The diophantine-equations tag has no wiki summary.

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### Transforming a Diophantine equation to an elliptic curve

I heard that the following problem lead to determine the rational points of an elliptic curve:
For which integers $n$ there are integers $x,y,z$ such that $x/y+y/z+z/x=n$. Could anyone show me why ...

**7**

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### Is there a solution to the a+b^m=b+c^n=c+a^l for l,m,n >1 and a, b, c distinct odd primes?

Is there a solution to:
$a+b^m=b+c^n=c+a^l$ for l,m,n >1 and a, b, c distinct odd primes?
I've had a play around with specific possible solutions and there are lots of possibilities that may be ...

**3**

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### 4900, a particularly square number

I read in "Letters to a young mathematician" that 4900 is the only square integer that is the sum of consecutive squares (I believe he meant by that "starting from 1", but maybe that's not even ...

**1**

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### Diophantine equation problem

How many positive integer solutions does the equation x^2+y^2+z^2-xz-yz = 1 have? My guess is (1,0,1), (0,1,1) and (1,1,1). What is proof of that fact that there are none other?

**1**

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### Integer points of an elliptic curve

I would like to find those integers $x,y$ that satisfies $y^2=x^3+1$. Is there some elementary way to find those?

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### Asymptotics related to the Erdos--Moser diophantine equation

I share the authorship of this question with Pieter Moree.
In our recent joint work with Y. Gallot (arXiv:0907.1356 [math.NT]) we attack
the Erdős--Moser diophantine equation
$$
...

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### When is $Pn^2-2an+\frac{a^2-k}{P}$ , with $P$ Prime, $k=a^2 mod P$, a square?

It is easy to show that the following problems are equivalent.
a. When is $Pn^2-2an+\frac{a^2-k}{P}$ , with $P$ Prime, $k=a^2 mod P$, and $n$ any integer, a square?
and
b. When is $X^2-PY^2=k$ ...