In this article we’ll be taking a look and generating “Lorem ipsum” dummy text with Python. Most of you have probably encountered dummy text before: all kinds of documents used it. If you have ever used a Google Docs template before, you have definitely seen this kind of text.

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Welcome back! In today’s article, we’ll be taking a look at the Collatz sequence. This article aims to explain the basic principles of the Collatz sequence and how to solve tasks involving it using Python programming.

The **Collatz Sequence** is an actively studied sequence, particularly because of its relationship to the **Collatz conjecture**. Let’s first explain what the **Collatz conjecture** is, before diving into the specifics about the sequence.

This particular conjecture has not been *disproved or proved* yet, so it remains a mathematical mystery in the world. The conjecture says the following:

The Collatz sequence will always eventually reach…

The Fibonacci Sequence shows up and presents itself in quite a lot of ways in mathematics and computer science/programming. This article aims to describe several ways you might see Fibonacci show up, and how to use Python to discover various aspects of the sequence.

The Fibonacci Sequence is a sequence of natural numbers, starting with `1`

. It goes like this:

`1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ....`

Do you see the pattern? The nth Fibonacci number is the sum of the n-1st and the n-2nd Fibonacci number.

This is…

In the first part of this two-article series on prime factorization, I talked about using the simple method of the **Sieve of Eratosthenes** to factor a number. At the end of that article, I hinted on a much more efficient (but more complicated as well) method: **Pollard’s Rho Algorithm**.

In case you haven’t read Part 1, you can access it here:

Last time, we talked about using the **Sieve of Eratosthenes** to find a list of primes, *then* use that list of primes to factor a number. …

There are lots of approaches to prime factorization, but which one is the best? I know best is a subjective term, but when I say best, I say the program that balances efficiency and less code.

Let’s first talk about how we will approach solving this problem. The most straightforward way is to use Python to generate a list of primes, then test the number against each of the primes in that list.

However, if we simply create a bunch of loops, the process will take a long time. We will need the make use of a mathematical algorithm.

Most…

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