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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
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
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
$$ \frac{x^{n} − 1}{x-1} = y^{q} \quad x > 1, \quad y>1, \quad n>2 \quad …
- 8,842
10
votes
Fermat's Last Theorem in finite fields
It is more or less easy to obtain, via exponential sums, an asymptotic formula for the number $J$ of solutions of the congruence
$$x^{n}+y^{n} \equiv z^{n} \pmod{p}$$
where $$1 \leq x, y, z \leq p- …
- 8,842
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