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The Jordan totient $J_k(n)$ can be defined by $J_k(n)=n^k\prod_{p\mid n}(1-p^{-k})$. In this paper, we study the average behavior of fractions $P/Q$ of two products $P$ and $Q$ of Jordan totients, which we call Jordan totient quotients. To this end, we describe two general and ready-to-use methods that allow one to deal with a larger class of totient functions. The first one is elementary and the second one uses an advanced method due to Balakrishnan and P\'etermann. As an application, we determine the average behavior of the Jordan totient quotient, the $k^{th}$ normalized derivative of the $n^{th}$ cyclotomic polynomial $\Phi_n(z)$ at $z=1$, the second normalized derivative of the $n^{th}$ cyclotomic polynomial $\Phi_n(z)$ at $z=-1$, and the average order of the Schwarzian derivative of $\Phi_n(z)$ at $z=1$., Comment: 16 pages

Bang (1886), Zsigmondy (1892) and Birkhoff and Vandiver (1904) initiated the study of the largest prime divisors of sequences of the form a n − b n , denoted P ( a n − b n ) , by essentially proving that for integers a > b > 0 , P ( a n − b n ) ≥ n + 1 for every n > 2 . Since then, the problem of finding bounds on the largest prime factor of Lehmer sequences, Lucas sequences or special cases thereof has been studied by many, most notably by Schinzel (1962), and Stewart (1975, 2013). In 2002, Murty and Wong proved, conditionally upon the abc conjecture, that P ( a n − b n ) ≫ n 2 − ϵ for any ϵ > 0 . In this article, we improve this result for the specific case where b = 1 . Specifically, we obtain a more precise result, and one that is dependent on a condition we believe to be weaker than the abc conjecture. Our result actually concerns the largest prime factor of the nth cyclotomic polynomial evaluated at a fixed integer a, P ( Φ n ( a ) ) , as we let n grow. We additionally prove some results related to the prime factorization of Φ n ( a ) . We also present a connection to Wieferich primes, as well as show that the finiteness of a particular subset of Wieferich primes is a sufficient condition for the infinitude of non-Wieferich primes. Finally, we use the technique used in the proof of the aforementioned results to show an improvement on average of estimates due to Erdős for certain sums.

We prove that certain basic hypergeometric series truncated at $k=n-1$ have the factor $\Phi_n(q)^2$, where $\Phi_n(q)$ is the $n$-th cyclotomic polynomial. This confirms two recent conjectures of the author and Zudilin. We also put forward some conjectures on $q$-congruences modulo $\Phi_n(q)^2$., Comment: 9 pages

Motivated by Zudilin's work, we give a q-analogue of a Ramanujan-type supercongruence of van Hamme and Mortenson via the q-WZ method. Meanwhile, we give a q-analogue of a related congruence of Sun in the same way. We also propose several related conjectures on congruences involving central q-binomial coefficients.

Let f ( x ) ∈ F q [ x ] be an irreducible polynomial of degree m and exponent e. For each positive integer n, such that ν p ( q − 1 ) ≥ ν p ( e ) + ν p ( n ) for all prime divisors p of n, we show a fast algorithm to determine the irreducible factors of f ( x n ) . Using this algorithm, we give the complete factorization of x n − 1 into irreducible factors in the case where n = d p t , p is an odd prime, q is a generator of the group Z p 2 ⁎ and either d = 2 m with m ≤ ν 2 ( q − 1 ) or d = r a , where r is a prime dividing q − 1 but not p − 1 .

In this paper, we establish a new q-congruence with parameters modulo the fourth power of a cyclotomic polynomial from Watson's ϕ 7 8 transformation. Taking parameters substitution in our new q-congruence, we present a q-analogue of the (D.2) supercongruence of Van Hamme and several different q-analogues of He's supercongruence for p ≡ 1 ( mod 3 ) . Finally, we propose a Dwork-type supercongruence conjecture on He's supercongruence.

With the help of the creative microscoping method recently introduced by Guo and Zudilin and the Chinese remainder theorem for coprime polynomials, we establish a q-supercongruence with two parameters modulo [ n ] Φ n ( q ) 3 . Here [ n ] = ( 1 − q n ) / ( 1 − q ) and Φ n ( q ) is the n-th cyclotomic polynomial in q. In particular, we confirm a recent conjecture of Guo and give a complete q-analogue of Long's supercongruence. The latter is also a generalization of a recent q-supercongruence obtained by Guo and Schlosser.

Given n ∈ N , we study the conditions under which a finite field of prime order q will have adjacent elements of multiplicative order n . In particular, we analyze the resultant of the cyclotomic polynomial Φ n ( x ) with Φ n ( x + 1 ) , and exhibit Lucas and Mersenne divisors of this quantity. For each n ≠ 1 , 2 , 3 , 6 , we prove the existence of a prime q n for which there is an element α ∈ Z q n where α and α + 1 both have multiplicative order n . Additionally, we use algebraic norms to set analytic upper bounds on the size and quantity of these primes.

In this paper, we establish a Rodriguez–Villegas type congruence for truncated q-hypergeometric functions. Using this result, we can confirm several conjectures of Guo and Zeng, such as ∑ k = 0 p − 1 ( q ; q 3 ) k ( q 2 ; q 3 ) k ( q 3 ; q 3 ) k 2 ≡ ( − 3 p ) q 1 − p 2 3 ( mod ( 1 + q + ⋯ + q p − 1 ) 2 ) , where p ⩾ 5 is a prime, ( a ; q ) n = ( 1 − a ) ( 1 − a q ) ⋯ ( 1 − a q n − 1 ) , and ( ⋅ p ) denotes the Legendre symbol modulo p.

Let n 1 , ⋯ , n r be any finite sequence of integers and let S be the set of all natural numbers n for which there exists a divisor d ( x ) = 1 + ∑ i = 1 d e g ( d ) c i x i of x n − 1 such that c i = n i for 1 ≤ i ≤ r . In this paper we show that the set S has a natural density. Furthermore, we find the value of the natural density of S.

In this paper, we give q-analogies of classical Kummer, Lucas and ASH (Anton, Stickelberger, Hensel)'s results on binomial coefficients modulo primes. Our results generalize the previous result by T. Cai (2001) [1] .

Let ξ be a discrete random variable (r.v.) with uniform distribution on the support set { 0 , 1 , … , N } . We study the problem of construction of non-degenerate independent r.v.’s ξ 1 and ξ 2 such that ξ = ξ 1 + ξ 2 , if these r.v.’s exist. We describe a general form for the solutions to this problem, offer some analytic constructions and develop algorithms for computing the distributions of ξ 1 and ξ 2 .

Let a(r,n) be rth coefficient of nth cyclotomic polynomial. Suzuki proved that {a(r,n)|r≥1,n≥1}=Z. If m and n are two natural numbers we prove an analogue of Suzuki's theorem for divisors of xn−1 with exactly m irreducible factors. We prove that for every finite sequence of integers n1,…,nr there exists a divisor f(x)=∑i=0deg(f)cixi of xn−1 for some n∈N such that ci=ni for 1≤i≤r. Let H(r,n) denote the maximum absolute value of rth coefficient of divisors of xn−1. In the last section of the paper we give tight bounds for H(r,n).

We provide a general formula and give an explicit expression of the Lefschetz zeta function for any quasi-unipotent map on the n -dimensional torus. The Lefschetz zeta function is used to characterize the minimal set of Lefschetz periods for Morse–Smale diffeomorphisms on the n -dimensional torus; we completely describe this set, for different families containing infinitely many Morse–Smale diffeomorphisms. Moreover we show that for any given odd integer, there are Morse–Smale diffeomorphisms such that the corresponding minimal set of Lefschetz periods consists of all square free divisors of the given number. The results of the present article generalize the previous results of Berrizbeitia–Sirvent [7] . The methods used are based on the arithmetical properties of the number n .

We introduce and study in detail so-called circulant (Coxeter-periodic) elements and circulant families in a bilinear lattice K as well as their dual versions, called anti-circulant. We show that they form a natural environment for a systematic explanation of certain cyclotomic factors of the Coxeter polynomial χ K of K and in consequence, of Coxeter polynomials of algebras of finite global dimension. We discuss the properties of quadratic forms induced by circulant and anti-circulant families. Moreover, we interpret the results in the language of representation theory of algebras and point out applications (facts concerning tubular families in Auslander–Reiten quivers and quadratic forms of algebras). Abstract considerations in bilinear lattices are illustrated with a collection of non-trivial examples arising from module and derived categories. The results show that techniques of linear algebra and number theory provide efficient tools for explaining various representation theoretic facts.

Define the n th real cyclotomic polynomial to be the minimal polynomial over Z of ζ n + ζ n − 1 , where ζ n = e 2 π i / n is a primitive n th root of unity. We prove that the real cyclotomic polynomials can be formed from compositions of polynomials closely related to the Chebyshev polynomials of the first kind. We use these relations to determine the resultant of two real cyclotomic polynomials.

Let n ? Z + , and ? n ( x ) be the nth classical cyclotomic polynomial. In 4, Theorem 1, D. Lehmer showed that the geometric mean of { ? s ( 1 ) : s , n ? Z + , s ? n } ? e ? 2.71828 , as n ? ∞ . Replacing Z by F q T , and the nth elementary cylcotomic polynomial ? n ( x ) by the Carlitz m-cyclotomic polynomial ? m ( x ) , where m ? F q T , we obtain an analogue to Lehmer's result. We also express ? m ( 0 ) ? F 2 T in terms of ? * ( ? ) , the Pillai polynomial function. The resulting expression is a function field analogue of Holder's formula for ? n ( 1 ) .

The article of record as published may be found at http://dx.doi.org/10.1016/j.disc.2015.05.017 The goal of this paper is two-fold. We first focus on the problem of deciding whether two monomial rotation symmetric (MRS) Boolean functions are affine equivalent via a permutation. Using a correspondence between such functions and circulant matrices, we give a simple necessary and sufficient condition. We connect this problem with the well known Ádám’s conjecture from graph theory. As applications, we reprove easily several main results of Cusick et al. on the number of equivalence classes under permutations for MRS in prime power dimensions, as well as give a count for the number of classes in pq number of variables, where p, q are prime numbers with p < q < p2. Also, we find a connection between the generalized inverse of a circulant matrix and the invertibility of its generating polynomial over F2, modulo a product of cyclotomic polynomials, thus generalizing a known result on nonsingular circulant matrices.

By applying Chinese remainder theorem for coprime polynomials and the "creative microscoping" method recently introduced by the author and Zudilin, we establish parametric generalizations of three $q$-supercongruences modulo the fourth power of a cyclotomic polynomial. The original $q$-supercongruences then follow from these parametric generalizations by taking the limits as the parameter tends to $1$ (l'H\^opital's rule is utilized here). In particular, we prove a complete $q$-analogue of the (J.2) supercongruence of Van Hamme and a complete $q$-analogue of a "divergent" Ramanujan-type supercongruence, thus confirming two recent conjectures of the author. We also put forward some related conjectures, including a $q$-supercongruence modulo the fifth power of a cyclotomic polynomial., Comment: 13 pages

In this paper, we derive explicit factorizations of generalized cyclotomic polynomials and generalized Dickson polynomials of the first kind of order 2 m 3 , over finite field F q .

By the Chinese remainder theorem, the canonical map Ψ n : R [ X ] / ( X n − 1 ) → ⊕ d | n R [ X ] / Φ d ( X ) is an isomorphism when R is a field whose characteristic does not divide n and Φ d is the dth cyclotomic polynomial. When R is the ring Z of rational integers, this map is injective but not surjective. In this paper, we give an explicit formula for the elementary divisors of the cokernel of Ψ n (when R = Z ) using the prime factorisation of n. We also give a pictorial algorithm using Young tableaux that takes O ( n 3 + ϵ ) bit operations for any ϵ > 0 to determine a basis of Smith vectors (see Definition 3.1 ) for Ψ n . In general when R is an integral domain, we prove that the determinant of the matrix of Ψ : R [ X ] / ( ∏ j f j ) → ⨁ j R [ X ] / ( f j ) written with respect to the standard basis is ∏ 1 ⩽ i j ⩽ n R ( f j , f i ) , where f i 's are monic polynomials and R ( f j , f i ) is the resultant of f j and f i .

Pairings on elliptic curves play an important role in cryptography. We provide an explicit formula for vectors of polynomials describing optimal ate pairings over cyclotomic family of elliptic curves. The explicit formula is simple in that it only involves partitioning a certain cyclotomic polynomial. The simplicity of the formula allows us to analyze the sparsity of the vector.

We consider monodromy groups of the generalized hypergeometric equation \begin{equation*} \big[z(\theta+\alpha_{1})\cdots (\theta+\alpha_{n})-(\theta+\beta_{1}-1)\cdots (\theta+\beta_{n}-1)\big]f(z) = 0\text{, where }\theta = z d/dz, \end{equation*} in a suitable basis, closely related to the Frobenius basis. We pay particular attention to the maximally unipotent case, where $\beta_{1}=\ldots=\beta_{n}=1$, and present a theorem that enables us to determine the form of the corresponding monodromy matrices in the case where $(X-e^{-2\pi i\alpha_{1}})\cdots (X-e^{-2\pi i\alpha_{n}})$ is a product of cyclotomic polynomials., Comment: 20 pages

Let A be a finite dimensional algebra over an algebraically closed field k. Assume A is a basic connected and triangular algebra with n pairwise non-isomorphic simple modules. We consider the Coxeter transformation ϕ A ( T ) as the automorphism of the Grothendieck group K 0 ( A ) induced by the Auslander–Reiten translation τ in the derived category Der b ( mod A ) of the module category mod A of finite dimensional left A-modules. We say that A is of cyclotomic type if the characteristic polynomial χ A of ϕ A is a product of cyclotomic polynomials, equivalently, if the Mahler measure M ( χ A ) = 1 . In [6] we have considered many examples of algebras of cyclotomic type in the representation theory literature. In this paper we study the Mahler measure of the Coxeter polynomial of accessible algebras. In 1933, D.H. Lehmer found that the polynomial T 10 + T 9 − T 7 − T 6 − T 5 − T 4 − T 3 + T + 1 has Mahler measure μ 0 = 1.176280 . . . , and he asked whether there exist any smaller values exceeding 1. In this paper we prove that for any accessible algebra A either M ( χ A ) = 1 or M ( χ B ) ≥ μ 0 for some convex subcategory B of A. We introduce interlaced tower of algebras A m , … , A n with m ≤ n − 2 satisfying χ A s + 1 = ( T + 1 ) χ A s − T χ A s − 1 for m + 1 ≤ s ≤ n − 1 . We prove that, if Spec ϕ A n ⊂ S 1 ∪ R + and A n is not of cyclotomic type, then M ( χ A m ) M ( χ A n ) . We construct several examples.

Let Λ be a k-algebra of finite global dimension. We study tubular families in the Auslander–Reiten quiver of the bounded derived category Db(Λ) satisfying certain natural axioms. In particular, we precisely describe their influence on the cyclotomic factors of the Coxeter polynomial χΛ of Λ and discuss several numerical limitations for their possible shapes. Moreover, we show that our results provide an alternative, relatively simple proof of non-trivial classical facts concerning tubular families in module categories, and also extend them slightly.

Text We consider the analogue, when Z is replaced with F q [ T ] of the elementary cyclotomic polynomials and prove an analogue of Suzuki's Theorem. Video For a video summary of this paper, please visit http://youtu.be/mVBw82N34NE .

We study a family of inverse ternary cyclotomic polynomials $\Psi_{pqr}$ in which $r\le\varphi(pq)$ is a positive linear combination of $p$ and $q$. We derive a formula for the height of such polynomial and characterize all flat polynomials in this family., Comment: 11 pages

One of the theorems in the original paper (Borisov, Nathanson, and Wang (2004) [1] ) only holds for polynomials and not for rational functions. We explain the mistake, and also prove a new theorem for the rational function case.

We examine two natural questions concerning the polynomial divisors of x n − 1 : “For a given integer n, how large can the coefficients of divisors of x n − 1 be?” and “How often does x n − 1 have a divisor of every degree between 1 and n?” We consider the latter question when x n − 1 is factored in both Z [ x ] and F p [ x ] . The primary tools used in our investigation arise the study of the anatomy of integers. We also make use of several results on the size of the multiplicative order function (which stem from Hooleyʼs conditional proof of Artinʼs Primitive Root Conjecture) in our work over F p [ x ] .

We present some variations on the Greene–Krammerʼs identity which involve q-Catalan numbers. Our method reveals an intriguing analogy between these new identities and some congruences modulo a prime.

Let a ( n , k ) be the kth coefficient of the nth cyclotomic polynomial. Ji, Li and Moree (2009) [2] showed that { a ( n , k ) | n ≡ 0 mod d , n ⩾ 1 , k ⩾ 0 } = Z . In this paper we will determine { a ( n , k ) | n ≡ a mod d , k ≡ b mod f , n ⩾ 1 , k ⩾ 0 } .

We find analogues of the primitive divisor results of Zsigmondy, Bang, Bilu-Hanrot-Voutier, and Carmichael in polynomial rings, following the methods of Carmichael.

Tiwari proved that termination of linear programs (loops with linear loop conditions and updates) over the reals is decidable through Jordan forms and eigenvectors computation. Braverman proved that it is also decidable over the integers. In this paper, we consider the termination of loops with polynomial loop conditions and linear updates over the reals and integers. First, we prove that the termination of such loops over the integers is undecidable. Second, with an assumption, we provide an complete algorithm to decide the termination of a class of such programs over the reals. Our method is similar to that of Tiwari in spirit but uses different techniques. Finally, we conjecture that the termination of linear programs with polynomial loop conditions over the reals is undecidable in general by %constructing a loop and reducing the problem to another decision problem related to number theory and ergodic theory, which we guess undecidable., 17pages, 0 figures

It is customary to define a cyclotomic polynomial Φ n ( x ) to be ternary if n is the product of three distinct primes, p q r . Let A ( n ) be the largest absolute value of a coefficient of Φ n ( x ) and M ( p ) be the maximum of A ( p q r ) . In 1968, Sister Marion Beiter (1968, 1971) [3] , [4] conjectured that A ( p q r ) ≤ p + 1 2 . In 2008, Yves Gallot and Pieter Moree (2009) [6] showed that the conjecture is false for every p ≥ 11 , and they proposed the Corrected Beiter conjecture: M ( p ) ≤ 2 3 p . Here we will give a sufficient condition for the Corrected Beiter conjecture and prove it when p = 7 .

We consider the problem of solving a linear system Ax=b over a cyclotomic field. Cyclotomic fields are special in that we can easily find a prime p for which the minimal polynomial m(z) for the field factors into a product of distinct linear factors. This makes it possible to develop fast modular algorithms.We give two output sensitive modular algorithms, one using multiple primes and Chinese remaindering, the other using linear p-adic lifting. Both use rational reconstruction to recover the rational coefficients in the solution vector. We also give a third algorithm which computes the solutions as ratios of two determinants modulo m(z) using Chinese remaindering only. Because this representation is d=degm(z) times more compact in general, we can compute it the fastest.We have implemented the algorithms in Maple. Our benchmarks show that the third method is fastest on random inputs, but on real inputs arising from problems in computational group theory, the first two methods are faster because the solutions have small rational coefficients.

We define a class of algebras over finite fields, called polynomially cyclic algebras, which extend the class of abelian field extensions. We study the structure of these algebras; furthermore, we define and investigate properties of Lagrange resolvents and Gauss and Jacobi sums.Natural examples of polynomially cyclic algebras are for instance algebras of the form Fp[X]/(Fq(X)) where p,q are distinct odd primes and Fq is the q-th cyclotomic polynomial. Further examples occur similarly on replacing the cyclotomic polynomials with factors of division polynomials of elliptic curves. Finally, Gauss and Jacobi sums over polynomially cyclic algebras are applied for improving current algorithms for counting the number of points of elliptic curves over finite fields.

We describe a reciprocity relation between the prime ideal factorization, and related properties, of certain cyclotomic integers of the type ϕ n ( c − ζ m ) in the cyclotomic field of the m-th roots of unity and that of the symmetrical elements ϕ m ( c − ζ n ) in the cyclotomic field of the n-th roots. Here m and n are two positive integers, ϕ n is the n-th cyclotomic polynomial, ζ m a primitive m-th root of unity, and c a rational integer. In particular, one of these integers is a prime element in one cyclotomic field if and only if its symmetrical counterpart is prime in the other cyclotomic field. More properties are also established for the special class of pairs of cyclotomic integers ( 1 − ζ p ) q − 1 and ( 1 − ζ q ) p − 1 , where p and q are prime numbers.

We confirm several conjectures of Guo, Jouhet and Zeng concerning the factors of alternating binomials sums.

Let m be a positive integer, let r be a prime such that 2 is a primitive root modulo r^m, and let q=2^(^r^-^1^)^r^^^m^^^-^^^1. In this paper a binomial exponential sum over F"q which assumes 32log"2q+2 distinct values is explicitly evaluated and its value distribution is determined.

In this paper abelian function fields are restricted to the subfields of cyclotomic function fields. For any abelian function field K / k with conductor an irreducible polynomial over a finite field of odd characteristic, we give a calculating formula of the relative divisor class number h K − of K. And using the given calculating formula we obtain a criterion for checking whether or not the relative divisor class number is divisible by the characteristic of k.

We introduce the notion of the descent set polynomial as an alternative way of encoding the sizes of descent classes of permutations. Descent set polynomials exhibit interesting factorization patterns. We explore the question of when particular cyclotomic factors divide these polynomials. As an instance we deduce that the proportion of odd entries in the descent set statistics in the symmetric group S_n only depends on the number on 1's in the binary expansion of n. We observe similar properties for the signed descent set statistics., 21 pages, revised the proof of the opening result and cleaned up notation

Let a(n,k) be the kth coefficient of the nth cyclotomic polynomial. In part I it was proved that {a(mn,k)|n>=1,k>=0}=Z, in the case when m is a prime power. In this paper we show that the result also holds true in the case when m is an arbitrary positive integer.

In this article, we study the cyclotomic polynomials of degree N − 1 with coefficients restricted to the set { + 1 , − 1 } . By a cyclotomic polynomial we mean any monic polynomial with integer coefficients and all roots of modulus 1. By a careful analysis of the effect of Graeffe's root squaring algorithm on cyclotomic polynomials, P. Borwein and K.K. Choi gave a complete characterization of all cyclotomic polynomials with odd coefficients. They also proved that a polynomial p ( x ) with coefficients ±1 of even degree N − 1 is cyclotomic if and only if p ( x ) = ± Φ p 1 ( ± x ) Φ p 2 ( ± x p 1 ) ⋯ Φ p r ( ± x p 1 p 2 ⋯ p r − 1 ) , where N = p 1 p 2 ⋯ p r and the p i are primes, not necessarily distinct. Here Φ p ( x ) : = x p − 1 x − 1 is the pth cyclotomic polynomial. Based on substantial computation, they also conjectured that this characterization also holds for polynomials of odd degree with ±1 coefficients. We consider the conjecture for odd degree here. Using Ramanujan's sums, we solve the problem for some special cases. We prove that the conjecture is true for polynomials of degree 2 α p β − 1 with odd prime p or separable polynomials of any odd degree.

Using Faltings' theorem on the Mordell conjecture, we prove that for any prime p ⩾ 5 and integer n ⩾ 2 , the cyclotomic symbols { a , Φ n ( a ) } do not form a subgroup of K 2 ( Q ) . This partially confirms a conjecture of Browkin.

We say that a cyclotomic polynomial Φ n has order three if n is the product of three distinct primes, p q r . Let A ( n ) be the largest absolute value of a coefficient of Φ n . For each pair of primes p q , we give an infinite family of r such that A ( p q r ) = 1 . We also prove that A ( p q r ) = A ( p q s ) whenever s > q is a prime congruent to ± r ( mod p q ) .

We present some simple observations on factors of the q-binomial coefficients, the q-Catalan numbers, and the q-multinomial coefficients. Writing the Gaussian coefficient with numerator n and denominator k in a form such that 2k⩽n by the symmetry in k, we show that this coefficient has at least k factors. Some divisibility results of Andrews, Brunetti and Del Lungo are also discussed.

If A is a set of positive integers, we denote by p(A,n) the number of partitions of n with parts in A. First, we recall the following simple property: let f(z)=1+@?"n"="1^~@e"nz^n be any power series with @e"n=0 or 1; then there is one and only one set of positive integers A(f) such that p(A(f),n)=@e"n(mod2) for all n>=1. Some properties of A(f) have already been given when f is a polynomial or a rational fraction. Here, we give some estimations for the counting function A(P,x)=Card{a@?A(P);a=

A sequence of functions {f_n(q)}_{n=1}^{\infty} satisfies the functional equation for multiplication of quantum integers if f_{mn}(q) = f_m(q)f_n(q^m) for all positive integers m and n. This paper describes the structure of all sequences of rational functions with rational coefficients that satisfy this functional equation., Comment: 11 pages; LaTex

In his 1964 paper, de Bruijn (Math. Comp. 18 (1964) 537) called a pair ða; bÞ of positive odd integers good, if Z ¼ aS~2bS; where S is the set of nonnegative integers whose 4-adic expansion has only 0’s and 1’s, otherwise he called the pair ða; bÞ bad. Using the 2-adic integers we obtain a characterization of all bad pairs. A positive odd integer u is universally bad if ðua; bÞ is bad for all pairs of positive odd integers a and b: De Bruijn showed that all positive integers of the form u ¼ 2 k þ 1 are universally bad. We apply our characterization of bad pairs to give another proof of this result of de Bruijn, and to show that all integers of the form u ¼ fpk ð4Þ are universally bad, where p is prime and fnðxÞ is the nth cyclotomic polynomial. We consider a new class of integers we call de Bruijn universally bad integers and obtain a characterization of such positive integers. We apply this characterization to show that the universally bad integers u ¼ fpk ð4Þ are in fact de Bruijn universally bad for all primes p42: Furthermore, we show that the universally bad integers f 2k ð4Þ; and more generally, those of the form 4 k þ 1; are not de Bruijn universally bad.

We study coefficients of ternary cyclotomic polynomials Φpqr(z)=∏ρ(z−ρ), where p, q, and r are distinct odd primes and the product is taken over all primitive pqrth roots of unity ρ.