A Class of Exponential Congruences in Several Variables

Choi, Geumlan; Zaharescu, Alexandru

doi:10.4134/jkms.2004.41.4.717

Cited by 2 publications

(7 citation statements)

References 7 publications

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“…Recently Bose [15] also generalized Selfridge's question. In 2004, Choi and Zaharescu [39] generalized Theorem 6.6 to the case of n variables as follows. Theorem 6.7 (Choi and Zaharescu [39]).…”

Section: Ayad Andmentioning

confidence: 99%

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A Survey on Fixed Divisors

Prasad¹,

Rajkumar²,

Reddy³

2019

Confluentes Mathematici

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In this article, we compile the work done by various mathematicians on the topic of the fixed divisor of a polynomial. This article explains most of the results concisely and is intended to be an exhaustive survey. We present the results on fixed divisors in various algebraic settings as well as the applications of fixed divisors to various algebraic and number theoretic problems. The work is presented in an orderly fashion so as to start from the simplest case of Z, progressively leading up to the case of Dedekind domains. We also ask a few open questions according to their context, which may give impetus to the reader to work further in this direction. We describe various bounds for fixed divisors as well as the connection of fixed divisors with different notions in the ring of integer-valued polynomials. Finally, we suggest how the generalization of the ring of integer-valued polynomials in the case of the ring of n × n matrices over Z (or Dedekind domain) could lead to the generalization of fixed divisors in that setting.keywords Fixed divisors, Generalized factorials, Generalized factorials in several variables, Common factor of indices, Factoring of prime ideals, Integer valued polynomials NotationsWe fix the notations for the whole paper. R = Integral Domain K = Field of fractions of R N (I) = Cardinality of R/I (Norm of an ideal I ⊆ R) W = {0, 1, 2, 3, . . .} A[x] = Ring of polynomials in n variables (= A[x 1 , . . . , x n ]) with coefficients in the ring A S = Arbitrary (or given) subset of R n such that no non-zero polynomial in K[x] maps it to zero S = S in case when n = 1 Int(S, R) = Polynomials in K[x] mapping S back to R ν k (S) = Bhargava's (generalized) factorial of index k k! S = k th generalized factorial in several variables M m (S) = Set of all m × m matrices with entries in S p = positive prime number Z p = p-adic integers ord p (n) = p-adic ordinal (valuation) of n ∈ Z.

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Section: Ayad Andmentioning

confidence: 99%

“…In 2004, Choi and Zaharescu [39] generalized Theorem 6.6 to the case of n variables as follows. Theorem 6.7 (Choi and Zaharescu [39]). Let R be the ring of integers in an algebraic number field and let b 1 , b 2 , .…”

Section: Ayad Andmentioning

confidence: 99%

A Survey on Fixed Divisors

Prasad¹,

Rajkumar²,

Reddy³

2019

Confluentes Mathematici

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“…Theorem 1.4 (Choi and Zaharescu [2]). Let R be the ring of integers in an algebraic number field and let b 1 , b 2 , .…”

Section: Introductionmentioning

confidence: 99%

“…Sun and M. Zhang [11] also answered Selfridge's question. Actually, there are fourteen such pairs which are solution of Selfridge's question and they are (1,0), (2,1), (3,1), (4,2), (5,1), (5,3), (6,2), (7,3), (8,2), (8,4), (9,3), (14,2), (15,3) and (16,4). Once Selfridge's question is answered completely, a natural question arises: what happens if we replace '2' by '3' or more generally by some other integer (other than ± 1).…”

Section: Introductionmentioning

confidence: 99%

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A generalization of Selfridge's question

Prasad

2019

Preprint

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Selfridge asked to investigate the pairs (m, n) of natural numbers for which 2 m − 2 n divides x m − x n for all integers x. This question was answered by different mathematicians by showing that there are only finitely many such pairs. Let R be the ring of integers of a number field K and Mn(R) be ring of all n × n matrices over R. In this article, we prove a generalization of Selfridge's question in the case of Mn(R).

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A Class of Exponential Congruences in Several Variables

Cited by 2 publications

References 7 publications

A Survey on Fixed Divisors

A Survey on Fixed Divisors

A generalization of Selfridge's question

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