On sums of polynomial-type exceptional units in $$\varvec{\mathbb {Z}}/\varvec{n\mathbb {Z}}$$

Anand,; Chattopadhyay, Jaitra; Roy, Bidisha

doi:10.1007/s00013-019-01413-7

Cited by 4 publications

(2 citation statements)

References 16 publications

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“…In view of Proposition 2.2, to prove Theorem 1.1, it is sufficient to prove that the class numbers of all the fields of Theorem 1.1 are indivisible by p. We accomplish this by using Dirichlet's class number formula for the aforementioned fields and showing that all of their discriminants have large square factors, which is essentially a modification of the arguments used in [17]. We state a proposition from [1] which will be used to produce infinitely many primes p with p − 1, . .…”

Section: Preliminariesmentioning

confidence: 99%

On the $p$-rationality of consecutive quadratic fields

Chattopadhyay¹,

Laxmi²,

Saikia³

2022

Preprint

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In 2016, in the work related to Galois representations, Greenberg conjectured the existence of multi-quadratic p-rational number fields of degree 2 t for any odd prime number p and any integer t ≥ 1. Using the criteria provided by him to check p-rationality for abelian number fields, certain infinite families of quadratic, biquadratic and triquadratic p-rational fields have been shown to exist in recent years. In this article, for any integer k ≥ 1, we build upon the existing work and prove the existence of infinitely many prime numbers p for which the imaginary quadratic fields Q( −(p − 1)), . . . , Q( −(p − k)) and Q( −p(p − 1)), . . . , Q( −p(p − k)) are all p-rational. This can be construed as analogous results in the spirit of Iizuka's conjecture on the divisibility of class numbers of consecutive quadratic fields. We also address a similar question of p-rationality for two consecutive real quadratic fields by proving the existence of infinitely many p-rational fields of the form Q( p 2 + 1) and Q( p 2 + 2). The result for imaginary quadratic fields is accomplished by producing infinitely many primes for which the corresponding consecutive discriminants have large square divisors and the same for real quadratic fields is proven using a result of Heath-Brown on the density of square-free values of polynomials at prime arguments.

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Section: Preliminariesmentioning

confidence: 99%

On the $p$-rationality of consecutive quadratic fields

Chattopadhyay¹,

Laxmi²,

Saikia³

2022

Preprint

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“…Let n ≥ 1 be an integer and let f (x) ∈ Z[x]. An integer a is an f-exunit in the ring Z n if gcd( f (a), n) = 1 (see [1]). We denote by E f (n) the set of all f -exunits in the ring Z n .…”

Section: Introductionmentioning

confidence: 99%

Sums of Polynomial-Type Exceptional Units Modulo

Zhao¹,

Hong²,

Zhu³

2021

Bull. Aust. Math. Soc.

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Let $f(x)\in \mathbb {Z}[x]$ be a nonconstant polynomial. Let $n\ge 1, k\ge 2$ and c be integers. An integer a is called an f-exunit in the ring $\mathbb {Z}_n$ of residue classes modulo n if $\gcd (f(a),n)=1$ . We use the principle of cross-classification to derive an explicit formula for the number ${\mathcal N}_{k,f,c}(n)$ of solutions $(x_1,\ldots ,x_k)$ of the congruence $x_1+\cdots +x_k\equiv c\pmod n$ with all $x_i$ being f-exunits in the ring $\mathbb {Z}_n$ . This extends a recent result of Anand et al. [‘On a question of f-exunits in $\mathbb {Z}/{n\mathbb {Z}}$ ’, Arch. Math. (Basel)116 (2021), 403–409]. We derive a more explicit formula for ${\mathcal N}_{k,f,c}(n)$ when $f(x)$ is linear or quadratic.

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