number-theory

Let $p_1, p_2, p_3, p_4, p_5 ... = 2, 3, 5, 7, 11, ...$ be the prime numbers in increasing order. Let $k > 5$ and $A_k = {1, 4, 5, 6, 7, 9, 11, p_6, ..., p_k}$. In $A_k$ all least common multiples ...

Let $f(z)$ be an entire function of exponential type $1$, and let $s,t>0$, $s+t=1$. Can we always find entire functions $g,h$ of exponential types $s,t$ such that $f=gh$ ?

I am looking for a reference—either a book or a specific paper—that contains the actual proof of the result that no three consecutive positive integers are perfect powers. While reading Wacław ...

A variation of Abel's summation formula is $$\sum_{n = 1}^N f(n)g(n) = f(N)G(N) - \sum_{n = 1}^{N - 1} (f(n + 1) - f(n))G(n),$$ where $$G(n) = \sum_{k = 1}^n g(k).$$ Suppose that $\alpha \in ...

In a project on Quantum Mechanics, we ran into the following identities with Bessel functions and their derivatives: ...

Wikipedia gives the following proof that the Cauchy-completeness of the real numbers implies Dedekind-completeness: Let $S$ be a nonempty set of real numbers. If $S$ has exactly one element, then its ...

Engineering Riemann Hypothesis This morning, I revisited the Riemann Hypothesis from a zero–pole perspective 🧮✨ and introduced a new reciprocal formulation called the Srichan Teza Function. https://lnkd.in/gkFRTfX3 The idea is simple 🔄: Start from the completed zeta function ξ(s) = 1/2 · s(s − 1)π⁻ˢᐟ² Γ(s/2)ζ(s) and define T_S(s) = 1/ξ(s) Then every zero of […]

The quest to approximate irrational numbers with fractions reveals hidden patterns, surprising hierarchies and enduring mathematical mysteries

If you've taken any course that covers number systems, you've probably learned the "repeated multiplication" method for converting a decimal fraction to binary. Multiply by 2, record the digit before the decimal point, keep the remainder, repeat. It works. You can pass an exam with it. But it also feels a little like a magic trick — you follow the steps, you get the right answer, and you have no …

Suppose $R$ is a reduced Noetherian ring. We know that $f \in R$ is a zero divisor if and only if $V(f)$ contains an irreducible component of $\mathrm{Spec} R$. I would like to know if one could use ...

I mean, if a function $F(s)$ is in Selberg class, and $F(s)$ has a pole of order m at $s=1$, is it true that there exists a function $G(s)$ in Selberg class such that $F(s)=\zeta^m(s)G(s)$, and $G$ is ...

Famously, the set of computable numbers is countable. That's pretty much a result of their definition: The decimal expansion of a computable number can be generated to any arbitrary length by a finite algorithm. And since the set of all possible finite algorithms is countable, so is the set of... Read more

I am working on this coding problem, Counting bits. For a given integer, $n$, I have to return a list of ones in the binary representation of all the numbers from $0$ to $n$. I understand that it is a ...

It is a fact that I find remarkable that the sum of the cubes of the first $n$ positive integers is equal to the square of the sum of the first $n$ positive integers. I wonder, does something like ...

I am happy to announce the third SAIR challenge, which is focused on obtaining numerical data for the infamous inverse Galois problem. This is a collaborative project with the L-functions and modular forms database (LMFDB), and is organized by John Jones, Jen Paulhus, David Roe, Andrew Sutherland, and myself. The challenge is somewhat similar to […]

I stumbled upon A065377, the list of primes which can't be written in the form $p+k^2$ ($p$ prime and $k>0$ being an integer), and it's $2, 5, 13, 31, 37, 61, 127, 379, 439, 571, 829, 991, 1549, ...

The nth pentagonal number Pn is the number of dots in diagrams like those below with n concentric pentagons. We have the formula Pn = (3n² − n)/2 where n is a positive integer. If n is an integer but not positive, the equation above defines a generalized pentagonal number. If you’re given an n, you can easily compute Pn. […] The post Testing pentagonal numbers first appeared on John D. Cook .

The ancient Egyptians had a terrible notation for fractions. They had notations for for each , for , but everything else was written as a sum of these, with repeats forbidden, so that for example had to be written as . ( Wikipedia ) In an older article about Egyptian fractions and the Rhind Mathematical Papyrus , I said: Getting the table of good-quality representations of is not trivial, and …

Given a $\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)$-conjugacy class of $q$-Weil numbers, we obtain the minimal polynomial $m_\pi(x)$ of Frobenius. However, the Weil polynomial is not ...

This manuscript proves the Birch and Swinnerton-Dyer endpoint by fixed-carrier exclusion of analytic-arithmetic mismatch. The earlier structural BSD role-compression paper established the role architecture: elliptic-curve carrier, analytic L-function standing, Mordell-Weil rank readout, and the analytic-arithmetic bridge. The present paper moves from role architecture to endpoint closure. It prov…