Questions tagged [computational-number-theory]

Computational Number Theory is for explicit calculations or algorithms involving anything of interest to number theorists.

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Possible contemporary improvement to bounded gaps between primes?

In his summary of his book Bounded gaps between primes: the epic breakthroughs of the early 21st century, Kevin Broughan writes Which brings me to my final remark: where to next in the bounded gaps ...
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Riemann-Siegel formula for Dirichlet characters

After unearthing and giving a proof of what is now known as the Riemann--Siegel formula for the Riemann zeta function enabling the computation of $\zeta(1/2+iT)$ in time $O(T^{1/2})$, in 1943 Siegel ...
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A Pell like equation

If one takes in general $(\star)\, \,x^2-dy^2=C$ where $d$, $C$ in $\mathbb{N}$. Taking $d=w^2p^2+p$ with $w\in \mathbb{Q}\ge 1$ and $p\in \mathbb{Z}$ which is verified (explained later), for the ...
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Best known primality test for the whole intervals of integers up to $10^{20}$ — like the sieve of Eratosthenes

What are the best known primality test(s) for the whole intervals of integers up to $N=10^{20}$ ? "Best" means "have minimal amortized time per tested integer". That is, the ...
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Elementary Iwasawa module

Let $k$ be a given number field. What is the importance and applications of knowing that the Iwasawa module $X_\infty$ of $k$ is an elementary $\Lambda$-module?
Integrality certification for product of two matrices $A B^{-1}$
Let's consider two non-singular integer matrices $A,B \in\mathbb{Z}^{n\times n}$. I want a test to check if $A\times B^{-1}$ is integral (or no denominators). I am referring the unimodular ...