All Questions
Tagged with nt.number-theory computational-number-theory
341 questions
1
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0
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116
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Can all congruences for a third-order recurrence relation hold for some composite $n$?
Let $p$ be a prime with $p \gt 3$. Consider the polynomial $f = x^3 - 3x -1$. Suppose $f$ is irreducible over $\mathbb{F}_{p}$. Let $E$ be the splitting field of $f$ over $\mathbb{F}_{p}$, and let $\...
2
votes
0
answers
161
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Compute generators for group of totally positive units of a number field?
Given a number field $K$, I would like to compute (in Sage) generators for the group of totally positive units of $K$.
Update: I've tried some code (details below), which I've received some help on in ...
- 101
-1
votes
0
answers
73
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Analog of ceil and floor of $\sqrt{a(a+1)}$ in modular arithmetic
If we take ceil and floor of $\sqrt{x(x+1)}$ (when it exists) we get $x$ and $x+1$ respectively. Is there an analog of this assuming roots exist in modular arithmetic (at least modulo primes)?
...
- 13.9k
0
votes
0
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78
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Factoring totient of a prime
Is it any easy to factor $p-1$ when $p$ is a prime compared to general factorization problem?
What about when $2p+1$ is also a prime?
- 13.9k
1
vote
1
answer
240
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The equation $ax^2 +by^2 =1 \mod P$ in cyclotomic field
Let $L$ be a cyclotomic field, and $P$ a prime ideal of $\mathcal{O}_L$.
is there any symbol for the equation $ax^2 + by^2 =1 \mod P$ and if so, is it computable in polynomial time?
if $a$ is ...
- 133
1
vote
0
answers
114
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Simultaneous elimination of variables in multiple polynomials
I have a system of $n=O(1)$ non-homogeneous polynomials of total degree $d=O(1)$ $p_1,\dots,p_r\in \mathbb Z[x_1,\dots,x_n]$. I would like to eliminate $n-1$ variables simultaneously from the $n$ ...
- 13.9k
3
votes
1
answer
203
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Chowla's theorem on class number of real quadratic field
Let $p\equiv1\bmod 4$ be a prime number and $h$
the class number of real quadratic field $\mathbb Q(\sqrt{p})$, $\epsilon=\frac{t+u\sqrt{p}}{2}$ its fundamental unit. In this paper https://www.pnas....
- 287
34
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0
answers
1k
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Is $2\uparrow\uparrow\infty + 3$ divisible by a prime number?
Define power tower using Knuth's arrow: $$a\uparrow\uparrow b=\left.a^{a^{a^{...^a}}}\right\}b\text{ layers}$$
It can be proved that for any positive integers $a, n, m\ \ $,
$\lim_{n \to \infty} a \...
- 631
2
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1
answer
157
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$f(x)\bmod p$ and decomposition of prime ideals
While reading Serre's beautiful book Lectures on $N_X(p)$, I thought of a related question.
Let $f(x)\in \mathbb{Z}[x]$ be a monic irreducible polynomial with integer coefficients. Let $K$ be the ...
- 709
5
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1
answer
303
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Efficiently computing $\prod_{i=1}^{n} A_i$
Let $k$ be a nonnegative integer, how to compute $\prod\limits_{i=1}^{n} A_i$ quickly and accurately, where $$A_i=\begin{bmatrix}
0 & 1\\
i^k & 1
\end{bmatrix}?$$
I know if $k=0$, we can use ...
- 696
3
votes
1
answer
513
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Regarding the digit expansion of $\sqrt 7$
Let $\sqrt 7=\sum_{i=0}^\infty a_i 7^{-i}, 0\le a_i \le 6$ be the expansion of $\sqrt 7$ in base $7$.
I am curious about the following question: Is there a $K\in \mathbb{N}$ such that for any $n\ge ...
- 31
2
votes
1
answer
108
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On square root modulo $2^t-1$
Is there a way to compute an $x$ satisfying $$x^2\equiv a\bmod(2^t-1)$$ where $a,t$ are integers given to us and factorization of $2^t-1$ is not given to us?
- 13.9k
5
votes
1
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172
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Isogenous elliptic curves and canonical modular polynomials
Let $\ell$ and $p$ be two primes. We are looking for a method for checking whether two supersingular elliptic curves over the finite field $F_p$, given through their $j$-invariants, are $\ell$-...
- 81
14
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2
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683
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Can four integer numbers $x$, $y$, $x-y$, $x+y$ be powerful numbers where $\gcd(x,y)=1$?
Can four integer numbers $x$, $y$, $x-y$, $x+y$ be powerful numbers where $\gcd(x,y)=1$ ?
- 1,776
3
votes
1
answer
261
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Could efficient solutions of $x^2+n y^2=A$ be related to integer factorization?
Let $n$ be positive integer with unknown factorization and $A$ integer with known
factorization.
According to pari/gp developers pari can efficiently find all solutions of:
$$x^2+n y^2=A \qquad (1)$$
...
- 25.4k
3
votes
3
answers
293
views
Finding a prime which is a square modulo all small primes
I want to find a small prime $p$ satisfying
$p\equiv 1 \pmod{8}$
$\left(\frac{p}{q} \right) = 1$ for all primes $3 \leq q \leq N$
where $N$ is a moderately large number (say, around $15,000$). I ...
- 26.9k
2
votes
0
answers
107
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Record for determining complete list of imaginary quadratic fields with small class number
In 2003, Mark Watkins (Class numbers of imaginary quadratic fields) determined all imaginary quadratic fields having class number at most 100.
Has this list been improved? That is, what is the largest ...
- 26.9k
2
votes
0
answers
146
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Reference for accelerated sum to compute the Meissel-Mertens constant
The Meissel-Mertens constant
$$ B_1 = \lim_{n \to \infty} \left(\sum_{p \leq n} \frac{1}{p} - \log\log n\right) $$
has the series representation
$$
\begin{equation} \tag{1}
B_1 = \gamma + \sum_{n=2}^{...
- 451
0
votes
0
answers
61
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Is generating semirandom blake256 hashes until packed points is on the curve, a safe algorithm to avoid the discrete log between the generated points?
I know there are more robust methods, but I wanted to know about this specific one
For any distinct said randomly generated point : $P_i,P_j\in \{P_1,...,P_k\}$ it should be hard to find $s$ such that ...
- 87
5
votes
0
answers
187
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Is there an effective way to compute the square root of an algebraic number?
For an algebraic number $\alpha$, let $f_\alpha$ denote its minimal polynomial. We can symbolically represent an algebraic number $\alpha$ by the tuple
$$
(f_\alpha, x, y, r) \in \mathbb{Q}[x] \times ...
- 1,087
4
votes
0
answers
821
views
One of the numbers $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational
I am reading an interesting paper One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational by Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive ...
- 11
16
votes
2
answers
1k
views
Is it decidable whether two real algebraic irrationals generate the same extension of the rationals?
For an algebraic number $\alpha$, let $f_\alpha$ denote its minimal polynomial. We can symbolically represent an algebraic number $\alpha$ by the tuple
$$
(f_\alpha, x, y, r) \in \mathbb{Q}[x] \times ...
- 1,087
2
votes
2
answers
1k
views
Sum of three square is a square and sum of their product taken two at a time is also a square
Let $a^2 + b^2 + c^2 = X^2$ and
$$(ab)^2 + (ac)^2 + (bc)^2 = Y^2$$
Such that $a,b,c,x,y$ are all non zero Integers.
How to find All solutions ?
Is there any parametrization which gives Infinitely ...
- 147
1
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1
answer
106
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Number of solutions for linear modular equations given GCD
We are currently investigating a problem involving number theory, an area outside our field of expertise.
Let $n$ be a positive integer. Consider two pairs of integers $(j,k)$ and $(j′,k′)$ as ...
13
votes
2
answers
1k
views
Using the Eichler-Selberg Trace formula to compute class numbers?
The Eichler-Selberg trace formula (Theorem 2.2 here) gives a relation between the trace of a Hecke operator acting on the space of cusp forms and sums of weighted class numbers of imaginary quadratic ...
- 133
4
votes
1
answer
222
views
Generators of the ideal class group
Theorem 4 of Eric Bach's "Explicit bounds for primality testing and related problems" states the following:
Let $K$ be a number field of degree greater than 1. Let $d$ be the absolute value ...
- 43
6
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0
answers
200
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Empirical bounds on $\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right|$
It is reasonable to expect that $$\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right| < 2 \log \log t$$
for all $t\geq 4$ (say): a somewhat stronger bound is known for $t\geq 10^{165}$ or so (Theorem 5 ...
- 20.2k
1
vote
2
answers
383
views
Is there any way to estimate this functions: $f(n)=\sum_{d|n}d\varphi(d)$ and $g(n)=\sum_{d|n}\frac{\varphi(d)}{d}$?
Let that $n$ be a natural number and $\varphi(n)$ be the Euler totient function. Is there any formula or estimation for computing functions $f,g$ such that:
$$
f(n)=\sum_{d\mid n}d\varphi(d)
$$
and
$$
...
- 139
3
votes
2
answers
287
views
When I know the two points on an elliptic curve, and the two points satisfy the relationship: $Q=e \cdot P$, is it possible for me to solve for e [closed]
When I know the two points on an elliptic curve, and the two points satisfy the relationship: $Q=e \cdot P$, is it possible for me to solve for $e$.
The equation of the curve is: $y^2 = x^3 + ax + b \...
- 31
0
votes
1
answer
126
views
Integer quadratic representation subject to discriminant minimization algorithm
Let $f(x)=ax^2+bx+c$ and $f(x)=n$. Is there an algorithm to choose $a,b,c$ such that the discriminant is minimized? Where $a,b,c,n,x$ are all integers.
More concretely, is there an algorithm to find $...
2
votes
0
answers
71
views
Simultaneous computation of the three Weber modular functions
Recall that the three classical Weber modular functions are defined by
$f(\tau)=e^{-\pi i/24}\eta((\tau+1)/2)/\eta(\tau)$,
$f_1(\tau)=\eta(\tau/2)/\eta(\tau)$, and
$f_2(\tau)=\sqrt{2}\eta(2\tau)/\eta(\...
- 13.1k
10
votes
2
answers
3k
views
Can every integer be written as a sum of squares of primes?
This question is mainly inspired from a different problem I was working on.
Is there a value of $k$ such that, for each $n\in \mathbb N$, the equation
$$\sum_{i=1}^{k}x_i^2=n$$
is solvable in $x_1,\...
- 791
0
votes
0
answers
145
views
Positive definite quadratic form algorithm
Let $f(x,y)= ax^2+bxy+cy^2$, or similarly denote it by $(a,b,c)$. This question is about the case $(1,0,p)$ where $p$ is prime. Suppose I have one solution $\bar{x}_1=(x_0,y_0)$ for $f(x,y)=m$ for ...
2
votes
1
answer
251
views
A weird property of odd positive integers $n$ with $\sigma(n)\sim2n $
When one looks at positive odd integers $n$ for which $|\sigma(n)-2n|\le\log n$, (sequence A088012) it appears that for all seven known numbers of this type the abundance, $\sigma(n)-2n$ is $\equiv 6\...
- 433
0
votes
0
answers
185
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Efficiently count the number of primitive roots in all moduli up to $n$
Let's define $f(n)$ as the number of primitive roots modulo $n$. That is, $f(n) = \begin{cases}\varphi(\varphi(n))&n=1,2,4,p^k,2p^k\\0&\text{otherwise}\end{cases}$. We want to efficiently ...
- 3,319
1
vote
0
answers
64
views
Condition on the minimality of Minkowski units
I am interested in to undrestand when the Minkowski units in real biquadratic number fields are minimal in the log unit lattices.
I have read some pieces of literature online which are investigating ...
- 11
0
votes
0
answers
363
views
Counting perfect powers using primes
Result
Let $n\in\mathbb{N}_{\geq1}$
$n$ is by definition a perfect power iff
$\,\ \exists m,k\in\mathbb{N}_{>1}:n=m^{\,k}$
Let $N(n)$ be the number of perfect powers $\leq n$
We define
$$\mathbb{...
- 53
0
votes
1
answer
109
views
Residues distribution modulo an interval
Given a number $n$ and an Interval $I = [ \; \lfloor n^{1/4} \rfloor, \lfloor n^{(1/3) \rfloor \;} ]$, can we say anything about the distribution of $\{ n \mod b \;\;| \; b \in I \}$?
In particular, ...
3
votes
1
answer
534
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abc-conjecture and positive definite kernels, again?
One formulation of the abc-conjecture is:
$$\forall a,b \in \mathbb{N}: \frac{a+b}{\gcd(a,b)}< \operatorname{rad}\left ( \frac{ab(a+b)}{\gcd(a,b)^3}\right )^2 $$
Let us define:
$$K(a,b) := \frac{2(...
- 3,064
0
votes
0
answers
319
views
Percent of rational coordinates that is a multiple of another point on the elliptic curve
Consider elliptic curves $E:= y^2=x^3+Ax+B $ (A, B are integers) which have points $P, Q$ with rational coordinates and satisfy $P=[n]Q, n>1$. Now consider the below problem:
Input: Rational ...
-2
votes
2
answers
149
views
Calculate the great common factor between $2^{2n+1}-1$ and $2^{4m+2}+1$ [closed]
How to calculate the great common factor between $2^{2n+1}-1$ and $2^{4m+2}+1$, where $n$ and $m$ are positive numbers.
We guess that: the great common factor is $1$.
- 577
1
vote
0
answers
136
views
Quadratic equations over Gaussian integers
Given an equation $x^2\equiv(a+ib)\bmod(c+id)$ where $a,b,c,d\in\mathbb Z$ holds, how to test if the equation has solutions and how to find the solutions in polynomial in $\log(|abcd|)$ time if $c+id$ ...
- 13.9k
0
votes
0
answers
74
views
Using coppersmith for bounded solution of a short linear Diophantine problem
I have a $3$-variable linear Diophantine equation
$$ax+by+cz=r$$ where $a,b,c,r\in\mathbb Z$ are known and can be as large in magnitude as needed and I know the equation has a solution $x,y,z\in\...
- 13.9k
93
votes
3
answers
6k
views
A little number theoretic game
I came up with this little two player game:
The players take turns naming a positive integer. When one player says the number $n$, the other player can only reply in two different ways: They can ...
- 861
2
votes
0
answers
221
views
Modular inverse computation - avoiding Euclidean algorithm
Modular inverse is known to be computable by Extended Euclidean algorithm which is the reaping the rewards of computing the GCD of two numbers or proving two numbers are coprime.
If we already know ...
- 13.9k
5
votes
1
answer
310
views
Parity of number of solutions to Diophantine equations
By $MRDP$ resolution of Hilbert's tenth, we infer, counting number of solutions to Diophantine equations is undecidable.
Is parity of number of solutions to Diophantine equations undecidable?
- 13.9k
6
votes
0
answers
125
views
Equivalence of primes based on the partition of their Pisano periods
The period of Fibonacci numbers modulo $m$ is called Pisano period and its length is denoted as $\pi(m)$. Define the Pisano partition of $m$ as the set partition of the indices $\{0,1,\dotsc,\pi(m)-1\}...
- 34.3k
12
votes
1
answer
2k
views
Primality of a number of more than 50k digits
With modern tecnology is it possible to prove the primality of a number of more than 50k digits?
Obviously not a prime for which specific methods for testing primality are known like Mersenne primes.
1
vote
1
answer
184
views
Deduce kernel of isogeny from action on torsion points
I'm stuck with the following problem:
In Petit's work "Faster Algorithms for Isogeny Problems using Torsion Point Images", p. 8, he says that we can deduce $\ker \psi_{N_2}$ knowing the ...
- 73
4
votes
1
answer
362
views
What is meant by a meet-in-the-middle approach?
I'm studing C. Petit's work "Faster algorithms for isogeny problems using torsion point images" (link) and he talks about meet-in-the-middle approach/strategy for solve some isogenies ...
- 73