In short, the ABC Conjecture states that “in a simple addition equation ($a + b = c$), it is extremely rare for the numbers involved to be made by multiplying the same small prime numbers over and over again.” Proposed in 1985, it has long been considered one of the ultimate master-puzzles in modern mathematics, exploring the profound relationship between addition… Read More »
The clash over “Corollary 3.12” is not a simple disagreement over a calculation error. Instead, it is a profound conflict between two fundamentally different mathematical visions regarding what the true identity of a “number” actually is. Here is a deeper look into the core of this dispute. 1. The Critics’ Stance (Prof. Scholze et al.): “A Logical Collapse”… Read More »
simple algorithm The Euclidean algorithm is a classical method for computing the greatest common divisor (GCD) of two non‑negative integers.The key idea is based on the following property: For any integers a and b (with a ≥ b), the GCD of a and b is the same as the GCD of b and a mod b. Because the remainder becomes strictly smaller each step, the process terminates after only a few iterations, even… Read More »
Simple Python Example The code below shows a recursive DFS on an undirected graph represented as an adjacency list.Vertices are numbered starting from 0. Expected output Iterative (stack‑based) version If you prefer an explicit stack (e.g., to avoid Python’s recursion limit), here’s a compact iterative version: Both implementations produce the same visitation order; choose the style that best… Read More »
Fourier analysis is a mathematical technique that represents complex signals as the sum of simple waves (sine and cosine waves) with different frequencies. Why is it important? Basic Concept Let’s take a complex waveform as an example and compare it to the timbre of a piano sound. A piano’s timbre is created by multiple strings vibrating at different… Read More »
Lengths and ratios of right-angled triangles (Basic knowledge) Trigonometric functions (sin, cos, tan, etc.) are frequently used in machine learning as tools to effectively work with angles, periods, rotations, and positions. Once the angles of a right triangle are determined, the ratio is also determined. Commercially available set squares have well-known trigonometric ratios. As shown in the figure, the… Read More »
Merge sort is an efficient sorting algorithm based on the divide-and-conquer paradigm. It recursively divides a large array into smaller sub-arrays, sorts those sub-arrays, and then merges them back together to create the final sorted array. Features: Algorithm Steps Example Let’s look at the process of sorting the array [8, 3, 1, 7, 0, 10, 2] using merge sort. Python… Read More »
What is the Fibonacci Sequence? The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. It’s defined as follows: The sequence looks like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,… Implementing with Recursion Recursion is a technique where a function calls itself to solve a… Read More »
The Levenshtein distance (also known as edit distance) is a metric for measuring the similarity between two strings. It’s defined as the minimum number of operations required to transform one string into the other using insertion, deletion, and substitution operations. We can efficiently calculate the Levenshtein distance using dynamic programming. Example: Levenshtein Distance between “kitten” and “sitting” Let’s… Read More »
Quicksort is an efficient sorting algorithm based on the divide-and-conquer paradigm. It works as follows: Example of Quicksort: Let’s consider sorting the following array [5, 2, 8, 1, 9, 4, 7]. Python Code Example Quicksort Characteristics: Pivot Selection Strategies: The choice of pivot significantly impacts Quicksort’s performance. Quicksort is widely used due to its relatively simple implementation and speed.… Read More »