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Results tagged with diophantine-equations
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user 5712
Diophantine equations are polynomial equations $F=0$, or systems of polynomial equations $F_1=\ldots=F_k=0$, where $F,F_1,\ldots,F_k$ are polynomials in either $\mathbb{Z}[X_1,\ldots,X_n]$ of $\mathbb{Q}[X_1,\ldots,X_n]$ of which it is asked to find solutions over $\mathbb{Z}$ or $\mathbb{Q}$. Topics: Pell equations, quadratic forms, elliptic curves, abelian varieties, hyperelliptic curves, Thue equations, normic forms, K3 surfaces ...
3
votes
Nontrivial solutions for $\sum x_i = \sum x_i^3 = 0$
If $N=6$ then we have a nice identity:
$$(x_1+x_2)(x_1+x_3)(x_2+x_3)=-(x_4+x_5)(x_4+x_6)(x_5+x_6).$$
(Loo-Keng Hua used such identities in his "Additive Theory of Prime Numbers".) In particular if $x_ …
- 12.3k
5
votes
Rational solutions of $x^4 \pm y_1^4 \pm .. y_n^4 = |k|$ for given rational $k$
This problem (for integers) is known as "An Easier Waring's Problem". In the article (1934) with the same name Wright proved that for degree $4$ one need $v(4)$ summands where $8\le v(4)\le 12$. This …
- 12.3k
3
votes
If $n=x^k+y^k$ then also $n=a^2+b^2=c^3+d^3=\ldots =x^k+y^k$
It is true for $n=2^{k!+1}$ and $a=b=2^{k!/2}$, $c=d=2^{k!/3}$,...
- 12.3k
5
votes
Accepted
solutions to special diophantine equations
This system is well studied. You can find full description of solutions in
"Introduction to the theory of numbers" by Leonard E. Dickson. (See Theorem 47).
If all the variables are between $1$ and …
- 12.3k
2
votes
2
answers
1k
views
Solve in positive integers $n!=m^2$
Is anybody know a solution of this problem?
(Sorry, correct question is here.)
- 12.3k
35
votes
3
answers
5k
views
Solve in positive integers: $n!=m(m+1)$
Does anybody know a solution to this problem? (Sorry, I've missed one summand in the previous post.)
- 12.3k
0
votes
Sharply Estimating Pythagorean Triples
For each type of Pythagorean Triples
$p=2xy$, $r=x^2-y^2$ and $p=x^2-y^2$, $r=2xy$ your inequalities describe a simple region on $Oxy$ plane. Boundary consits from the straight line $2xy=x^2-y^2$ and …
- 12.3k
1
vote
Finding Pythagorean quadruples on a given plane?
It is not the answer but some relevant information.
In the paper "Cubes in an Integer Lattice" Ivan Horozov gave parameriyation of all mutually perpendicular integer vectors $A_{1}=\left(x_{1}, y_{1} …
- 12.3k
4
votes
English translation of Voronoi's dissertation
"The Theory of Irrationalities of the Third Degree" is much more clere than original Voronoi's doctoral dissertation. There are some recent works on Voronoi's and Minkowski's algoritms. You can start …
- 12.3k
5
votes
Request for an exact formula related to a partition in number theory
You can take generating function $$f(z):=\frac{1}{1-z^{a_1}}\frac{1}{1-z^{a_2}}\cdots \frac{1}{1-z^{a_n}}$$ as in Max Alekseyev's answer and calculate $F (a_1, \dots, a_n; b, n)$ as $$
\frac{1}{2 \pi …
- 12.3k