Questions tagged [computational-number-theory]
Computational Number Theory is for explicit calculations or algorithms involving anything of interest to number theorists.
504 questions
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Special $x$-values in $k^2 ≡ 64(x^3-n^2x) \pmod{p}$: connection to elliptic curves?"
I am investigating a system of congruences related to the elliptic curve $y^2 = x^3 - n^2x $ (congruent number curve) with the following conditions:
System:
$$k^2 ≡ 64(x^3 - n^2x) \pmod{p}$$ where 'p' ...
- 319
14
votes
1
answer
711
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A functional equation on $\mathbb{Z}/p\mathbb{Z}$
Let $p$ be a prime such that $2$ is a primitive root of $p$.
We want to find a bijective function $f: (\mathbb{Z}/p\mathbb{Z})^× \to (\mathbb{Z}/p\mathbb{Z})^× $ s.t.
$$f(2k) = f(k) + f(f(k)) $$
$$f(-...
- 141
1
vote
0
answers
27
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Probability of Generating an Ideal by Two Bounded-Coefficient Random Combinations of Given Generators
Let $K = \mathbb{Q}(\zeta_{p^k})$ be a prime-power cyclotomic field with ring of integers $\mathcal{O}_K$. Fix a $\mathbb{Z}$-basis $\underline{\omega} = (\omega_1, \dots, \omega_d)$ of $\mathcal{O}_K$...
- 11
3
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0
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60
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Does a Motzkin-prefix greedy construction for distinct subset sums recover the Conway–Guy sequence?
A finite set $P=\{p_1>\dots>p_n\}\subset\mathbb{N}$ has distinct subset sums (DSS) if all $2^n$ subset sums are different. A problem of Erdős asks for $f(n)$, the minimal possible value of ...
- 131
1
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0
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53
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Effective bounds for degree and height in algebraic number enumeration
I am considering a specific enumeration of all complex algebraic numbers within the unit ball, defined as follows:
Enumerate all primitive, irreducible polynomials in $\mathbb{Z}[x]$ with positive ...
- 545
8
votes
1
answer
393
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On sums of finitely many distinct unit fractions of the form $\frac1{p_k+p_{k+1}}$
A well-known result on Egyptian fractions states that any positive rational number can be written as a sum of finitely many distinct unit fractions.
For each prime $p$, let $p'$ be the first prime ...
- 18.1k
2
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0
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165
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Connected components of the graph whose vertices are the primes and in which two vertices are connected by an edge if they differ by a power of 2
Let $\Gamma$ be the graph whose vertices are the prime numbers and in which
two vertices are connected by an edge if they differ by a power of 2.
Questions:
Is it true that $\Gamma$ has exactly one ...
- 19.9k
24
votes
2
answers
970
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Is this sequence eventually $\operatorname{lcm}(1,2,\dotsc,n)$?
Here is a simple game: Start with the set $\{1,2,3,\dotsc,n\}$ of natural numbers. At any turn of the game, you may pick two numbers from this set, $a$ and $b$, then replace them with their product $a\...
- 1,402
24
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1
answer
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A mysterious recurrence for primes
For any positive integer $n$, define $s(n)$ as the smallest positive integer $m$ such that the $n$ distinct numbers
$$ (p_1-1)^2,\ (p_2-1)^2,\ \ldots,\ (p_n-1)^2$$
are pairwise incongruent modulo $m$,...
- 18.1k
4
votes
1
answer
242
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On the sets $\{\frac{m^2-1}{n^2-1}:\ m,n=2,3,\ldots\}$ and $\{\frac{m(m+1)}{n(n+1)}:\ m,n\in\mathbb Z^+\}$
Let us introduce two subsets of the field $\mathbb Q$ of rational numbers:
$$S:=\left\{\frac{m^2-1}{n^2-1}:\ m,n=2,3,\ldots\right\}\tag{1}$$
and
$$T:=\left\{\frac{m(m+1)}{n(n+1)}:\ m,n=1,2,3,\ldots\...
- 18.1k
82
votes
10
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Is the least common multiple sequence $\text{lcm}(1, 2, \dots, n)$ a subset of the highly abundant numbers?
I've been comparing the sequence of the Least Common Multiple of the first $n$ integers, $L_n = \text{lcm}(1, 2, \dots, n)$, with the sequence of Highly Abundant Numbers (HA).
The two sequences in ...
- 1,089
17
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0
answers
740
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How to compute A263996?
Consider the sequence
$$a_n:=\min_{\substack{A\subseteq\mathbb{Z}\\|A|=n}}|(A+A)\cup (A\cdot A)|.$$
This is A263996 in OEIS. (Actually, they restrict to $\mathbb{N}$, but it makes little difference to ...
- 8,150
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172
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Algorithms for searching equal sums of like powers (taxicab problem)
I am developing a brute-force algorithm to search for solutions to the generalized taxicab equation, $a^k + b^k = c^k + d^k$ between a lower and upper bound. My current approach is a priority queue-...
0
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0
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124
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Describing imaginary abelian fields in terms of Dirichlet characters
I am reading the paper, titled "The imaginary abelian number fields with class numbers equal to their genus class numbers" in which the authors describe an imaginary abelian field $N$ by the ...
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1
answer
186
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Is it always true that $n<(max(rad(n−2),rad(n−1),rad(n)))^2$ for $n>3$?
Upon observing three consecutive integer numbers $n-2$, $n-1$, and $n$ for $n > 3$, I discovered that most of them satisfy the inequality: $n < X^2$ where $X= \max(rad(n-2), rad(n-1), rad(n))$. ...
- 2,148
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0
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230
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Two conjectures concerning the distribution of small numbers in coprime pairs (a, b)
The research findings of mathematician Ivan Niven imply that the distribution of the smallest exponents in the prime factorization of natural numbers is $1$, while the average largest exponent in the ...
- 2,148
3
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0
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154
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Brute-force bounds on $\psi(x) = \sum_{n\leq x} \Lambda(n)$
Up to what $x$ do we have brute-force computational bounds of the form
$$|\psi(x)-x|\leq c\sqrt{x},$$
where $\psi(x) = \sum_{n\leq x} \Lambda(n)$?
I know we have bounds of the above form (with $c$ not ...
- 22k
4
votes
1
answer
399
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On the growth of an algorithm
The origin of the following question is irrelevant and will hence be skipped. Hopefully, it will be interesting without context.
Consider the following algorithm. We will use $p_i$ to denote the $i$-...
- 1,266
2
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0
answers
176
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Non vanishing of modular L functions on real line
In M. Ram Murty's paper "Oscillations of Fourier coefficients of modular forms", Math. Ann. 262, 431-446 (1983), MR696516, Zbl 0489.10020 (an offprint can be found here), I see a conjecture (...
- 121
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2
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674
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Is $a(n)=(34^n+1)/(34+1)$ ever prime for odd $n>3$?
Is $a(n)=(34^n+1)/(34+1)$ ever prime for odd $n>3$?
$n$ must be prime because of polynomial factorization.
There are no counterexamples up to $n=11,000$ and there are no
congruence obstructions ...
- 25.8k
3
votes
1
answer
338
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A more general conjecture about palindromic numbers in the comments section of the page of the OEIS sequence A266577
I posed a conjecture which is a generalization of Conjecture on palindromic numbers
My question is to find a proof or a disproof of it.
I have put it in the comments section of the page of the OEIS ...
1
vote
0
answers
111
views
Is the NH hash family $\varepsilon$-AXU?
As context, I'll start with summarizing and simplifying the section of "UMAC: Fast and Secure Message Authentication", by Black et al.(https://www.cs.ucdavis.edu/~rogaway/papers/umac-full....
- 111
0
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0
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116
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LLL algorithm for $\mathbb{Z}[X]$-lattices
I understand the LLL algorithm for finding approximate shortest vector in $\mathbb{Z}$-lattices (where the norm function is either $\ell_\infty$ or $\ell_2$), as well as finding the shortest vector in ...
6
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0
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462
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(New?) summation formulae for continued fractions, reference request
Consider a simple continued fraction
$$a_{0}\in \mathbb{Z} , a_{n}\in \mathbb{Z}_{\geq 0}, \quad \xi=a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\ddots}}}=\left[a_0, a_1, \dotsc\right] .$$
Define
\...
- 5,163
2
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0
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154
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Is it possible to transform a univariate congruence equation into a multivariable system of congruence equations for solving?
Considering a quadratic congruence equation
$$
x^2 = d \pmod{p} \label{1}\tag{1}
$$
I have a strange idea: what if we construct a system of congruence equations in multiple variables based on the ...
11
votes
1
answer
783
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On the positive integer solutions to $1+a^4+b^4=c^4+d^4$
Let $1+a^4+b^4=c^4+d^4$ such
that $a$,$b$,$c$ and $d$ are all positive Integers.
How to find Infinitely many solutions ?
By computer search up to $a$,$b$,$c$,$d$ $<3600$ , I found two ...
- 514
1
vote
1
answer
341
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Polynomial Time Algorithm for Solving Diophantine Equation with large integer number
My question is
Is there an algorithm in polynomial time that can find solutions $(x,y,k)$ to the Diophantine equation
$$ x^2 + k y^2 = N$$
where $x,y,k$ are unknow integers, $N$ is known, but its ...
7
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3
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608
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Cornacchia's algorithm with too many prime factors
Given quadratic diophantine equation $x^2+dy^2=m$ where $d,m> 0$ and $d$ is square-free, Cornacchia's algorithm: https://en.wikipedia.org/wiki/Cornacchia%27s_algorithm, solves the problem in ...
- 387
2
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1
answer
213
views
Finding divisors close to the cube root of an integer
For a given integer $N = p\cdot q$ with $p \leq q$, if $q$ is close to $N^{1/2}$, say
$$N^{1/2} \leq q \leq N^{1/2} + (4N)^{1/4}$$
then $p,q$ can be recovered very quickly using Fermat's factoring ...
- 181
8
votes
2
answers
631
views
Testing if a positive definite quadratic form over $\mathbb Z$ represents 1
I am looking for algorithms that can be used to test if a given positive definite $n$-ary ($n\geq 3$) quadratic form over $\mathbb Z$, whose factorization of the discriminant is known, represents 1.
I ...
- 514
1
vote
1
answer
211
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Zero-density results and partial RH
In a couple of papers regarding explicit zero-density results of $\zeta(s)$, such as Kadiri, Lumley, and Ng - Explicit zero density for the Riemann zeta function, I have seen that they take a ...
- 11
3
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2
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351
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How to calculate the double sine function via Sage or Pari/GP to high precision?
From a paper of Ruijsenaars, we can extract the following integral representation for the logarithm of the double sine function:
$$\log(S_2(w,z)) = \int_0^\infty \left( \frac{e^{-y(1+w-z)}-e^{-yz}}{(1-...
- 2,115
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0
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133
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Is the new method used by the GIMPS project applicable to non-Mersenne primes?
For years, there was a simple reason why the largest known prime is of the form $2^{p}-1$: We had the Lucas-Lehmer test which was specific to Mersenne numbers, and faster than all other known methods.
...
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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 $\...
- 143
4
votes
1
answer
378
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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 ...
- 131
0
votes
1
answer
220
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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?
- 1
1
vote
1
answer
296
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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
6
votes
1
answer
395
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New Mersenne prime and compute time [closed]
GIMPS has just announced that $2^{136,279,841}-1$ is prime. Does anyone have a sense of the scale of the computational resources involved in finding this? (And maybe how it compares to, say, ...
- 143
1
vote
0
answers
172
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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$ ...
- 1
4
votes
1
answer
287
views
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....
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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}$$
It can be proved that for any positive integers $a, n, m\ \ $,
$\lim_{n \to \infty} a \...
- 671
2
votes
1
answer
219
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$f(x)\bmod p$ and decomposition of prime ideals
While reading Serre's beautiful book Lectures on $N_X(p)$, I thought of a related question.
Let $f(x)\in \mathbb{Z}[x]$ be a monic irreducible polynomial with integer coefficients. Let $K$ be the ...
- 721
3
votes
1
answer
391
views
Efficiently computing $\prod_{i=1}^{n} A_i$
Let $k$ be a nonnegative integer, how to compute $\prod\limits_{i=1}^{n} A_i$ quickly and accurately, where $$A_i=\begin{bmatrix}
0 & 1\\
i^k & 1
\end{bmatrix}?$$
I know if $k=0$, we can use ...
- 418
4
votes
1
answer
587
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Regarding the digit expansion of $\sqrt 7$
Let $\sqrt 7=\sum_{i=0}^\infty a_i 7^{-i}, 0\le a_i \le 6$ be the expansion of $\sqrt 7$ in base $7$.
I am curious about the following question: Is there a $K\in \mathbb{N}$ such that for any $n\ge ...
- 41
5
votes
0
answers
192
views
Compute weight of modular form from its Fourier coefficients
It is known that Hecke eigenform $f \in S_{k}(\Gamma_0(N), \chi)$ is uniquely determined by first $C_{k,N}$ many Fourier coefficients, where $C_{k,N}$ is a constant only depends on $k$ and $N$. For ...
- 2,255
2
votes
1
answer
296
views
On square root modulo $2^t-1$
Is there a way to compute an $x$ satisfying $$x^2\equiv a\bmod(2^t-1)$$ where $a,t$ are integers given to us and factorization of $2^t-1$ is not given to us?
- 1
5
votes
1
answer
363
views
Isogenous elliptic curves and canonical modular polynomials
Let $\ell$ and $p$ be two primes. We are looking for a method for checking whether two supersingular elliptic curves over the finite field $F_p$, given through their $j$-invariants, are $\ell$-...
- 81
2
votes
0
answers
192
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Conductor of hyperelliptic curve after adding a rational root
Let $r\geq 5$ be a prime. Suppose I have a specified hyperelliptic curve $C: y^2=f(x)$ defined over $\mathbb{Q}$, where $f\in\mathbb{Q}[x]$ has degree $r$.
Note: the roots of $f$ are not rational but ...
- 83
15
votes
2
answers
811
views
Can four integer numbers $x$, $y$, $x-y$, $x+y$ be powerful numbers where $\gcd(x,y)=1$?
Can four integer numbers $x$, $y$, $x-y$, $x+y$ be powerful numbers where $\gcd(x,y)=1$ ?
- 2,148
3
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1
answer
345
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Could efficient solutions of $x^2+n y^2=A$ be related to integer factorization?
Let $n$ be positive integer with unknown factorization and $A$ integer with known
factorization.
According to pari/gp developers pari can efficiently find all solutions of:
$$x^2+n y^2=A \qquad (1)$$
...
- 25.8k