Related papers: Legendre-Teege Reciprocity
This article proves the following theorem, first enunciated by Roger Penrose about 70 years ago but never published: In $\mathbb{R}P^{2}$, if conics are assigned to seven of the vertices of a combinatorial cube such that (i) conics…
We note that an argument by Rogers (1958) gives a proof of Vaaler's theorem (1979) about sections of the cube and allows certain generalizations of the theorem.
Given a bridgeless graph $G$, the Cycle Double Cover Conjecture posits that there is a list of cycles of $G$, such that every edge appears in exactly two cycles on the list. This conjecture was originally posed independently in 1973 by…
Between 1941 and 1962, scalar-tensor theories of gravitation were suggested four times by different scientists in four different countries. The earliest originator, the Swiss mathematician W. Scherrer, was virtually unknown until now…
An interesting episode in the history of the prime number theorem concerns a formula proposed by Legendre for counting the primes below a given bound. We point out that arithmetic bias likely played an important role in arriving at that…
We provide a simple proof of the general rational quartic reciprocity law due to Williams, Hardy and Friesen.
Quaternary Legendre pairs are pertinent to the construction of quaternary Hadamard matrices and have many applications, for example in coding theory and communications. In contrast to binary Legendre pairs, quaternary ones can exist for…
We continue investigating rational quartic reciprocity laws and, at the suggestion of the editor of AA, provide details of a proof of a remark in the first article with this title.
The Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovered as a consequence of an asymptotic formula in mathematical physics. Here we give a simple…
Legendre's relation for the complete elliptic integrals of the first and second kinds is generalized. The proof depends on an application of the generalized trigonometric functions and is alternative to the proof for Elliott's identity.
Much has been written on reciprocity laws in number theory and their connections with group representations. In this paper we explore more on these connections. We prove a "reciprocity Law" for certain specific representations of semidirect…
D'Alembert made the first serious attempt to prove the Fundamental Theorem of Algebra (FTA) in 1746. An elementary proof of (FTA) based on the same idea is given in Proofs from THE BOOK. We give a shorter and more transperant version of…
Feferman proved in 1962 that any arithmetical theorem is a consequence of a suitable transfinite iteration of full uniform reflection of $\mathsf{PA}$. This result is commonly known as Feferman's completeness theorem. The purpose of this…
An influential argument for scientific realism posits that, if scientific theories were not true, their empirical success would be a coincidence. Here, I show that the false Drude's theory succeeds in explaining the Wiedemann-Franz law by…
The Legendre conjecture has resisted analysis over a century, even under assumption of the Riemann Hypothesis. We present, a significant improvement on previous results by greatly reducing the assumption to a more modest statement called…
The world of primes has many gaps between evidence and theorems. Here, we review Legendre's conjecture on primes between consecutive squares and recent progress on the weaker question of primes between consecutive larger powers. Assuming…
Legendre's theorem states that every irreducible fraction $\frac{p}{q}$ which satisfies the inequality $\left |\alpha-\frac{p}{q} \right | < \frac{1}{2q^2}$ is convergent to $\alpha$. Later Barbolosi and Jager improved this theorem. In this…
I present the proof of Goedel's First Incompleteness theorem in an intuitive manner, while covering all technically challenging steps. I present generalizations of Goedel's fixed point lemma to two-sentence and multi-sentence versions,…
The reciprocity principle is that, when an emitted wave gets scattered on an object, the scattering transition amplitude does not change if we interchange the source and the detector - in other words, if incoming waves are interchanged with…
Motivated by two Legendre-type formulas for overpartitions, we derive a variety of their companions as Legendre theorems for overpartition pairs. This leads to equalities of subclasses of overpartitions and overpartition pairs.