Posts

Special-purpose factorization algorithms

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In this post, we will take a brief look at a nice collection of special-purpose factorization algorithms. Most of them old and well-known. Some of them new.

Partial sums of arithmetical functions

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In this post we're going to look at some very interesting generalized formulas for computing the partial sums of some arithmetical functions.

Continued fraction factorization method

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The factorization method that we'll discuss in this post, it's called the continued fraction factorization method (CFRAC), and is quite an old method, but still pretty interesting, sharing many concepts and ideas with other modern factorization methods.

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.

Representing integers as the sum of two squares

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In this post we present a recursive algorithm for finding all the possible representations, as a sum of two squares, for any given integer that can be expressed this way.

Representing integers as the difference of two squares

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Most integers can be represented as a difference of two squares, where each square is a non-negative integer.

Various representations for famous mathematical constants

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In this unusual post, much like in the older post, The beauty of Infinity, we're listing the most famous mathematical constants as representations of infinite seriesinfinite products and limits.

Thoughts on programming language notations

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Some posts ago, we looked at what it's required in creating a new programming language. In this post we're going a little bit more into it, trying to find ways to effectively express meanings in natural ways, similar to what we can express in a natural language.

Bacovia: a symbolic math library

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Named after the great symbolist poet, George Bacovia, I created this library to symbolically manipulate mathematical expressions in a simple and elegant way.

Mandelbrot set

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The Mandelbrot set and its complex beauty.

RSA algorithm

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RSA is a practical public-key cryptographic algorithm, which is widely used on modern computers to communicate securely over large distances.


The acronym of the algorithm stands for Ron Rivest, Adi Shamir and Leonard Adleman, which first published the algorithm in 1978.

# Algorithm overviewChoose `p` and `q` as distinct prime numbersCompute `n` as `n = p*q`Compute `\phi(n)` as `\phi(n) = (p-1) * (q-1)`Choose `e` such that `1 < e < \phi(n)` and `e` and `\phi(n)` are coprimeCompute the value of `d` as `d ≡ e^(-1) mod \phi(n)`Public key is `(e, n)`Private key is `(d, n)`The encryption of `m` as `c`, is `c ≡ m^e mod n`The decryption of `c` as `m`, is `m ≡ c^d mod n`
# Generating `p` and `q` In order to generate a public and a private key, the algorithm requires two distinct prime numbers `p` and `q`, which are randomly chosen and should have, roughly, the same number of bits. By today standards, it is recommended that each prime number to have at least 2048 bits.

In Perl, there is a…