The strong Fermat-Catalan Equation
We give a cyclotomic proof of the fact that the equation $\frac{x^p + y^p}{x+y} = p^e z^q$ has no solutions in coprime integers $x,y,z$ and $p > 3; q$, a pair of distinct odd primes.
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arXiv subjects
Preda Mihailescu
Publications and source records attributed to Preda Mihailescu.
Beyond and slightly aside of the Fermat Desert
We give a cyclotomic proof of the fact that the equation $\frac{x^p + y^p}{x+y} = p^e z^p$ has no solutions in coprime integers $x,y,z$ and $p > 3$, a prime. This implies in particular Fermat's Last Theorem.
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Leopoldt's Conjecture for CM fields
The conjecture of Leopoldt states that the $p$ - adic regulator of a number field does not vanish. It was proved for the abelian case in 1967 by Brumer, using Baker theory. We prove this conjecture for CM number fields $\K$. The proof uses Iwasawa's methods -- especially Takagi Theory -- for deriving his skew symmetric pairing, together with Kummer- and Class Field Theory.
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On the vanishing of Iwasawa's constant $μ$ for the cyclotomic $\Z_p$-extensions of $CM$ number fields
We prove that $μ= 0$ for the cyclotomic $\Z_p$-extensions of CM number fields.
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On the equation X^n-1=B.Z^n
We consider the Diophantine equation X^n - 1 = B.Z^n, where B in Z is understood as a parameter. We prove that if the equation has a solution, then either the Euler totient of the radical, phi(rad (B)), has a common divisor with the exponent n, or the exponent is a prime and the solution stems from a solution to the diagonal case of the Nagell-Ljunggren equation: (X^n-1)/(X-1) = n^e.Y^n; e = 0 or 1. This allows us to apply recent results on this equation to the binary Thue equation in question. In particular, we can then display parametrized families for which the Thue equation has no solution. The first such family was proved by Bennett in his seminal paper on binary Thue equations.
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The $T$ and $T^*$ components of $Λ$ - modules and Leopoldt's conjecture
The conjecture of Leopoldt states that the $p$ - adic regulator of a number field does not vanish. It was proved for the abelian case in 1967 by Brumer, using Baker theory. A conjecture, due to Gross and Kuz'min will be shown here to be in a deeper sense a dual of Leopoldt's conjecture with respect to the Iwasawa involution. We prove both conjectures for arbitrary number fields $\K$. The main ingredients of the proof are the Leopoldt reflection, the structure of quasi - cyclic $\Z_p[ \Gal(\K/\Q) ]$ - modules of some of the most important $Λ[ \Gal(\K/\Q) ]$ - modules occurring ($T$ acts on them like a constant in $\Z_p$), and the Iwasawa skew symmetric pairing. There a simplified presentation of the Iwasawa linear space and the proofs of the Conjectures of Leopoldt and Gross-Kuz'min can be found, together with a proof of $lambda^+ = 0$ for CM fields. The present paper is at present the only one which presents the approach for non CM extensions. This will be in time incorporated in the exposition of Snoqit, allowing the proofs of all mentioned conjectures for general number fields. Only then will the present paper become obsolete.
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The Artin Symbol as a Canonical Capitulation Map
We show that there is a canonical, order preserving map $ψ$ of lattices of subgroups, which maps the lattice $\Sub(A)$ of subgroups of the ideal class group of a galois number field $\K$ into the lattice $\Sub(\KH/\K)$ of subfields of the Hilbert class field. Furthermore, this map is a capitulation map in the sense that all the primes in the classes of $A' \subset A$ capitulate in $ψ(A')$. In particular we have a new, strong version of the generalized Hilbert 94 Theorem, which confirms the result of Myiake and adds more structure to (part) of the capitulation kernel of subfields of $\KH$.
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Seminar Notes on Open Questions in Iwasawa Theory - SNOQIT I: The $Λ[ G ]$-modules of Iwasawa theory II: Units and Kummer theory in Iwasawa extensions
In Part I we review some specific properties of the $Λ$-modules in Iwasawa theory, which add structure to the general properties of Noetherian $Λ$-torsion modules. Part II deals with Kummer theory and gives a detailed construction of the Iwasawa linear space. This provides a new, simpler proof of the conjectures of Leopoldt and Gross for CM extensions. Using a construction of Thaine, we then prove that $λ^+ = 0$ in these fields, thus proving a part of Greenberg's conjecture - the fact $μ^+ = 0$ still has to be shown. In the Appendices we give some elementary partial proofs of the main facts proved using Iwasawa's linear space. These two papers do not give proofs for non CM fields, and the reader interested in methods for dealing with this case is referred to the "$T and T^*$" paper on this arxive. This methods will be integrated in the Snoqit series.
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Iwasawa's constant $μ$ vanishes in cyclotomic $\Z_p$-extensions of CM fields
Let $\K$ be a galois CM extension of $\Q$ and $\K_{\infty}$ its cyclotomic $\Z_p$-extension. Let $A_n$ be the $p$-parts of the class groups in the intermediate subfields $\K_n \subset \K_{\infty}$ and $\rg{A} = \varprojlim_n A_n$. We show that the $p$-rank of $\rg{A}$ is finite, which is equivalent to the vanishing of Iwasawa's constant $μ$ for $\rg{A}$. (Currently withdrawn)
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The Gross - Kuz'min conjecture for CM fields
Let $A' = \varprojlim_n$ be the projective limit of the $p$-parts of the ideal class groups of the $p$ integers in the $\Z_p$-cyclotomic extension $\K_{\infty}/\K$ of a CM number field $\K$. We prove in this paper that the $T$ part $(A')^-(T) = 0$. This fact has been explicitly conjecture by Kuz'min in 1972 and was proved by Greenberg in 1973, for abelian extensions $\K/\Q$. Federer and Gross had shown in 1981 that $(A')^-(T) = 0$ is equivalent to the non-vanishing of the $p$-adic regulator of the $p$-units of $\K$.
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The Leopoldt and Iwasawa Conjectures for CM fields
This is a two - part paper, in which we prove the following fact: let K be a CM field and L/K be a CM Z_p-extension. Then the Iwasawa mu-invariant of L vanishes. For the case when L is the cyclotomic Z_p extension, this is the Iwasawa conjecture stating that mu = 0 - a fact which had been proven for abelian fields by Ferrero and Washington. If L is not cyclotomic, than the Leopoldt conjecture fails for K. In this case we show that there is some auxiliar CM Z_p-extension for which mu does not vanish. This is a contradiction, showing that the Leopoldt conjecture must hold for CM fields. Concerning the ideas of the proof, they use class field theory and a concept of stability of Lambda modules under deformation by Thaine shifts, which is developed explicitly in the papers. This combined paper makes obsolete earlier versions and attempts to prove parts of this result, which are found on this arxiv.
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SNOQIT I: Growth of $Λ$-modules and Kummer theory
The paper contains at the end a proof of the conjecture of Gross - Kuz'min, for CM extensions of Q. The main topic of the paper is the investigation of the growth of order and ranks at finite levels of some Lambda modules (p-parts of ideal class groups).
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Cyclotomy Primality Proofs and their Certificates
The first efficient general primality proving method was proposed in the year 1980 by Adleman, Pomerance and Rumely and it used Jacobi sums. The method was further developed by H. W. Lenstra Jr. and more of his students and the resulting primality proving algorithms are often referred to under the generic name of Cyclotomy Primality Proving (CPP). In the present paper we give an overview of the theoretical background and implementation specifics of CPP, such as we understand them in the year 2007.
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Dual Elliptic Primes and Applications to Cyclotomy Primality Proving
Two rational primes p, q are called dual elliptic if there is an elliptic curve E mod p with q points. They were introduced as an interesting means for combining the strengths of the elliptic curve and cyclotomy primality proving algorithms. By extending to elliptic curves some notions of galois theory of rings used in the cyclotomy primality tests, one obtains a new algorithm which has heuristic cubic run time and generates certificates that can be verified in quadratic time. After the break through of Agrawal, Kayal and Saxena has settled the complexity theoretical problem of primality testing, some interest remains for the practical aspect of state of the art implementable proving algorithms.
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The Fuzzy Vault for fingerprints is Vulnerable to Brute Force Attack
The \textit{fuzzy vault} approach is one of the best studied and well accepted ideas for binding cryptographic security into biometric authentication. The vault has been implemented in connection with fingerprint data by Uludag and Jain. We show that this instance of the vault is vulnerable to brute force attack. An interceptor of the vault data can recover both secret and template data using only generally affordable computational resources. Some possible alternatives are then discussed and it is suggested that cryptographic security may be preferable to the one - way function approach to biometric security.
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A Cylcotomic Investigation of the Fermat - Catalan Equation
We give some new, simple results on the equation X^p + Y^p = Z^q.
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