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Results tagged with diophantine-equations
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user 21743
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 ...
2
votes
Impossible Heronian Triangles (Ratio of 2 Sides)
I did some calculations this morning on the elliptic curve
$W^2=Z(Z-m^2)(Z-n^2)$
I have several codes in Ubasic or Pari which use the Birch & Swinnerton-Dyer conjecture to predict ranks.
For $1 \l …
- 276
10
votes
Accepted
Impossible Heronian Triangles (Ratio of 2 Sides)
Suppose the ratio is called $k$, and let the sides be $g, kg, h$, with the angle between sides $g$ and $kg$ called $A$, with all these variables rational.
Then $\Delta = \frac{1}{2}kg^2 \sin A$, so i …
- 276
5
votes
Diophantine equation $2(x - 1/x) = y - 1/y$
This question is actually the first example, in a different formulation, of the following problem:
Find two integer right-angled triangles with a common base and altitudes in the integer ratio $N:1$, …
- 276