory, along with some curious formulas and identities for some number-theoretic functions.

Investigating the Fibonacci numbers modulo m

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The Fibonacci sequence is, without doubt, one of the most popular sequences in mathematics and in popular culture, named after Italian mathematician Leonardo of Pisa (also known as Fibonacci, Leonardo Bonacci, Leonardo of Pisa, Leonardo Pisano Bigollo, or Leonardo Fibonacci), who first introduced the numbers in Western European with his book Liber Abaci, in 1202.