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Let Fq be the finite field of q elements, and Fqn its extension of degree n. A normal basis of Fq n over Fq is a basis of the form {α, αq, · · ·, αqn−1 }. Some problems on normal bases can be finally reduced to the determination of the irreducible factors of the polynomial x n − 1 in Fq, while the latter is closely related to the cyclotomic polynomials. Denote by δ(xn − 1) the set of all distinct monic irreducible factors of xn − 1 in Fq. The criteria for [δ(xn-1)] ≤2 have been studied in the literature. In this paper, we provide the sufficient and necessary conditions for [δ(xn-1)]=s where s is a positive integer by using the properties of cyclotomic polynomials and results from the Diophantine equations. As an application, we obtain the sufficient and necessary conditions for [δ(xn-1)]=3,4,5 [ABSTRACT FROM AUTHOR]

A Diophantine m-tuple over a finite field F q is a set { a 1 , ... , a m } of m distinct elements in F q ∗ such that a i a j + 1 is a square in F q whenever i ≠ j . In this paper, we study M(q), the maximum size of a Diophantine tuple over F q , assuming the characteristic of F q is fixed and q → ∞ . By explicit constructions, we improve the lower bound on M(q). In particular, this improves a recent result of Dujella and Kazalicki by a multiplicative factor. [ABSTRACT FROM AUTHOR]

Let $ \mathbb{F}_q $ be the finite field of $ q $ elements, and $ \mathbb{F}_{q^{n}} $ its extension of degree $ n $. A normal basis of $ \mathbb{F}_{q^{n}} $ over $ \mathbb{F}_q $ is a basis of the form $ \{\alpha, \alpha^{q}, \cdots, \alpha^{q^{n-1}}\} $. Some problems on normal bases can be finally reduced to the determination of the irreducible factors of the polynomial $ x^{n}-1 $ in $ \mathbb{F}_q $, while the latter is closely related to the cyclotomic polynomials. Denote by $ \mathfrak{F}(x^{n}-1) $ the set of all distinct monic irreducible factors of $ x^{n}-1 $ in $ \mathbb{F}_q $. The criteria for $ |\mathfrak{F}(x^{n}-1)|\leq 2 $ have been studied in the literature. In this paper, we provide the sufficient and necessary conditions for $ |\mathfrak{F}(x^{n}-1)| = s, $ where $ s $ is a positive integer by using the properties of cyclotomic polynomials and results from the Diophantine equations. As an application, we obtain the sufficient and necessary conditions for $ |\mathfrak{F}(x^{n}-1)| = 3, 4, 5. $

We prove two single-parameter q-supercongruences which were recently conjectured by Guo, and establish their further extensions with one more parameter. Crucial ingredients in the proof are the terminating form of the q-binomial theorem, a Karlsson–Minton-type summation formula due to Gasper, and the method of "creative microscoping" developed by Guo and Zudilin. Incidentally, an assertion of Li, Tang and Wang is also confirmed by establishing its q-analogue. [ABSTRACT FROM AUTHOR]

We give a characterization when a nonzero square matrix with integer entries is periodic using cyclotomic polynomials. We also determine the set of all possible periods of periodic matrices with integer entries via Euler Phi functions. [ABSTRACT FROM AUTHOR]

In terms of a very-well-poised 6ϕ5 summation formula, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials, we establish four new q-supercongruences for truncated basic hypergeometric series. One of these results is a new q-analogue of the (F.2) supercongruence of Van Hamme. [ABSTRACT FROM AUTHOR]

In this paper, in view of the transformation formulas for basic hypergeometric series and the creative microscoping method introduced by Guo and Zudilin, we establish new q-analogues of Van Hamme’s (A.2) and (H.2) supercongruences with two parameters [ABSTRACT FROM AUTHOR]

Recently, Guo and Tang independently established some q-supercongruences from Rahman’s quadratic transformation. In this paper, by applying the method of creative microscoping devised by Guo and Zudilin together with Rahman’s quadratic transformation again, we provide proofs for eight conjectures on q-supercongruences proposed by Guo and by Tang. [ABSTRACT FROM AUTHOR]

We prove that the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems over the cyclotomic field Q (ζ n) are not equivalent. Precisely, we show that reducing one problem to the other increases the noise by a factor that is more than polynomial in n. We do so by providing a lower bound, holding for infinitely many positive integers n, for the condition number of the Vandermonde matrix of the nth cyclotomic polynomial. [ABSTRACT FROM AUTHOR]

The integrality of the numbers $A_{n,m}={(2n)!(2m)!}/{n!m!(n+m)!}$ was observed by Catalan as early as 1874 and Gessel named $A_{n,m}$ the super Catalan numbers. The positivity of the q -super Catalan numbers (q -analogue of the super Catalan numbers) was investigated by Warnaar and Zudilin ['A q -rious positivity', Aequationes Math. 81 (2011), 177–183]. We prove the divisibility of sums of q -super Catalan numbers, which establishes a q -analogue of Apagodu's congruence involving super Catalan numbers. [ABSTRACT FROM AUTHOR]

We give some q-supercongruences from a q-analogue of Watson’s F 2 3 {{}_{3}F_{2}} summation and the method of “creative microscoping”, introduced by the author and Zudilin. These q-supercongruences may be considered as further generalizations of the (A.2) supercongruence of Van Hamme modulo p 3 {p^{3}} or p 2 {p^{2}} for any odd prime p. Meanwhile, we confirm a supercongruence conjecture of Wang and Yue through establishing its q-analogue. [ABSTRACT FROM AUTHOR]

Let F q be the finite field of q elements, and F q n its extension of degree n. A normal basis of F q n over F q is a basis of the form { σ i (α) : i = 0 , ⋯ , n - 1 } where σ denotes the Frobenius automorphism of F q n over F q and α ∈ F q n . Normal bases over finite fields have proved very useful for fast arithmetic computations with potential applications to coding theory and to cryptography. Some problems on normal bases are characterized in terms of linearized polynomials. We show that such problems can be finally reduced to the determination of the irreducible factors of x n - 1 over F q . Using this approach, we extend the known results with alternative as well as relatively elementary proofs. And we believe that this approach can be used to deal with other similar problems. [ABSTRACT FROM AUTHOR]

Long proved that Van Hamme's (C.2) supercongruence is also true modulo p 4 for any prime p > 3 . By making use of the q-WZ method, the author and Wang gave a q-analogue of Long's supercongruence. In this paper, employing the method of 'creative microscoping', introduced by the author and Zudilin in 2019, we obtain a generalization of this q-supercongruence. A limiting case of our result implies that, for 0 ⩽ t ⩽ s ⩽ 10 and any odd prime p ⩾ 4 s + 1 and integer r ⩾ 1 , ∑ k = s (p r - 1) / 2 + s 4 k + 1 256 k 2 k - 2 s k - s 2 k + 2 s k + s 2 k - 2 t k - t 2 k + 2 t k + t ≡ p r (mod p r + 3). [ABSTRACT FROM AUTHOR]

We show various criteria to verify if a given nested fractal has a good labeling property, inter alia we present a characterization of GLP for fractals with an odd number of essential fixed points. We show a convenient reduction of the area to be investigated in the verification of GLP and give examples that further reduction is impossible. We prove that if the number of essential fixed points is a power of two, then a fractal must have GLP and that there are no values other than primes or powers of two guaranteeing GLP. For all other numbers of essential fixed points, we are able to construct examples having and others not having GLP. [ABSTRACT FROM AUTHOR]

We complete the recent research by Akiyama and Kaneko on the higher-order derivative values Φ n (k) (1) of the cyclotomic polynomials. This article focuses on Lehmer's explicit formula of Φ n (k) (1) / Φ n (1) as a polynomial of the Euler and Jordan totient functions over Q . Then we prove Akiyama–Kaneko's conjecture that the polynomials have a specific simple factor. [ABSTRACT FROM AUTHOR]

The trinomial coefficients n k are given by ∑ k = - n n n k x k = (1 + x + x - 1) n. Andrews and Baxter listed six kinds of q-trinomial coefficients (q-analogues of the trinomial coefficients). In this paper, we obtain some supercongruences on these q-trinomial coefficients. As a conclusion, we obtain the following new supercongruence: ap b p ≡ a b (mod p 2) , where a, b are positive integers subject to a > b and p > 3 is an odd prime. [ABSTRACT FROM AUTHOR]

Cyclotomic polynomials are an interesting topic and play an important role for other topics in Number Theory. For special values of n, computing a cyclotomic polynomial is not difficult; this can be done by using properties of the polynomial. For a prime value of n, the polynomial can be written quickly. However, for values which are multiples of odd prime numbers, say 85 = 17 ¢ 5, the task can be quite difficult if it must be done manually. The polynomial has 41 terms and the degree of 64. Software for cyclotomic polynomials is available; Maple for example, can solve the problem of cyclotomic polynomials very easily. However, understanding of how to compute polynomials is very important for a student in applying various properties of the cyclotomic polynomial. Here, we will use Maple to help students understand cyclotomic polynomials from the basic. Answer is not obtained directly but step-by-step using properties of the polynomial. So, Maple is used to simulate the process of obtaining the polynomial. Once the students grasp the skill, they will be able to use the software for advanced applications. [ABSTRACT FROM AUTHOR]

We give a lower bound of the degree and the number of distinct prime divisors of the index of special perfect polynomials. More precisely, we prove that ω(d) ≥ 9, and deg(d) ≥ 258, where d := gcd(Q2 , σ(Q² )) is the index of the special perfect polynomial A := p²1Q², in which p1 is irreducible and has minimal degree. This means that σ(A) = A in the polynomial ring F2[x]. The function σ is a natural analogue of the usual sums of divisors function over the integers. The index considered is an analogue of the index of an odd perfect number, for which a lower bound of 135 is known. Our work use elementary properties of the polynomials as well as results of the paper. [ABSTRACT FROM AUTHOR]

In this paper, a couple of q-supercongruences for truncated basic hypergeometric series are proved, most of them modulo the cube of a cyclotomic polynomial. One of these results is a new q-analogue of the (E.2) supercongruence by Van Hamme, another one is a new q-analogue of a supercongruence by Swisher, while the other results are closely related q-supercongruences. The proofs make use of special cases of a very-well-poised 6 ϕ 5 summation. In addition, the proofs utilize the method of creative microscoping (which is a method recently introduced by the first author in collaboration with Wadim Zudilin), and the Chinese remainder theorem for coprime polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we give a general form of ideals over R [ X ] / 〈 f (X) p s 〉 when f is a monic irreducible polynomial in R[X] and R = F p m + u F p m + ... + u n F p m with un+ 1 = 0. For f(X) = X − 1 this gives a general form of cyclic codes over R. Furthermore, we apply the result to classify completely the cyclic codes of length 5ps over F p m + u F p m + u 2 F p m when p ≡ 2 (mod 5) or p ≡ 3 (mod 5) and m is odd. [ABSTRACT FROM AUTHOR]

In this paper, we use a method inspired by the valuation theory to classify completely cyclic codes of length 3 p s over p m + u p m + u 2 p m . Then, we compute their separation parameters which was useful to compute the Hamming distance in [8]. [ABSTRACT FROM AUTHOR]

Musiker and Reiner in [9] studied coefficients of cyclotomic polynomial in terms of topology of associated simplicial complexes. They determined homotopy type of associated complexes for all cyclotomic polynomials, except for cyclotomic polynomials whose degree is a product of three prime numbers. Using discrete Morse theory for simplicial complexes we partially answer a question posed by the two authors regarding homotopy type of the associated complexes when degree of the cyclotomic polynomial is a product of three prime numbers. [ABSTRACT FROM AUTHOR]

For a suitable irreducible base polynomial f (x) ∈ Z [ x ] of degree k, a family of polynomials F m (x) depending on f(x) is constructed with the properties: there is exactly one irreducible factor Φ d , f (x) for F m (x) for each divisor d of m; deg (Φ d , f (x)) = φ (d) · deg (f) generalizing the factorization of x m − 1 into cyclotomic polynomials; when the base polynomial f (x) = x − 1 this F m (x) coincides with x m − 1. As an application, irreducible polynomials of degree 12, 24, 24 are constructed having Galois groups of order matching their degrees and isomorphic to S 3 ⊕ C 2 , S 3 ⊕ C 2 ⊕ C 2 and S 3 ⊕ C 4 , respectively. [ABSTRACT FROM AUTHOR]

In this paper we prove supercongruence relations for truncated N-tuple sums of basic hypergeometric series. As an application, we give double, triple, and quadruple sum analogues of some Ramanujan-type supercongruences. [ABSTRACT FROM AUTHOR]

In this paper, we classify completely the negacyclic codes of length 3 p s over p m + u p m + u 2 p m when u 3 = 0 by using six-cyclotomic codes. [ABSTRACT FROM AUTHOR]

In this paper, we present sufficient conditions to guarantee the invertibility of rational circulant matrices with any given size. These sufficient conditions consist of linear combinations of the entries in the first row with integer coefficients. Our result is general enough to show the invertibility of circulant matrices with any size and arrangement of entries. For example, using these conditions, we show the invertibility of the family of circulant matrices with particular forms of integers generated by a primitive element in Z p . Also, using a combinatorial structure of these sufficient conditions, we show invertibility for circulant 0, 1-matrices. [ABSTRACT FROM AUTHOR]

Let Φ n (q) denote the n-th cyclotomic polynomial in q. Recently, Guo and Schlosser (Constr Approx 53:155–200, 2021) put forward the following conjecture: for any odd integer n > 1 , ∑ k = 0 n - 1 [ 8 k - 1 ] (q - 1 ; q 4) k 6 (q 2 ; q 2) 2 k (q 4 ; q 4) k 6 (q - 1 ; q 2) 2 k q 8 k ≡ 0 (mod [ n ] Φ n (q) 2) , if n ≡ 1 (mod 4) , 0 (mod [ n ]) , if n ≡ 3 (mod 4). where (a ; q) k = (1 - a) (1 - a q) ... (1 - a q k - 1) , [ n ] = (1 - q n) / (1 - q) , and Φ n (q) denotes the n-th cyclotomic polynomial in q. Applying the 'creative microscoping' method and several summation and transformation formulas for basic hypergeometric series and the Chinese remainder theorem for coprime polynomials, we confirm the above conjecture, as well as another similar q-supercongruence conjectured by Guo and Schlosser. [ABSTRACT FROM AUTHOR]

We give a geometric description of the parabolic bifurcation locus in the space Rat d of all rational functions on P 1 of degree d > 1, generalising the study by Morton and Vivaldi in the case of monic polynomials. The results are new even for quadratic rational functions. [ABSTRACT FROM AUTHOR]

In 2015, Swisher generalized the (G.2) supercongruence of Van Hamme to the modulus p 4 . In this paper, we first propose two q-analogues of Swisher's supercongruence, and then a parameter extension of them is presented. Furthermore, we prove a q-congruence modulo the fourth power of a cyclotomic polynomial, which was conjectured by the authors earlier. [ABSTRACT FROM AUTHOR]

The n th cyclotomic polynomial Φ n (X) is the minimal polynomial of ζ n : = e 2 π i / n. Given an integer m ≥ 1 and a prescribed set S of arithmetic progressions modulo m , we define n x as the product of the primes p ≤ x lying in those progressions. Let d (n) denote the number of divisors of n. It turns out that under certain conditions on S and m there exists j x such that log ⁡ | Φ n x (ζ m j x ) | / d (n x) tends to a positive limit. Our aim is to determine those conditions. We use the arithmetic of cyclotomic number fields, non-standard properties of character tables of finite abelian groups and a recent theorem of Bzdȩga, Herrera-Poyatos and Moree. After developing some generalities, we restrict to the case where m is a prime. Our motivation comes from a paper of Vaughan (1975). He studied the case where S = { ± 2 (mod 5) } and used it to show that the maximum coefficient in absolute value of Φ n can be very large. [ABSTRACT FROM AUTHOR]

We first give a q -analogue of a supercongruence of Sun, which is a generalization of Van Hamme's (H.2) supercongruence for any prime p ≡ 3 (mod 4). We also give a further generalization of this q -supercongruence, which may also be considered as a generalization of a q -supercongruence recently conjectured by the second author and Zudilin. Then, by combining these two q -supercongruences, we obtain q -analogues of the following two results: for any integer d > 1 and prime p with p ≡ − 1 (mod 2 d) ∑ k = 0 (p 2 − 1) / d 1 d k 3 k ! 3 ≡ p 2 (mod p 4) , ∑ k = 0 p 2 − 1 1 d k 3 k ! 3 ≡ p 2 (mod p 4) , which are generalizations of Swisher's (H.3) conjecture modulo p 4 for r = 2. The key ingredients in our proof are the 'creative microscoping' method, the q -Dixon sum, Watson's terminating 8 ϕ 7 transformation, and properties of the p -adic Gamma function. [ABSTRACT FROM AUTHOR]

We investigate a variation of the q-Wolstenholme theorem, which extends the q-analogue of Wolstenholme's theorem due to Shi and Pan (Am Math Mon 114:529–531, 2007). The proof relies on the Ramanujan sum and higher-order logarithmic derivatives of cyclotomic polynomials. [ABSTRACT FROM AUTHOR]

Let [ n ] = (1 - q n) / (1 - q) denote the q-integer and Φ n (q) the nth cyclotomic polynomial in q. Recently, Guo and Schlosser provided two conjectures: For any odd integer n > 3 , modulo [ n ] Φ n (q) (1 - a q n) (a - q n) , ∑ k = 0 (n + 1) / 2 [ 4 k + 1 ] (a q - 1 ; q 2) k (q - 1 / a ; q 2) k (q ; q 2) k 2 (a q 4 ; q 2) k (q 4 / a ; q 2) k (q 2 ; q 2) k 2 q 4 k ≡ 0 , and modulo Φ n (q) 2 (1 - a q n) (a - q n) , ∑ k = 0 (n + 1) / 2 [ 4 k + 1 ] (a q - 1 ; q 2) k (q - 1 / a ; q 2) k (q - 1 ; q 2) k (q ; q 2) k (a q 4 ; q 2) k (q 4 / a ; q 2) k (q 4 ; q 2) k (q 2 ; q 2) k q 6 k ≡ 0 , where (a ; q) k = (1 - a) (1 - a q) ⋯ (1 - a q k - 1) . In this paper, we confirm these two conjectures and further give their generalizations involving two free parameters. Our proof uses Guo and Zudilin's 'creative microscoping' method and the Chinese remainder theorem for coprime polynomials. [ABSTRACT FROM AUTHOR]

We establish a family of q-supercongruences modulo the cube of a cyclotomic polynomial for truncated basic hypergeometric series. This confirms a weaker form of a conjecture of the present authors. Our proof employs a very-well-poised Karlsson–Minton type summation due to Gasper, together with the 'creative microscoping' method introduced by the first author in recent joint work with Zudilin. [ABSTRACT FROM AUTHOR]

Using Andrews' multiseries generalization of Watson's 8 ϕ 7 transformation, we give a new extension of the (H.2) supercongruence of Van Hamme for primes p ≡ 3 (mod 4) , as well as its q-analogue. Meanwhile, applying the method of 'creative microscoping', recently introduced by the author and Zudilin, we establish some further q-supercongruences modulo Φ n (q) 3 , where Φ n (q) denotes the nth cyclotomic polynomial in q. [ABSTRACT FROM AUTHOR]

A distinguishing attack on the NTRUCipher symmetric encryption scheme defined over the residue ring modulo a cyclotomic polynomial over a finite field of prime order is proposed. The attack is based on the existence of a homomorphism from this ring into the specified field and can be quite effective under sufficiently general conditions. [ABSTRACT FROM AUTHOR]

Inspired by the recent work on q-congruences and a quadratic transformation formula of Rahman, we provide some new q-supercongruences. By taking parameters specialization in one of our results, we obtain a new Ramanujan-type supercongruence, which has the same right-hand side as Van Hamme’s (G.2) supercongruence for p ≡ 1 (mod 4) . We also formulate some related challenging conjectures on supercongruences and q-supercongruences. [ABSTRACT FROM AUTHOR]