Joshua Bald

Classical plane geometry by coordinate certificates: Desargues, Monge, Simson, Napoleon and Lami in Lean 4

We formalise five classical theorems of plane geometry in Lean 4 with Mathlib: Desargues’stheorem in the projective plane, together with its converse; Monge’s theorem that the threeexternal centres of similitude of three circles are collinear; the Wallace–Simson theorem on theped...

The Cayley–Bacharach theorem for plane cubics, formalised in Lean 4, with finite-field thresholds

We formalise the Cayley–Bacharach theorem for plane cubics in Lean 4 with Mathlib: underhypotheses made precise below, if two cubics meet in nine points, every cubic through eight ofthem passes through the ninth. We prove three versions. The first assumes that the eight pointsimp...

Dandelin spheres and Marden's theorem, formalised in Lean 4

We formalise two classical theorems about conics in Lean 4 with Mathlib. The first isDandelin’s theorem. A plane that misses the apex of a right circular cone meets it in an ellipse,a parabola or a hyperbola. The spheres inscribed in the cone and tangent to the plane touch it att...

Casey's theorem and Fuss's theorem, formalised in Lean 4 with Mathlib

We report two small Lean 4 formalisations, built on Mathlib, of classical theorems aboutcircles. The first is Casey’s theorem (1866), the generalisation of Ptolemy’s theorem in which thefour points on a circle Γ are replaced by four circles tangent to Γ and distances are replaced...

Baranyai's theorem on factorisations of complete uniform hypergraphs, formalised in Lean 4 with Mathlib

For positive integers $n,k$, Baranyai's theorem (1975) says that if $k$ divides $n$, the $\binom nk$ $k$-element subsets of an$n$-element set can be split into $\binom{n-1}{k-1}$ classes, each of which is a partition of the$n$-set into $n/k$ blocks. For $k=2$ this is the classica...

Casey's theorem and Fuss's theorem, formalised in Lean 4 with Mathlib

We report two small Lean 4 formalisations, built on Mathlib, of classical theorems aboutcircles. The first is Casey’s theorem (1866), the generalisation of Ptolemy’s theorem in which thefour points on a circle Γ are replaced by four circles tangent to Γ and distances are replaced...

Dandelin spheres and Marden's theorem, formalised in Lean 4

We formalise two classical theorems about conics in Lean 4 with Mathlib. The first isDandelin’s theorem. A plane that misses the apex of a right circular cone meets it in an ellipse,a parabola or a hyperbola. The spheres inscribed in the cone and tangent to the plane touch it att...

Baranyai's theorem on factorisations of complete uniform hypergraphs, formalised in Lean 4 with Mathlib

For positive integers $n,k$, Baranyai's theorem (1975) says that if $k$ divides $n$, the $\binom nk$ $k$-element subsets of an$n$-element set can be split into $\binom{n-1}{k-1}$ classes, each of which is a partition of the$n$-set into $n/k$ blocks. For $k=2$ this is the classica...

Casey's theorem and Fuss's theorem, formalised in Lean 4 with Mathlib

We report two small Lean 4 formalisations, built on Mathlib, of classical theorems aboutcircles. The first is Casey’s theorem (1866), the generalisation of Ptolemy’s theorem in which thefour points on a circle Γ are replaced by four circles tangent to Γ and distances are replaced...

CORRELATIONS BETWEEN PRIMITIVE ROOT STATUSES OF CONSECUTIVE PRIMES

Abstract. Call a prime p an Artin prime for base 10 if 10 is a primitive root modulo p. Weinvestigate the dependence between the Artin statuses of consecutive primes. Computing Artinstatus for all 50,847,531 primes 7 ≤ p ≤ 109, we find that consecutive primes are measurablyantico...

High-Order Telescoping Sequences for Function Approximation: A Framework Using Optimized Exponential Approximants

We develop a framework for constructing telescoping sequences that approximatea class of analytic functions --- those built from exponentials by finite sumsand products --- at a fixed third-order convergence rate. The engine is afamily of \emph{optimized exponential approximants}...

QUADRATIC EXCLUSION LAWS FOR CONSECUTIVE ARTIN PRIMES IN ARBITRARY BASES

Abstract. Let a ≥ 2 be a non-square integer with squarefree part d, and say that an oddprime p is Artin base a if a is a primitive root modulo p. (We restrict to odd primes throughout:modulo 2 every odd integer trivially has full multiplicative order, and the quadratic obstructio...

QUADRATIC EXCLUSION LAWS FOR CONSECUTIVE ARTIN PRIMES IN ARBITRARY BASES

Abstract. Let a ≥ 2 be a non-square integer with squarefree part d, and say that an oddprime p is Artin base a if a is a primitive root modulo p. (We restrict to odd primes throughout:modulo 2 every odd integer trivially has full multiplicative order, and the quadratic obstructio...

CORRELATIONS BETWEEN PRIMITIVE ROOT STATUSES OF CONSECUTIVE PRIMES

Abstract. Call a prime p an Artin prime for base 10 if 10 is a primitive root modulo p. Weinvestigate the dependence between the Artin statuses of consecutive primes. Computing Artinstatus for all 50,847,531 primes 7 ≤ p ≤ 109, we find that consecutive primes are measurablyantico...