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We show that the set of composite positive integers n ≤ x satisfying the congruence is of cardinality at most x as x → ∞.

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A unit x in a commutative ring R with identity is called exceptional if 1 − x is also a unit in R . For any integer n ≥ 2, define φ e ( n ) to be the number of exceptional units in the ring of integers modulo n . Following work of Shapiro, Mills, Catlin and Noe on iterations of Euler's φ -function, we develop analogous results on iterations of the function φ e , when restricted to a particular subset of the positive integers.

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In this paper, we give a new method to derive a binomial series identity discovered by J. M. Borwein and R. Girgensohn.

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For positive integers a, k , let denote the sequence a k , a k + 1, a k + a , . . . , a k + a k −1 . Let denote the set of integers that are expressible as a linear combination of elements of with non-negative integer coefficients. We determine and which denote the largest (respectively, the number of) positive integer(s) not in . We also determine the set of positive integers not in which satisfy , where = \ {0}.

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An -sequence is defined by , with initial conditions a 0 = 0, a 1 = 1. These -sequences play a remarkable role in partition theory, allowing -generalizations of the Lecture Hall Theorem [Bousquet-Mélou and Eriksson, Ramanujan J. 1: 101–111, 1997, Bousquet-Mélou and Eriksson, Ramanujan J. 1: 165–185, 1997] and Euler's Partition Theorem [Bousquet-Mélou and Eriksson, Ramanujan J. 1: 165–185, 1997, Savage and Yee, J. Combin. Theory Ser. A 115: 967–996, 2008]. These special properties are not shared with other sequences, such as the Fibonacci sequence, defined by second-order linear recurrences. The -sequence gives rise to the , which is known to be an integer [Lucas, Amer. J. Math. 1: 197–240, 1878]. In this paper, we use algebraic and combinatorial properties of -sequences to interpret the -nomial coefficients in terms of weighted lattice paths, integer partitions, and probablility distributions. We show how to use these interpretations to uncover -generalizations of familiar hypergeometric identities involving binomial coefficients. This leads naturally to an -analogue of the q -binomial coefficients (Gaussian polynomials) and a corresponding generalization of the “partitions in a box” interpretation of ordinary q -binomial coefficients.

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An almost square of type 2 is an integer n that can be factored in two different ways as n = a 1 b 1 = a 2 b 2 with a 1 , a 2 , b 1 , . In this paper, we continue the study of almost squares of type 2 in short intervals and improve the 1/2 upper bound. We also draw connections with almost squares of type 1.

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A nonempty finite set of positive integers A is relatively prime if gcd( A ) = 1 and it is relatively prime to n if gcd( A ∪ { n }) = 1. The number of nonempty subsets of A which are relatively prime to n is Φ( A, n ) and the number of such subsets of cardinality k is Φ k ( A, n ). Given positive integers l 1 , l 2 , m 2 , and n such that l 1 ≤ l 2 ≤ m 2 we give Φ([1, m 1 ] ∪ [ l 2 , m 2 ], n ) along with Φ k ([1, m 1 ] ∪ [ l 2 , m 2 ], n ). Given positive integers l, m , and n such that l ≤ m we count for any subset A of { l, l + 1, . . . , m } the number of its supersets in [ l, m ] which are relatively prime and we count the number of such supersets which are relatively prime to n . Formulas are also obtained for corresponding supersets having fixed cardinalities. Intermediate consequences include a formula for the number of relatively prime sets with a nonempty intersection with some fixed set of positive integers.

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We study a card game called Mousetrap , together with its generalization He Loves Me, He Loves Me Not . We first present some results for the latter game, based, on one hand, on theoretical considerations and, on the other one, on Monte Carlo trials. Furthermore, we introduce a combinatorial algorithm, which allows us to obtain the best result at least for French card decks (52 cards with 4 suits). We then apply the algorithm to the study of Mousetrap and Modular Mousetrap , improving recent results. Finally, by means of our algorithm, we study the reformed permutations in Mousetrap, Modular Mousetrap and He Loves Me, He Loves Me Not , attaining new results which give some answers to several questions posed by Cayley and by Guy and Nowakowski in their papers.

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Taking a combinatorial point of view on cyclotomic polynomials leads to a larger class of polynomials we shall call the inclusion-exclusion polynomials. This gives a more appropriate setting for certain types of questions about the coefficients of these polynomials. After establishing some basic properties of inclusion-exclusion polynomials we turn to a detailed study of the structure of ternary inclusion-exclusion polynomials. The latter subclass is exemplified by cyclotomic polynomials Φ pqr , where p < q < r are odd primes. Our main result is that the set of coefficients of Φ pqr is simply a string of consecutive integers which depends only on the residue class of r modulo pq .

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Let A and B be finite subsets of ℂ such that | B | = C | A |. We show the following variant of the sum product phenomenon: If | AB | < α | A | and α ≪ log | A |, then | k A + l B | ≫ | A | k | B | l . This is an application of a result of Evertse, Schlickewei, and Schmidt on linear equations with variables taking values in multiplicative groups of finite rank, in combination with an earlier theorem of Ruzsa about sumsets in . As an application of the case A = B we give a lower bound on | A + |+| A × |, where A + is the set of sums of distinct elements of A and A × is the set of products of distinct elements of A .