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Results tagged with diophantine-equations
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user 1593
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 ...
4
votes
Integer solutions to $x^2 + x + 1 = y^z$?
You should definitely take a look at this previous answer of mine:
https://mathoverflow.net/a/251637/1593
- 8,842
7
votes
3
answers
603
views
Question on a crucial lemma in Euler's approach to Fermat's Last Theorem for $n=3$
As many of you may know, the illustrious L. Euler put forward a proof of the case $n=3$ of Fermat's Last Theorem via infinite descent. The thing is that, at a certain point, he resorted to the follo …
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2
votes
Perfect powers in the solutions of a certain Pell equation
I consider that an answer for the second question is recurrence relations for solutions of Pell's equations. In point of fact, by resorting to them I am going to prove below the OP's original conjectu …
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6
votes
2
answers
700
views
Origin and variations of problem on $4xy-x-y$ being square
One of the forms in which the Diophantine equation in question can be found in the literature is this:
Solve the equation \begin{eqnarray}z^{2} = 4xy-x-y \qquad \qquad (\ast)\end{eqnarray} in positi …
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10
votes
Accepted
Solve this diophantine equation: $m^4+n^4=10m^2n^2+1$
You have already listed all the possible solutions $(m,n)$ in which either $m=0$ or $n=0$.
Let us suppose that $(m,n) \in \mathbb{N} \times \mathbb{N}$ is a solution of your equation. Then, the discr …
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9
votes
1
answer
679
views
On the exact reference of a cute Diophantine problem
The problem asks to prove that the Diophantine equation $x^{3}+y^{3} = (x+y)^{2}+(xy)^{2}$ does not have any solutions in natural numbers $x, y$.
I believe that this problem appeared in the section За …
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41
votes
Is $x^2+x+1$ ever a perfect power?
Your equation can be rewritten as
$$\frac{x^{3}-1}{x-1} = y^{N}.$$
As Gerhard Paseman commented above, the Diophantine equation
$$ \frac{x^{n} − 1}{x-1} = y^{q} \quad x > 1, \quad y>1, \quad n>2 \quad …
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16
votes
Diophantine equation $3^n-1=2x^2$
W. Ljunggren proved in 1 that the Diophantine equation
$$\frac{x^{n}-1}{x-1} = y^{2}$$
doesn't admit solutions in integers $x>1, y>1, n>2$, except when $n=4, x=7$ and $n=5, x=3$. Since your equation …
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5
votes
Trost's Discriminant Trick
As I was browsing through the problems & solutions department of the American Mathematical Monthly the other day, I found out that, in his solution to problem E-2332 [1972, 87], which was published on …
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20
votes
2
answers
2k
views
On a result attributed to W. Ljunggren and T. Nagell
I've read in a number of places that, building on previous work of T. Nagell, W. Ljunggren proved in 1 that the Diophantine equation
$$\frac{x^{n}-1}{x-1} = y^{2}$$
doesn't admit solutions in intege …
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