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Results tagged with algebraic-number-theory
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user 1593
Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces
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
$$ \frac{x^{n} − 1}{x-1} = y^{q} \quad x > 1, \quad y>1, \quad n>2 \quad …
- 8,842
7
votes
3
answers
603
views
Question on a crucial lemma in Euler's approach to Fermat's Last Theorem for $n=3$
As many of you may know, the illustrious L. Euler put forward a proof of the case $n=3$ of Fermat's Last Theorem via infinite descent. The thing is that, at a certain point, he resorted to the follo …
- 8,842
4
votes
An old paper of S.Chowla on unit equations
For what it is worth, I reproduce below the relevant paragraphs in Chowla's aforementioned paper.
The number of solutions of the equation
$$\epsilon - \epsilon^{\prime} =1 \qquad \qquad \qquad(1)$$ …
- 8,842
10
votes
Fermat's Last Theorem in finite fields
It is more or less easy to obtain, via exponential sums, an asymptotic formula for the number $J$ of solutions of the congruence
$$x^{n}+y^{n} \equiv z^{n} \pmod{p}$$
where $$1 \leq x, y, z \leq p- …
- 8,842