### Representing integers as the sum of two squares

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.

Showing posts from 2017

By
Daniel Șuteu

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 overview Choose `p` and `q` as distinct prime numbers Compute `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 coprime Compute 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 a

By
Daniel Șuteu

In this post we're going to take a look at what infinitesimals are and why they are important. Infinitesimals are an abstract concept of very small values that are impossible to represent quantitatively in a finite system. # Definition We define one infinitesimal as: `ε = lim_{n to \infty}\frac{1}{n}` with the inequality: `ε > 0`. In general, the following inequalities hold true: `\frac{0}{n} < \frac{1}{n} < \frac{2}{n} < ... < \frac{n}{n}` as `n -> \infty`. # Appearance The infinitesimals appear in some fundamental limits, one of which is the limit for the natural exponentiation function: `lim_{n to \infty}(1 + \frac{\x}{n})^n = \exp(\x)` Using our infinitesimal notation, we can rewrite the limit as: `lim_{n to \infty}(1 + ε*\x)^n = \exp(\x)` where, for `x=1`, we have: `lim_{n to \infty}(1 + ε)^n = \e`. # Debate There was (and, probably, still is) a debate in mathematics whether the following limit: `lim_{n to \infty}\frac{1