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Results tagged with computational-number-theory
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user 12481
Computational Number Theory is for explicit calculations or algorithms involving anything of interest to number theorists.
17
votes
Can four integer numbers $x$, $y$, $x-y$, $x+y$ be powerful numbers where $\gcd(x,y)=1$?
Using the infinitely many rational points on the curve
$x^4-y^4=(2^4-1) y^2 z^2$, the smallest solution we found
is with $x$ having 146 decimal digits (smaller solutions may exist).
Using the same cur …
- 25.4k
11
votes
Efficient computation of integer representation as a sum of three squares
A modification of Dror's comment.
This probabilistic algorithm worked for me.
The main idea is to pick some $z$, compute $m=n - z^2$, factor $m$ with trial division and express it as a sum of two sq …
- 25.4k
7
votes
Accepted
Calculating (n ^ fibonacci(k)) MOD m for a large value of k
Assuming $n$ and $m$ are coprime and $m$ is factored, first compute $a=\text{fibonacci}(k) \mod \varphi(m)$. Computing linear recurrence efficiently modulo $n$ is possible, e.g. via matrix exponentiat …
- 25.4k
4
votes
On Pell's equation
Partial answer to (2).
The smallest solution in some cases might be prohibitively large.
Check this paper pp 3,4.
It uses negative Pell, but its solution is smaller that the wanted.
The paper show …
- 25.4k
4
votes
What are the solutions in numbers of $xyz \mid x^n + y^n + z^n$, $x,y,z$ globally coprime
Some solutions for $n=7$:
n=7;y=2;z=3;x_i=[1, 463, -5, -2315]
n=7;y=2;z=5;x_i=[78253, -7, -1597]
n=7;y=2;z=7;x_i=[-9, -639, -11601, -823671]
n=7;y=2;z=9;x_i=[-11, -4783097]
n=7;y=2;z=11;x_i=[559, 1499 …
- 25.4k
2
votes
Equivalence between Diffie Hellman and Discrete Log
If the Discrete Log is easy, so is computationally DH.
According to:
Difiie-Hellman is as Strong as Discrete Log for Certain Primes
they [DH] conjectured that breaking their scheme would be as ha …
- 25.4k
2
votes
On the mixed sum of three k-th powers
EDIT This answers previous revision, quite different question.
Let $k = 5$. I think a single representation outside
your forbidden congruences would violate "Vojta's more general abc conjecture".
…
- 25.4k
0
votes
Transformation of a bivariate polynomial into a homogeneous one
Edit corrected major mistake
One approach is to work symbolically and solve a system over the rationals.
Choose bounds for the degrees of $S,T,H$ and write them as $\sum a_m x^i y^j$ where each
$a_m …
- 25.4k
0
votes
Accepted
Explicit bivariate quadratic polynomials where Coppersmith is better than standard solver?
Partial answer about "regular diophantine solver".
Finding points on conics in general require integer factorization.
Several papers deal with points on $a x^2+b y^2=c z^2$
and they require factoriz …
- 25.4k
0
votes
Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number...
First we observed that $\zeta_K$ numerically has double zero at $1/2$
and asked question about it.
In comment, David E Speyer suggested $L(s,\chi)^2$ should divide $\zeta_K$.
We believe that all zeros …
- 25.4k
-1
votes
A quadrant of residues
Partial answer giving probabilistic algorithm under
assumptions which can be greatly relaxed.
wlog assume $AB$ is the smallest product.
Write
$$ r_1= xz \mod A \qquad (1)$$
$$ r_2= xy \mod B \qquad …
- 25.4k