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Fermat Numbers F n = 2 2n + 1

1919
*
The American mathematical monthly
*

By means of this result

doi:10.2307/2971748
fatcat:azv2qzojrrbnpknwrnjcgs4dtm
*it*is easy to establish Pepin's theorem concerning prime numbers*F*.: For*n*> 1,*F*, is prime if*and*only if*it*divides kl(Fn-l) + 1, where k is any quadratic non-residue of*F*,, as ... Consider the possibility that a =*n*-1. Then from (1)*it*follows that k2 + 1 is divisible by p*and*hence that k > 22"; whence we have a contradiction. Therefore, a _*n*-*2*. ...##
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n-Transitivity and the complementation property

2001
*
Linear Algebra and its Applications
*

We investigate the structure of (minimal)

doi:10.1016/s0024-3795(01)00246-4
fatcat:gsmonicvrjabfk6ewslvc4b3wi
*n*-transitive semigroups of matrices of rank at most*n*in M k (*F*), by considering an equivalent problem regarding certain families of (*n*− k)-dimensional subspaces ... of*F*k . ...*It*follows that L has the complementation*property*in*F*4 if*and*only if for every choice of non-zero x, z in*F**2*there exists B ∈ B such that Bx is not a multiple of z. ...##
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On AGT description of $ \mathcal{N} = 2 $ SCFT with N f = 4

2010
*
Journal of High Energy Physics
*

We consider Alday-Gaiotto-Tachikawa (AGT) realization of the Nekrasov partition function of

doi:10.1007/jhep01(2010)097
fatcat:hshq2d2emna4znk4zbratyn5xe
*N*=*2*SCFT. ... We also make some comments on elementary surface operators*and*WZW theory. ... The*N**f*= 4 theory has SO(8) global symmetry,*and**it*turns out that the SO(8) triality mixes with the SL(*2*, Z) duality symmetry of the theory in a funny way. ...##
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Fibonacci groups, F(2,n), are hyperbolic for n odd and n >= 11
[article]

2020
*
arXiv
*
pre-print

We do this by applying a curvature argument to an arbitrary van Kampen diagram of

arXiv:2005.10653v1
fatcat:etnqyg4lbzcxxkdc5a4dzkih74
*F*(*2*,*n*)*and*show that*it*satisfies a linear isoperimetric inequality.*It*then follows that*F*(*2*,*n*) is hyperbolic. ... We prove that the Fibonacci group,*F*(*2*,*n*), for*n*odd*and**n*>= 11 is hyperbolic. ... I would also like to thank Caitlin for her generous help*and*advice. ...##
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The quasiconformal subinvariance property of John domains in ^n and its application
[article]

2013
*
arXiv
*
pre-print

The main aim of this paper is to give a complete solution to one of the open problems, raised by Heinonen from 1989, concerning the subinvariance of John domains under quasiconformal mappings in ^

arXiv:1108.5550v2
fatcat:grjuoszycbejjgtmiaynartwzu
*n*. ... But we get from (3.8), (3.19)*and*Theorem H that Mod*f*(G 1 ),*f*(B*2*);*f*B(z 1,0 , d D (z 1,0 )) ≤ Mod (*f*(G 1 ),*f*(B*2*); R*n*) ≤ ψ*n*1 4 ρ 18 , which implies φ*n*4 7 cρ 11 ≤ µ 3 (*n*, ρ)Kψ*n*1 4 ρ ... Then Mod (E,*F*; R*n*) ≤ µ 3 (*n*, c)Mod (E,*F*; G) for every pair of continua E*and**F*in G, where µ 3 (*n*, c) is a constant depending on*n**and*c. Definition 2.2. ...##
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Some properties of S(n)-θ-closed spaces

1999
*
Topology and its Applications
*

Recently Dikranjan

doi:10.1016/s0166-8641(98)00015-7
fatcat:7y53byyilzcfbkf5w2kcvsowla
*and*Giuli characterized S(*n*)-θ-closed spaces in terms of special covers*and*filters. They also gave some interesting results*and*examples, as well as listing several open problems. ... We give conditions that make an S(*n*)-θ-closed space Katetov S(*n*). Furthermore, we provide examples which answer three of their questions in the general case. ... To prove that (X, τ ) is an S(*n*)-space,*it*is enough to show that ad θ*n*{*F*α } = ad θ*n*{V ∩*F*: V is a τ neighbourhood of x*and**F*∈*F*α } = {p}.*It*is clear that ad θ*n*{*F*α } ⊃ {p}. ...##
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Exact partition functions for deformed N = 2 $$ \mathcal{N}=2 $$ theories with N f = 4 $$ {\mathcal{N}}_f=4 $$ flavours

2016
*
Journal of High Energy Physics
*

combinations of Eisenstein series

doi:10.1007/jhep12(2016)029
fatcat:wr2uvczzgjctnftiu5qy4fy3ia
*and*Jacobi theta functions with well defined modular*properties*. ... We consider the Ω-deformed*N*=*2*SU(*2*) gauge theory in four dimensions with N_f=4 massive fundamental hypermultiplets. ... The total prepotential is therefore remarkably simple*and*reads*F*(α,β) (ν) = −β ∆ m log ν Λ − β log 1 +*N**n*=1 M 2n (q) ν*2**n*. (2.6) The*properties*of this result have been discussed in the introduction ...##
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Drazin-monotonicity characterizations for property-n matrices

1985
*
Linear Algebra and its Applications
*

A square matrix A is said to have prmerty

doi:10.1016/0024-3795(85)90259-9
fatcat:w65uahxqazfslidenmmj7q7p2a
*n*if there exists a nonnegative power of A. ... In this paper, necessary*and*sufficient conditions for such matrices to have a nonnegative Drazin inverse are presented. ... Then A has*property**n*if*and*only if A has*property**n*(w) for each integer w >, ind(A). Proof. Suppose that A is Drazin-monotone,*and*let A have*property**n*. ...##
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On the relevance of the differential expressions f^2+f'^2, f+f" and f f"- f'^2 for the geometrical and mechanical properties of curves
[article]

2011
*
arXiv
*
pre-print

We present a unified approach to known

arXiv:1102.1579v1
fatcat:m2zxzstnxvhffcqgr27qvecp44
*and*new*properties*of curves by showing the ubiquity of the expressions in the title in the analytic treatment of their mechanical*and*geometric*properties*... cubic),*n*= 1 3 Cayley's sextic,*n*= 1*2*is the cardioid*and**n*=*2*the leminiscate. ... Then if we compute the coefficients of the two fundamental forms, we find that E =*f**2*,*F*= 0, G =*f**2*+*f*′*2**and*L =*f**2*/(*f**2*+*f*′*2*) 1/*2*, M = 0,*N*= (*f*′*2*−*f**f*′′ )/(*f**2*+*f*′*2*) 1/*2*. ...##
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Iteration of the number-theoretic function f(2n) = n, f(2n + 1) = 3n + 2

1977
*
Advances in Mathematics
*

If so, the list of parity sequences for m = 1,

doi:10.1016/0001-8708(77)90087-1
fatcat:pwzefdmmubb55pll4wmuhcfidu
*2*,... in (3) would be rather remarkable, in view of the following*property*, which*it*does indeed have; namely, the*2** parity sequences for the integers m ... If we set A, = A(29*and*define TZN = 2N + {(2N+1 -2N) -(AN+1 -AN)} = AN + (2N+1 -AN+,) 3 2N, (10) then for M = 2N + l,..., nN,*it*is For the function*f*defined in Eq. ...##
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Topological properties of some algebraically defined subsets ofβN

2017
*
Topology and its Applications
*

ideal;

doi:10.1016/j.topol.2017.02.001
fatcat:kk6wc6yozvbu7nmabmltbp5u6m
*and**N** +*N** . ... Let S be a discrete semigroup*and*let the Stone-Čech compactification βS of S have the operation extending that of S which makes βS a right topological semigroup with S contained in*its*topological center ...*F*contains the open subsets of β*N*, because β*N*has a basis of c clopen sets,*and**F*obviously contains the closed subsets of β*N*.*It*is also obvious that*F*is closed under countable unions. ...##
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Weaknesses in the SL2($$ \mathbb{F}_{2^n } $$) Hashing Scheme
[chapter]

2000
*
Lecture Notes in Computer Science
*

We show that for various choices of the parameters in the SL2(IF2n ) hashing scheme, suggested by Tillich

doi:10.1007/3-540-44598-6_18
fatcat:hdfdsqtfv5ferjze5i2aq3rnvq
*and*Zémor, messages can be modified without changing the hash value. ... Moreover, we would like to thank the referees for several useful remarks*and*references. ... Further*properties*of the group SL*2*(IF p*n*) are summarized as follows: Remark 1. For any subfield IF p m ≤ IF p*n*of IF p*n*, SL*2*(IF p m )*and*all*its*conjugates are subgroups of SL*2*(IF p*n*). ...##
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On the Analytic Solutions of the Functional Equations w 1 f(a 1 z) + w 2 f(a 2 z) + ... + w n f(a n z) = 0

2014
*
Mediterranean Journal of Mathematics
*

R(z) = cw 1 a z 1 e γz + cw

doi:10.1007/s00009-014-0444-8
fatcat:vrouftgdyjclrf54zqez5pjg6i
*2*a z*2*e γz + . . . + cw*n*a z*n*e γz*and**it*is associated to the functional equation v 1*f*(b 1 z) + v*2**f*(b*2*z) + . . . + v*n**f*(b*n*z) = 0, z ∈ C, (4.4) with v j = cw ... to*it*. ...##
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The Synthesis of Sodium(and Potassium) N, N'-bis-(o-hydro xybenzoyl)-ethylenediaminato-Cuprate(II) and Its Properties

1967
*
Nippon kagaku zassi
*

3) PH(電 解 液) ○:[CuHBen】

doi:10.1246/nikkashi1948.88.3_329
fatcat:lfvgrwp525gb3jle4ptoznvq2u
*2*㍉ △:[NiHBen]2ww 電 解 波:HNO、 一 一N4NO・-NaOH系(0・05*F*) 図3[MHBcn]*2*一 の ロ 紙 電 気 泳 動 ③ ㌻ き*N*l C O / 捕 ＼ 一 ＼ / 瑞 O*N*- C 牟 ' ◎ ね)くc、 ll cH2・ 一閲一CH2 ゆ*2*一 図5〔NiHBen]*2*お よ び 類 似 錯 体 のd→4吸 収 ... な お,Na2[CuHBen]の0・005*F*溶 液(O」*N*硝 酸 ナ ト リウ ム 中)のPH値 は9・0± 二 〇・01を 示 し ・平 衡(3)が OO.5停 >0 .4 詮 費 ごo・3 醤0.2 蕊 α1 図4 OlO÷ 《4筆8101214 PH Na2[CuHBen]のd→d吸 収 帯(540mμ)のpH依 存 性 0 bD o 一 4 ウ 一 15*2*◎2530 ヲ(cm-1×10- ...##
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Duadic Codes over F 2 + u F 2

2001
*
Applicable Algebra in Engineering, Communication and Computing
*

Duadic codes over

doi:10.1007/s002000100079
fatcat:btccoloy4vhgtdoy76vkfcarvi
*F**2*+uF*2*are introduced as abelian codes by their zeros. This is the function field analogue of duadic codes over Z 4 introduced recently by Langevin*and*Solé. ... We classify them in modest lengths*and*exhibit interesting non-cyclic examples. ... Notations*and*Definitions Let R be the commutative ring*F**2*+ uF*2*:=*F**2*[X]/(X*2*). This ring is endowed with the obvious addition*and*multiplication, with the*property*that u*2*= 0. ...
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