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### The Modular Theory of Polyadic Numbers

Albert A. Bennett
1921 Annals of Mathematics
as a 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.  ...

### On singular moduli that are S-units [article]

Francesco Campagna
2019 arXiv   pre-print
K\"uhne proved that no 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  ...

### Strong Pseudo Primes to Base 2 [article]

Kubra Nari, Enver Ozdemir, Neslihan Aysen Ozkirisci
2019 arXiv   pre-print
Then, we provide theoretical and heuristics evidences showing that the resulting algorithm catches all composite 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.  ...

### Nonsingularity of least common multiple matrices on gcd-closed sets

Shaofang Hong
2005 Journal of Number Theory
For an 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  , 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  ...

### Idempotent factorizations of singular 2× 2 matrices over quadratic integer rings [article]

Laura Cossu, Paolo Zanardo
2019 arXiv   pre-print
Let D be the ring of 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.  ...

### Designing commutative cascades of multidimensional upsamplers and downsamplers

B.L. Evans
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 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.  ...

### On Criteria Concerning Singular Integers in Cyclotomic Fields

H. S. Vandiver
1938 Proceedings of the National Academy of Sciences of the United States of America
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;  ...  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;  ...

### Arithmetic of double series

D. H. Lehmer
1931 Transactions of the American Mathematical Society
The first has for basis the Cauchy multiplication of power series and is appropriately 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.  ...

### Arithmetic of Double Series

D. H. Lehmer
1931 Transactions of the American Mathematical Society
The first has for basis the Cauchy multiplication of power series and is appropriately 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.  ...

### A New Attack on Three Variants of the RSA Cryptosystem [chapter]

Martin Bunder, Abderrahmane Nitaj, Willy Susilo, Joseph Tonien
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 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.  ...

### On the Number of Permutations Admitting an m-th Root

Nicolas Pouyanne
2001 Electronic Journal of Combinatorics
Let \$m\$ be a positive 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!  ...

### Study of rational cubic forms via the circle method [after D.R. Heath-Brown, C. Hooley, and R.C. Vaughan]

Jean-Marc Deshouillers
1990 Séminaire de Théorie des Nombres de Bordeaux
computed data Let 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.  ...

### Partial total least squares algorithm

Sabine Van Huffel
1990 Journal of Computational and Applied Mathematics
efficiency of the classical TLS algorithm (a 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.  ...

### The method of Darboux

R Wong, M Wyman
1974 Journal of Approximation Theory
We assume, as did Darboux, that on the circle of convergence F(t) has only a finite 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.  ...

### Design and Performance Analysis of Fixed-point Jacobi SVD Algorithm on Reconfigurable System

Ramanarayan Mohanty, Gonnabhaktula Anirudh, Tapan Pradhan, Bibek Kabi, Aurobinda Routray
2014 Information Engineering Research Institute procedia
Accuracy is compared on the basis of 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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