Related papers: A well-motivated proof that pi is irrational

This article introduces, informally, the substance and the spirit of Grothendieck's theory of the Picard scheme, highlighting its elegant simplicity, natural generality, and ingenious originality against the larger historical record.

Let $\xi$ be a real number and $b \ge 2$ an integer. We study the relationship between the irrationality exponent of $\xi$ and the subword complexity $p(n, \mathbf{x})$ of the $b$-ary expansion $\mathbf{x}$ of $\xi$, where $p(n,…

Glivenko's theorem says that, in propositional logic, classical provability of a formula entails intuitionistic provability of double negation of that formula. We generalise Glivenko's theorem from double negation to an arbitrary nucleus,…

We conjecture that the exceptional set in Manin's Conjecture has an explicit geometric description. Our proposal includes the rational point contributions from any generically finite map with larger geometric invariants. We prove that this…

Consider the following story: A teacher announces to her students a test for the following week, such that the test will be ``surprising''. The students use this as the basis for a ``logical derivation'' and reach a contradiction, which…

By employing certain extended classical summation theorems, several surprising \pi and other formulae are displayed.

The degree of irrationality $irr(X)$ of a $n$-dimensional complex projective variety $X$ is the least degree of a dominant rational map $X\dashrightarrow \mathbb{P}^n$. It is a well-known fact that given a product $X\times \mathbb{P}^m$ or…

When formalized, some diagonal arguments do not show the diagonal object to be impossible but rather reveal some other anomaly (e.g., that one of the relevant sets is ill-defined). This raises the possibility that some diagonal arguments…

We present a proof-theoretical study of the interpretability logic IL, providing a wellfounded and a non-wellfounded sequent calculus for IL. The non-wellfounded calculus is used to establish a cut elimination argument for both calculi. In…

There is no trivial mathematics, there are only trivial mathematicians! A mathematician is trivial if he or she believes that there exists trivial mathematics. Being a non-trivial mathematician myself, I will describe ten different proofs…

We prove a recent conjecture of Sean A. Irvine about a nonlinear recurrence, using mechanized guessing and verification. The theorem-prover Walnut plays a large role in the proof.

We propose an approach for showing rationality of an algebraic variety $X$. We try to cover $X$ by rational curves of certain type and count how many curves pass through a generic point. If the answer is $1$, then we can sometimes reduce…

We produce an infinite family of transcendental numbers which, when raised to their own power, become rational. We extend the method, to investigate positive rational solutions to the equation $x^x = \alpha$, where $\alpha$ is a fixed…

The famous G\"odel incompleteness theorem says that for every sufficiently rich formal theory (containing formal arithmetic in some natural sense) there exist true unprovable statements. Such statements would be natural candidates for being…

Halin [1965] proved that if a graph has $n$ many pairwise disjoint rays for each $n$ then it has infinitely many pairwise disjoint rays. We analyze the complexity of this and other similar results in terms of computable and proof theoretic…

We prove a general divisibility theorem that implies, e.g., that, in any group, the number of generating pairs (as well as triples, etc.) is a multiple of the order of the commutator subgroup. Another corollary says that, in any associative…

Let $b \ge 2$ be an integer and $\xi$ an irrational real number. We prove that, if the irrationality exponent of $\xi$ is equal to $2$ or slightly greater than $2$, then the $b$-ary expansion of $\xi$ cannot be `too simple', in a suitable…

Unexpectedness is a central concept in Simplicity Theory, a theory of cognition relating various inferential processes to the computation of Kolmogorov complexities, rather than probabilities. Its predictive power has been confirmed by…

Building upon ideas of the second and third authors, we prove that at least $2^{(1-\varepsilon)\frac{\log s}{\log\log s}}$ values of the Riemann zeta function at odd integers between 3 and $s$ are irrational, where $\varepsilon$ is any…

We introduce the Consistent Reasoning Paradox (CRP). Consistent reasoning, which lies at the core of human intelligence, is the ability to handle tasks that are equivalent, yet described by different sentences ('Tell me the time!' and 'What…