The hottest Number Theory Substack posts right now

Simple rational numbers (like p/q) have only a few very-close different fractions with small denominators; once denominators reach q or larger you can’t get new inexact approximations that beat the 1/b^2 error threshold.
Irrational numbers, by contrast, admit infinitely many surprisingly accurate rational approximations; Dirichlet’s pigeonhole argument guarantees infinitely many fractions a/b with error on the order of 1/b^2 (for example 22/7 and 355/113 for π).
Intuitively, rationals form a uniform grid so their gaps limit how close other fractions can get, while irrationals sit inside those gaps and repeated multiples plus the pigeonhole principle produce arbitrarily close rational hits, which is the essence of Diophantine approximation.

Matula arborification is a recursive recipe that turns any positive integer into a rooted forest: 1 is the empty forest, 2 is a single node, primes become trees by attaching a new root to the forest of their index, and composites are represented by juxtaposing the trees of their prime factors.
This correspondence is useful in number theory and combinatorics — it can help prove relationships between primes and encodes integer sequences (for example the primeth sequence appears as vertical chains of trees).
The idea also has practical applications in chemistry for canonically labeling alkane structures (with valence limits ruling out some forests), and there are online tools that generate and visualize Matula trees for given integers.

You can build a graph by placing n vertices in a cycle and linking them according to the rank order of the first n terms of a real sequence, and as n grows these sequence graphs reveal striking geometric patterns.
Graphs coming from the Kronecker sequence (multiples of the golden ratio mod 1) can be drawn on a torus without crossings, typically after removing the edge from n−1 to 0.
Graphs from the van der Corput sequence embed into the Chamanara surface — a highly singular, infinite‑handle (“Loch Ness monster”) surface made by identifying shrinking boundary segments of a square — and finite approximations avoid the worst singularities so they can be visualized.

The Fibonacci sequence has a simple closed-form (Binet's) formula that uses the golden ratio phi (φ) and its conjugate psi (ψ) to compute Fn directly. It yields exact integers even though φ and ψ are irrational.
Generating functions turn the recurrence into the rational function F(x)=x/(1-x-x^2), and partial fraction decomposition expresses F(x) as a sum of two geometric series, which leads to Binet's formula.
The recurrence can also be encoded by a 2×2 Fibonacci matrix whose eigenvalues are φ and ψ; diagonalizing that matrix and raising it to the nth power gives the same closed-form result, and computing matrix powers is often numerically more stable than directly evaluating Binet's formula.

Repunits are numbers made only of the digit 1 and can appear in different bases. For example, the number 31 can be written as 111 in base 5 and base 2.
Mersenne primes are special numbers of the form 2^p - 1 that can be prime, where p is also a prime. However, it's rare for these to actually be prime numbers.
One interesting link is between Mersenne primes and perfect numbers, which are those that equal the sum of their divisors. Each Mersenne prime corresponds to a perfect number, like how 31 corresponds to the perfect number 496.

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Imaginary numbers like $i$ are not real because there isn't a real number whose square is -1, pushing the boundaries of mathematical concepts beyond reality.
The rule stating that multiplying two negative numbers results in a positive number, like $(-1) * (-1) = +1$, is a construct by mathematicians to maintain consistency in arithmetic.
Mathematicians create and manipulate rules in pure math to explore interesting results and sometimes stumble upon practical applications, demonstrating the power of abstraction in mathematics.

The book "A computational introduction to number theory and algebra" is recommended as an excellent resource for those interested in number theory, algebra, and cryptography, particularly from a computer science perspective.
The book emphasizes computational aspects, presents algorithms, and discusses complexity analysis, making it a valuable resource for cryptography applications.
The author has created a solutions manual for some chapters of the book, focusing on exercises related to basic properties of integers, congruences, and computing with large integers.

Check for divisibility between two numbers by comparing their multiples with another common multiple, often using integers.
Composite integers have prime divisors that are less than or equal to the square root of the composite integer.
The number of multiples of a given integer within a specified interval can be calculated using floor functions and division.

26 is a unique natural number sandwiched between a square and a cube, a discovery by Fermat, a notable French mathematician.
Euler's proof on Fermat's equation $y^2 = x^3 - 2$ showcases the power of abstract algebra and group theory in solving complex mathematical problems.
Understanding algebraic structures like groups, rings, and unique factorization plays a crucial role in various fields, from cryptography to machine learning.