Adding Units Mod n

Deaconescu, Marian

doi:10.1007/s000170050078

Cited by 15 publications

(12 citation statements)

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“…The equivalent formula The special case of k = 2 was treated, independently, by Alder [1], Deaconescu [11], and Sander [32]. For k = 2 the function N n (2, b) coincides with Nagell's totient function ( [27]) defined to be the number of integers x (mod n) such that (x, n) = (b − x, n) = 1.…”

Section: Interestingly This Classical Results Of D N Lehmer Has Beementioning

confidence: 99%

“…It is also worth mentioning that the case of k = 2 is related to a long-standing conjecture due to D. H. Lehmer from 1932 (see [11,12]), and also has interesting applications to Cayley graphs (see [32,33]). …”

Section: Interestingly This Classical Results Of D N Lehmer Has Beementioning

confidence: 99%

“…The problem in the case of k variables can be interpreted as a 'restricted partition problem modulo n' ( [28]), or an equation in the ring Z n , where the solutions are its units ( [11,32,33]). …”

Section: Interestingly This Classical Results Of D N Lehmer Has Beementioning

confidence: 99%

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Restricted linear congruences

Bibak

Kapron

Srinivasan

et al. 2017

Journal of Number Theory

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In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence, where a 1 , t 1 , . . . , a k , t k , b, n (n ≥ 1) are arbitrary integers. As a consequence, we derive necessary and sufficient conditions under which the above restricted linear congruence has no solutions. The number of solutions of this kind of congruence was first considered by Rademacher in 1925 and Brauer in 1926, in the special case of a i = t i = 1 (1 ≤ i ≤ k). Since then, this problem has been studied, in several other special cases, in many papers; in particular, Jacobson and Williams [Duke Math. J. 39 (1972), 521-527] gave a nice explicit formula for the number of such solutions when (a 1 , . . . , a k ) = t i = 1 (1 ≤ i ≤ k). The problem is very well-motivated and has found intriguing applications in several areas of mathematics, computer science, and physics, and there is promise for more applications/implications in these or other directions.

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Section: Interestingly This Classical Results Of D N Lehmer Has Beementioning

confidence: 99%

Section: Interestingly This Classical Results Of D N Lehmer Has Beementioning

confidence: 99%

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Restricted linear congruences

Bibak

Kapron

Srinivasan

et al. 2017

Journal of Number Theory

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“…which also follows immediately from results of Deaconescu [1] or Sander [13]. In [13, Theorem 1.1], the number of representations of an element c ∈ Z n as the sum of two units was determined to be ϕ * (n, c) := n p|n, p|c…”

Section: Introductionmentioning

confidence: 97%

On the sumsets of exceptional units in $$\mathbb {Z}_n$$ Z n

Yang

Zhao

2016

Monatsh Math

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Let R be a commutative ring with 1 ∈ R and R * be the multiplicative group of its units. In 1969, Nagell introduced the concept of an exceptional unit, namely a unit u such that 1 − u is also a unit. Let Z n be the ring of residue classes modulo n. In this paper, given an integer k ≥ 2, we obtain an exact formula for the number of ways to represent each element of Z n as the sum of k exceptional units. This generalizes a recent result of J. W. Sander for the case k = 2.

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“…Therefore it was labelled "Goldbach for residues" (cf. [1] We present a straightforward proof using multiplicativity of |S * * m (n)| with respect to m. The reader is invited to take a look at the interesting remarks in [5] on the relation of this problem with Lehmer's longstanding conjecture [9] saying that for an integer n > 1 Euler's totient function ϕ(n) divides n − 1 only if n is prime.…”

Section: Introduction and Resultsmentioning

confidence: 98%