Related papers: Two encounters with the p-adic Stark conjecture
John Goodricke and Edward Pigott, working in York, England, between 1781 and 1786, determined the periods of variation of eclipsing binaries such as Algol and Beta Lyrae and speculated that the eclipses of Algol might be caused by a "dark…
We give a systematic method of providing numerical evidence for higher order Stark-type conjectures such as (in chronological order) Stark's conjecture over $\mathbb{Q}$, Rubin's conjecture, Popescu's conjecture, and a conjecture due to…
The Firoozbakht, Nicholoson, and Farhadian conjectures can be phrased in terms of increasingly powerful conjectured bounds on the prime gaps $g_n := p_{n+1}-p_n$. \[ g_n \leq p_n \left(p_n^{1/n} -1 \right)\qquad\qquad\qquad (n \geq 1; \;…
$p$-Adic mathematical physics is a branch of modern mathematical physics based on the application of $p$-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a…
In 1952 P. F. Strawson proposed a logic of presuppositions. It is an interpretation of Aristotelian logic, i.e. of the logic of the traditional syllogism. In 1981 Richard Diaz published a monograph in which he presented truth-relevant…
A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which are known…
We derive heuristically the approximate formula for the difference $\sqrt{p_{n+1}} - \sqrt{p_n}$, where $p_n$ is the n-th prime. We find perfect agreement between this formula and the available data from the list of maximal gaps between…
The said paper [2] entitled "Proof Of Two Dimensional Jacobian Conjecture" is with gaps.
The Diophantine equation 4/n=1/x+1/y+1/z for a Pythagorean prime n is split into two independent Diophantine equations, which correspond to two different types of solution. The solvability of these equations forces certain restrictions on…
I discuss how modern cosmology illustrates under-determination of theoretical hypotheses by data, in ways that are different from most philosophical discussions. I emphasize cosmology's concern with what data could in principle be collected…
For a compact and convex window, Mecke described a process of tessellations which arise from cell divisions in discrete time. At each time step, one of the existing cells is selected according to an equally-likely law. Independently, a line…
It is now about 10 years since the evidence, based on Type Ia supernovae, for the acceleration of the expansion of the Universe was discovered. I will discuss some aspects of the work and events in the Supernova Cosmology Project (SCP),…
The main purpose of the present article is to establish the real case of "Karoubi's conjecture" in algebraic K-theory. The complex case was proved in 1990-91 by the second author and Andrei Suslin. Compared to the case of complex algebras,…
We prove a conjectural formula for the Brumer--Stark units. Dasgupta--Kakde have shown the formula is correct up to a bounded root of unity. In this paper we resolve the ambiguity in their result. We also remove an assumption from…
In this paper we establish an exponential covering theorem implying a conjecture formulated by A. Zygmund circa 1935 whose three-dimensional case was obtained by the first named author in 1978.
In 1904, Dickson [5] stated a very important conjecture. Now people call it Dickson's conjecture. In 1958, Schinzel and Sierpinski [14] generalized Dickson's conjecture to the higher order integral polynomial case. However, they did not…
Benjamini, Shinkar, and Tsur stated the following conjecture on the acquaintance time: asymptotically almost surely $AC(G) \le p^{-1} \log^{O(1)} n$ for a random graph $G \in G(n,p)$, provided that $G$ is connected. Recently, Kinnersley,…
We study the special values of the triple product $p$-adic $L$-function constructed by Darmon and Rotger at all classical points outside the region of interpolation. We propose conjectural formulas for these values that can be seen as…
A review of the development of the concept of dark matter is given. I begin the review with the description of the discovery of the mass paradox in our Galaxy and in clusters of galaxies. In mid 1970s the amount of observational data was…
The compelling original idea of a time crystal has referred to a structure that repeats in time as well as in space, an idea that has attracted significant interest recently. While obstructions to realize such structures became apparent…