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The Modular Theory of Polyadic Numbers

1921
*
Annals of Mathematics
*

as a

doi:10.2307/1967788
fatcat:bem4s2cr7vhtnblxlfdnuiuqsm
*factor*. ... It may be readily proved that every positive rational*number*P/Q when Q is not a*factor*of any power of the base, 10, may be written as a decadic*integer*. ...##
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On singular moduli that are S-units
[article]

2019
*
arXiv
*
pre-print

K\"uhne proved that no

arXiv:1904.08958v1
fatcat:4npocxowh5dgfavt76ybvuecse
*singular*modulus can be a unit in the ring of algebraic*integers*. In this paper we study for which sets S of prime*numbers*there is no*singular*modulus that is an S-units. ... We then give some remarks on the general case and we study the norm*factorizations*of a special family of*singular*moduli. ... Another possible research path is the following: let S be a finite set of prime*numbers*. What is the*number*of*singular*moduli that are S-units? By the theorem above, when Acknowledgments ...##
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Strong Pseudo Primes to Base 2
[article]

2019
*
arXiv
*
pre-print

Then, we provide theoretical and heuristics evidences showing that the resulting algorithm catches all composite

arXiv:1905.06447v1
fatcat:zsqzhtwt7bcz3ahoxjttmkfcqi
*numbers*. ... Our method is based on the structure of*singular*cubics' Jacobian groups on which we also define an effective addition algorithm. ... Let n be an odd*integer*. Let*us*define a*singular*cubic E with the equation E : y 2 = x(x − a) 2 on Z n . If the*integer*n is prime, then the order of E(Z n ) must be n + 1. ...##
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Nonsingularity of least common multiple matrices on gcd-closed sets

2005
*
Journal of Number Theory
*

For an

doi:10.1016/j.jnt.2005.03.004
fatcat:fsrtnxtnpne5vh367vpp7kphha
*integer*m > 1, let (m) denote the*number*of distinct prime*factors*of m. Define (1) = 0. ... For each*integer*r 3, there exists a gcd-closed set S satisfying max x∈S { (x)} = r, such that the LCM matrix [S] is*singular*... Applying the methods*used*in this paper and in [7] , one can prove that each positive*integer*less than 180 is a nonsingular*number*, thus establishing that x = 180 is the least primitive*singular**number*...##
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Idempotent factorizations of singular 2× 2 matrices over quadratic integer rings
[article]

2019
*
arXiv
*
pre-print

Let D be the ring of

arXiv:1910.01893v1
fatcat:wzenozzrqjempe3dlscyyjo424
*integers*of a quadratic*number*field Q[√(d)]. We study the*factorizations*of 2 × 2 matrices over D into idempotent*factors*. ... When d < 0 there exist*singular*matrices that do not admit idempotent*factorizations*, due to results by Cohn (1965) and by the authors (2019). We mainly investigate the case d > 0. ... In this paper we investigate idempotent*factorizations*of*singular*2 × 2 matrices over quadratic*integer*rings. ...##
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Designing commutative cascades of multidimensional upsamplers and downsamplers

1997
*
IEEE Signal Processing Letters
*

In multiple dimensions, the cascade of an upsampler by L and a downsampler by L commutes if and only if the

doi:10.1109/97.641397
fatcat:xug3e56eqzb3ljvls42dhxpku4
*integer*matrices L and M are right coprime and LM = ML. ... So, L and M are always right coprime provided that each rational*number*on the diagonal of is reduced to l m where l and m are coprime*integers*. ... We h a v e freedom to choose a non-*singular*diagonal rational matrix and a non-*singular**integer*matrix U, which are both m m matrices. ...##
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On Criteria Concerning Singular Integers in Cyclotomic Fields

1938
*
Proceedings of the National Academy of Sciences of the United States of America
*

Necessary conditions that an

doi:10.1073/pnas.24.8.330
pmid:16588232
pmcid:PMC1077106
fatcat:d7idtpb22fcillitpqs34cq5ca
*integer*in k(¢) be*singular*were given by Takagi,2 and when the field k(¢) is properly irregular, that is to say, when the second*factor*of its class*number*is prime to 1; ... As elsewhereI a*singular**integer*is defined as an*integer*a in the field k(W); = e2 '/, 1 an odd prime, such that a = a' where a is an ideal in k(r) which is not principal. ... Necessary conditions that an*integer*in k(¢) be*singular*were given by Takagi,2 and when the field k(¢) is properly irregular, that is to say, when the second*factor*of its class*number*is prime to 1; ...##
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Arithmetic of double series

1931
*
Transactions of the American Mathematical Society
*

The first has for basis the Cauchy multiplication of power series and is appropriately

doi:10.1090/s0002-9947-1931-1501625-3
fatcat:ibh3qjhtwnfnbbs5nsphkzjkly
*used*in considering functions sensitive to additive properties of*integers*. ... Properties of*integers*other than additive and multiplicative can be studied by constructing the appropriate theory without regard to the corresponding infinite series § (if it exists). ... We may consider for example the set of all simple*numbers*, i.e., all*numbers*not divisible by a square >1. The L.C.M.*Factorable*i/'-functions. ...##
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Arithmetic of Double Series

1931
*
Transactions of the American Mathematical Society
*

The first has for basis the Cauchy multiplication of power series and is appropriately

doi:10.2307/1989517
fatcat:i75nxdt7yzcfbbbwaybbxbvc2y
*used*in considering functions sensitive to additive properties of*integers*. ... Properties of*integers*other than additive and multiplicative can be studied by constructing the appropriate theory without regard to the corresponding infinite series § (if it exists). ... We may consider for example the set of all simple*numbers*, i.e., all*numbers*not divisible by a square >1. The L.C.M.*Factorable*i/'-functions. ...##
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A New Attack on Three Variants of the RSA Cryptosystem
[chapter]

2016
*
Lecture Notes in Computer Science
*

Then, in 2002, Elkamchouchi, Elshenawy and Shaban introduced an extension of the RSA scheme to the field of Gaussian

doi:10.1007/978-3-319-40367-0_16
fatcat:tci2vx2qkbcuxbnaryfvqfzc2u
*integers**using*a modulus N = PQ where P and Q are Gaussian primes such that p = |P| ... Then, in 2002, Elkamchouchi, Elshenawy and Shaban introduced an extension of the RSA scheme to the field of Gaussian*integers**using*a modulus N = P Q where P and Q are Gaussian primes such that p = |P ... The existence of prime*factorization*in Z[i] allows*us*to consider Gaussian*integers*of the form N = P Q where P and Q are Gaussian primes with large norm. ...##
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On the Number of Permutations Admitting an m-th Root

2001
*
Electronic Journal of Combinatorics
*

Let $m$ be a positive

doi:10.37236/1620
fatcat:5afmdqh6hjhitkudxvdsdyhmn4
*integer*, and $p_n(m)$ the proportion of permutations of the symmetric group $S_n$ that admit an $m$-th root. ... For every positive*integer*m, the*number*c n (m)×n! ... Let m be a positive*integer*. The*number*p n (m) × n! ...##
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Study of rational cubic forms via the circle method [after D.R. Heath-Brown, C. Hooley, and R.C. Vaughan]

1990
*
Séminaire de Théorie des Nombres de Bordeaux
*

computed data Let

doi:10.5802/jtnb.37
fatcat:swpayqhibjcmtbxpkw26stwcv4
*us*look at a table which gives, for every*integer*N up to 40,000 the minimal*number*of summands needed to writer N as a sum of positive integral cubes : the*numbers*23 and 239 are the ... In this way, a*factor*of the order p2 may be won. ...##
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Partial total least squares algorithm

1990
*
Journal of Computational and Applied Mathematics
*

efficiency of the classical TLS algorithm (a

doi:10.1016/0377-0427(90)90261-w
fatcat:ge6kobt7jbcprakqjkvmis36ya
*factor*2 approximately). ... ABSTRACT : The Partial Total least Squares CPTLS) subroutine solves the Total Least Squares CTLS) problem AX z B by*using*a Partial*Singular*Value Decomposition CPSVD), hereby improving the computational ... L -*INTEGER*. The*number*of columns in the observation matrix B. RANK -*INTEGER*. The rank of the TLS approximation CA+DA;B+DBl. ...##
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The method of Darboux

1974
*
Journal of Approximation Theory
*

We assume, as did Darboux, that on the circle of convergence F(t) has only a finite

doi:10.1016/0021-9045(74)90114-2
fatcat:4n4lmiy7ifbiva22ajoslhciua
*number*of*singularities*. ... Therefore,*using*the canonical form, it is assumed that F(t) has a*singularity*at t = 1, and is regular within and on the contour C shown below. ...##
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Design and Performance Analysis of Fixed-point Jacobi SVD Algorithm on Reconfigurable System

2014
*
Information Engineering Research Institute procedia
*

Accuracy is compared on the basis of

doi:10.1016/j.ieri.2014.08.005
fatcat:cvw7kcxhvrevlf3zer75wgmykm
*number*of accurate fractional bits, SQNR, orthogonality and*factorization*errors.*Number*of accurate fractional bits is computed*using*(7). From Fig. 2. ... Primary reasons for the evolution and accumulation of these errors in FPGA implementation are -( ) i*Use*of 16-bit multipliers as a correction*factor*and ( )ii Insufficient*integer*wordlength and fractional ...
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