An Elementary Formal Proof of the Group Law on Weierstrass Elliptic Curves in Any Characteristic

Elliptic curves are fundamental objects in number theory and algebraic geometry, whose points over a field form an abelian group under a geometric addition law. Any elliptic curve over a field admits a Weierstrass model, but prior formal proofs that the addition law is associative in this model involve either advanced algebraic geometry or tedious computation, especially in characteristic two. We formalise in the Lean theorem prover, the type of nonsingular points of a Weierstrass curve over a field of any characteristic and a purely algebraic proof that it forms an abelian group

Fusible numbers and Peano Arithmetic

Inspired by a mathematical riddle involving fuses, we define the "fusible numbers" as follows: $0$ is fusible, and whenever $x,y$ are fusible with $|y-x|<1$, the number $(x+y+1)/2$ is also fusible. We prove that the set of fusible numbers, ordered by the usual order on $\mathbb R$, is well-ordered, with order type $\varepsilon_0$. Furthermore, we prove that the density of the fusible numbers along the real line grows at an incredibly fast rate: Letting $g(n)$ be the largest gap between consecutive fusible numbers in the interval $[n,\infty)$, we have $g(n)^{-1} \ge F_{\varepsilon_0}(n-c)$ for some constant $c$, where $F_\alpha$ denotes the fast-growing hierarchy. Finally, we derive some true statements that can be formulated but not proven in Peano Arithmetic, of a different flavor than previously known such statements: PA cannot prove the true statement "For every natural number $n$ there exists a smallest fusible number larger than $n$." Also, consider the algorithm "$M(x)$: if $x<0$ return $-x$, else return $M(x-M(x-1))/2$." Then $M$ terminates on real inputs, although PA cannot prove the statement "$M$ terminates on all natural inputs."Comment: Minor improvements. 26 pages, 5 figures, 3 table