OpenAI uses internal Astra model to solve ten long-standing math problems, with Lean 4 formalizations and a paper, sparking debate on AI's role in mathematics.
A mathematician finds a counterexample to the Jacobian conjecture using Claude, disproving it for n=3.
An exploration of extending Fourier series to non-periodic functions, leading to the Fourier transform concept.
Explores the n queens problem on a toroidal chessboard when n is prime, discussing solutions with queens on lines of constant slope.
Notes on the trigonometric Fourier series, including coefficient derivation and connection to linear algebra in Hilbert space.
Explains how circular and hyperbolic functions relate through rotation, showing identities like cosh(z)=cos(iz).
Explores the formula for the nth derivative of a quotient, a lesser-known counterpart to the product rule.
Explores Newton's theorem of diameters for algebraic curves, using elliptic curves as an example.
Explores a mathematical paper showing how to derive all elementary functions from just the function e^x and the constant 1.
Explores the relationship between pentagonal and triangular numbers through visual and algebraic proofs.
Explores refined mathematical bounds for the remainder term in alternating series, relevant for numerical computing.
Explores Langford's problem: arranging two copies of numbers 1-n so that k numbers separate the two k's, with solutions for n ≡ 0 or 3 mod 4.
Explores the simplification of sin(arccos(x)) and its extension to complex numbers, discussing branch cuts and analytic continuation.
Explains how to compute a square root of -1 modulo a prime p, using number theory and Python code examples.
Explores prime gaps, their mathematical merit, and their use as proof-of-work in the Gapcoin cryptocurrency.
Explores prime clusters, their mathematical definition, and their use as proof-of-work in the Riecoin cryptocurrency.
Explores bi-twin prime chains, a mathematical pattern where sequences of twin primes follow a doubling rule, with examples and Python verification code.
A new record for the largest known 'compositorial prime' has been set, a prime number with over 3.7 million digits.
Analyzes the rational approximation of log2(3) and log2(5) using continued fractions, comparing their approximation errors.
Analysis of a unique knight's tour with minimal obtuse angles, presented by Donald Knuth in his annual Christmas lecture.
