Largest fake prime number weighs in at 300 billion digits

2013 ◽  
Vol 217 (2904) ◽  
pp. 14
Author(s):  
Jacob Aron
Keyword(s):  
Prime Number

scholarly journals Sign changes in the prime number theorem

2021 ◽  
Author(s):  
Thomas Morrill ◽  
Dave Platt ◽  
Tim Trudgian
Keyword(s):  
Prime Number ◽  
Prime Number Theorem ◽  
Sign Changes

3126. A Prime Number Conjecture

1965 ◽  
Vol 49 (369) ◽  
pp. 299
Author(s):  
T. Mitsopoulos
Keyword(s):  
Prime Number

Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiuya Wang
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Finite Groups ◽  
Upper Bound ◽  
Abelian Groups ◽  
Class Groups ◽  
Abelian Extensions ◽  
Pointwise Bound ◽  
First Case

AbstractElementary abelian groups are finite groups in the form of {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p. For every integer {\ell>1} and {r>1}, we prove a non-trivial upper bound on the {\ell}-torsion in class groups of every A-extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G, the {\ell}-torsion in class groups are bounded non-trivially for every G-extension and every integer {\ell>1}. When r is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg and Venkatesh under GRH.

scholarly journals The restricted Burnside problem for Moufang loops

2021 ◽  
pp. 1-11
Author(s):  
ALEXANDER GRISHKOV ◽  
LIUDMILA SABININA ◽  
EFIM ZELMANOV
Keyword(s):  
Prime Number ◽  
Burnside Problem ◽  
Moufang Loops ◽  
Restricted Burnside Problem ◽  
Positive Integers

Abstract We prove that for positive integers $m \geq 1, n \geq 1$ and a prime number $p \neq 2,3$ there are finitely many finite m-generated Moufang loops of exponent $p^n$ .

Brauer characters and fields of values

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Dan Rossi
Keyword(s):  
Prime Number ◽  
Brauer Characters

AbstractFor a prime number

scholarly journals Le complémentaire des puissances -ièmes dans un corps de nombres est un ensemble diophantien

2015 ◽  
Vol 151 (10) ◽  
pp. 1965-1980 ◽  
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Jan Van Geel
Keyword(s):  
Prime Number ◽  
Rational Points ◽  
Parameter Family ◽  
Symmetric Power ◽  
Symmetric Powers

For $n=2$ the statement in the title is a theorem of B. Poonen (2009). He uses a one-parameter family of varieties together with a theorem of Coray, Sansuc and one of the authors (1980), on the Brauer–Manin obstruction for rational points on these varieties. For $n=p$, $p$ any prime number, A. Várilly-Alvarado and B. Viray (2012) considered analogous families of varieties. Replacing this family by its $(2p+1)$th symmetric power, we prove the statement in the title using a theorem on the Brauer–Manin obstruction for rational points on such symmetric powers. The latter theorem is based on work of one of the authors with Swinnerton-Dyer (1994) and with Skorobogatov and Swinnerton-Dyer (1998), work generalising results of Salberger (1988).

scholarly journals ABELIAN GROUPS WITH A p2-BOUNDED SUBGROUP, REVISITED

2011 ◽  
Vol 10 (03) ◽  
pp. 377-389
Author(s):  
CARLA PETRORO ◽  
MARKUS SCHMIDMEIER
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Maximal Ideal ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Uniserial Ring

Let Λ be a commutative local uniserial ring of length n, p be a generator of the maximal ideal, and k be the radical factor field. The pairs (B, A) where B is a finitely generated Λ-module and A ⊆B a submodule of B such that pmA = 0 form the objects in the category [Formula: see text]. We show that in case m = 2 the categories [Formula: see text] are in fact quite similar to each other: If also Δ is a commutative local uniserial ring of length n and with radical factor field k, then the categories [Formula: see text] and [Formula: see text] are equivalent for certain nilpotent categorical ideals [Formula: see text] and [Formula: see text]. As an application, we recover the known classification of all pairs (B, A) where B is a finitely generated abelian group and A ⊆ B a subgroup of B which is p2-bounded for a given prime number p.

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