Questions tagged [computational-number-theory]

Question 1

Choose rational numbers $p$, $q$, $r$, and choose a zero $\alpha$ of the cubic polynomial $\alpha^3+p\cdot\alpha^2+q\cdot\alpha+r$. What is an efficient algorithm for deciding, given nonzero rational ...

Question 2

My Mathematica Codes: ...

Question 3

This is a follow-up question to this question. In that question, we learned that if, $T(y) = \mathrm{Res}_{x}(f(x), y - g(x))$ , then $T(g(z))$ is divisible by $f(z)$. Now, my question is: If $T(y) = ...

Question 4

I analyzed primes generated by applying four linear offsets to the first element of twin prime pairs ( 𝑝 , 𝑝 + 2 ), for 𝑝 ≤ 1,000,000: 𝑞1 = 2𝑝 + 1, 𝑞2 = 2𝑝 + 7, 𝑞3 = 2𝑝 − 3, 𝑞4 = 2𝑝 + 3. ...

Question 5

I am currently reading Henri Cohen's excellent book, A Course in Computational Number Theory (1993). I have a small question regarding Zassenhaus's round 2 algorithm in this book, namely Algorithm 6.1....

Question 6

I'm attempting to verify the main computational claim from "Integer partitions detect the primes" by Craig, van Ittersum, and Ono (PNAS 2024, doi: 10.1073/pnas.2409417121). Their Theorem 1.1(...

Question 7

Just for curiosity and fun, I was trying to calculate the discriminant of some unusual fields! Let $p=-3\cdot 11\cdot 29\cdot 31 = -29667$ and $q=2\cdot 5\cdot 11\cdot 19\cdot 29\cdot 31 = 1878910$. ...

Question 8

Let $a$ and $x$ be Gaussian integers. How can I calculate the value of $x$ so that $x^2 \equiv a \pmod p$ where $p$ is a regular integer prime and $p \equiv 3 \pmod 4$ ? Example: $a = 2 + 5i$ $p = 7$ ...

Question 9

Curve25519 has the well-known equation $E: y^2 = x^3 + 486662x^2 + x$ which is normally considered over the finite field $\mathbb{F}_p$, for $p = 2^{255}-19$. We can lift $E/\mathbb{F}_p$ to $E/\...

Question 10

The book Primes of the form $x^2+ny^2$ by Cox discusses in detail the problem of when a prime $p$ can be written as $x^2+ny^2$, and along the way introduces more and more sophisticated number theory ...

Question 11

Yesterday I asked Google if 403 is prime. The result showed that it has factors 13 and 31. So I wondered and wrote a little code that finds prime numbers which is also a prime when their digits are ...

Question 12

This is a follow-up question to this question. In that question, we learned that if $T(y) = \mathrm{Res}_{x}(f(x), y - g(x))$ , then $T(g(z))$ is divisible by $f(z)$. More specifically, we can say ...

Question 13

How can I check whether two finite sets of $n$ independent vectors in $\Bbb Z^n$ generate the same additive subgroup of $\Bbb Z^n$? For example, $\{(2,7),(5,3)\}$ and $\{(-3,4),(5,3)\}$ have the same ...

Question 14

I have a very little knowledge about Sage and this may be a dumb question, but I could not find anything about this. Is there any way to define a sage function of several variables whose arguments are ...

Question 15

I am reading an article "Well-poised hypergeometric service for diophantine problems of zeta values" by W. Zudilin. Consider the quantities defined here in pg. $617$ $$\tilde{F_n}:= \frac{1}{...

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Questions tagged [computational-number-theory]

When a cubic irrational can be written in terms of two nonnegative powers?

Efficient algorithm for solving Diophantine equation $x ^ 2+y ^ 3+z ^ 5=w ^ 7$ with $\gcd (x, y, z)=1$.

If resultant $T(y) = \mathrm{Res}_{x}(f(x), y - g(x))$ have a nontrivial factors then can $f(x)$ also have a nontrivial factors?

Statistical patterns in primes generated from twin prime offsets

A question regarding Zassenhaus's Round 2 algorithm

Prime-detecting equation from PNAS 2024 fails computational verification

Ideas and methods to compute absolute discriminant of fields.

Modular square root of a gaussian integer [duplicate]

Rank of Curve25519 lifted to $\mathbb{Q}$

Computing primes of the form $x^2+ny^2$ [closed]

A specific pattern occurs on numbers whose factors are coupled emirp.

Finding the root of the polynomial $T(y) = \mathrm{Res}_{x}(f(x), y \cdot h(x) - g(x))$.

How can I check whether two sets of integer vectors generate the same subgroup?

Is it possible to define a function of several matrix variables whose inputs are variables in Sage?

Linear forms in $1,\zeta(5)$ and $\zeta(7)$

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