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Results tagged with diophantine-equations
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user 163830
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 ...
0
votes
On the equation $x^3 + y^3 = z^4$
$x^3+y^3=z^4$ ----$(1)$
In the solution given by @Nulhomologous,
if we put $(s,t)=(2,1)$ we get:
$9^3+18^3=9^4$
Where the integer 'nine' is a common factor.
Eqn.(1) has another parametrization & is s …
- 9
0
votes
Rational points on the elliptic curve $y^2 = x^{3} - t^{2}z^3$
Above equation shown below:
$y^2=x^3-t^2z^3$ ----(1)
Equation $(1)$ has parametric solution given below:
$x=m^4-3m^2+3$
$y=m(m^4-3m^2+3)^2$
$z=(m^2-1)(3m^2-m^4-3)$
$t=1$
For, $m=2$, we get:
$(x,y,z,t …
- 9