Posts

Pseudoprimes: construction methods

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In this post we're going to look at several methods for generating various kinds of pseudoprimes and we're also listing some open-problems that motivate the generation of such numbers.

Primality testing algorithms

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In this post we're going to look at some algorithms for checking if a given number is either prime or composite.


Prime numbers play an important role in public-key cryptography (e.g. RSA algorithm) which require fast generation of large random prime numbers that have around `600` digits (or more).

The simplest way to generate a prime number, is to pick a random odd integer and check its primality, repeating this process until a prime is found. This approach requires a fast algorithm for checking the primality of a number.

In 2002, Manindra Agrawal, Neeraj Kayal, and Nitin Saxena published the AKS primality test, unconditionally proving that the primality of a number can be checked in polynomial time, with an asymptotic time complexity of `Õ(log(n)^12)`.

Later on, in 2005, Carl Pomerance and Hendrik Lenstra improved the complexity of the AKS algorithm to run in just `Õ(log(n)^6)` steps.

While being of great theoretical importance, the AKS algorithm is not actually used in practic…

Primorial in algorithms

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In this post we present several practical algorithms based on primorials.

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.

Curiosities 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…