Volume 9, Issue 2

The symmetric numerical semigroups S( F a , F b , F c ) and S( L k , L m , L n ) generated by three distinct Fibonacci ( F a , F b , F c ) and Lucas ( L k , L m , L n ) numbers are considered. Based on divisibility properties of the Fibonacci and Lucas numbers we establish necessary and sufficient conditions for both semigroups to be symmetric and calculate their Hilbert generating series, Frobenius numbers, and genera.

In 1980, Carl Pomerance and J. L. Selfridge proved D. J. Newman's coprime mapping conjecture: If n is a positive integer and I is a set of n consecutive integers, then there is a bijection ƒ : {1, 2, . . . , n } → I such that gcd( i , ƒ( i )) = 1 for 1 ≤ i ≤ n . The function ƒ described in their theorem is called a coprime mapping . Around the same time, Roger Entringer conjectured that all trees are prime , that is, that if T is a tree with vertex set V , then there is a bijection L : V → {1, 2, . . . , | V |} such that gcd( L ( x ), L ( y )) = 1 for all adjacent vertices x and y in V . There has been little progress so far towards a proof of this conjecture. In this paper, we extend Pomerance and Selfridge's theorem by replacing the set I with a set S of n integers in arithmetic progression and determining when there exist coprime mappings ƒ : {1, 2, . . . , n } → S and g : {1, 3, . . . , 2 n – 1} → S . The rest of the paper is devoted to using coprime mappings to prove that various families of trees are prime, including palm trees, banana trees, binomial trees, and certain families of spider colonies.

In this paper we study a generalization of the Fibonacci sequence in which rabbits are mortal and take more that two months to become mature. In particular we give a general recurrence relation for these sequences (improving the work in [Hoggat and Lind, Fibonacci Quart. 7: 482–487, 1969]) and we calculate explicitly their general term (extending the work in [Miles, Amer. Math. Monthly 67: 745–752, 1960]). In passing, and as a technical requirement, we also study the behavior of the positive real roots of the characteristic polynomial of the considered sequences.

We show that a theorem obtained by K. R. Slavin can be easily deduced from the q -Lucas theorem.

The 3 x + 1 map T is defined on the 2-adic integers ℤ 2 by T ( x ) = x /2 for even x and T ( x ) = (3 x + 1)/2 for odd x . Under iteration of T , the sequence ( T k ( x ) mod 2), called the parity vector of x ∈ ℤ 2 , can be interpreted as an infinite word over the alphabet {0, 1} or as the digits of the 2-adic integer . For any υ ∈ ℤ 2 (or equivalently for the infinite word υ ), the inverse map Φ (called the 3 x + 1 conjugacy map ) yields the unique x ∈ ℤ 2 with parity vector υ . It is unknown if there exists any aperiodic υ with an eventually periodic Φ( υ ). In this paper we compute Φ( υ ) for a class of aperiodic infinite words υ of minimal complexity, the mechanical words with irrational slope and intercept 0. Our main result is a generalized continued fraction expansion of –1/Φ( x ), convergent under the 2-adic metric of ℤ 2 . The given examples suggest that Φ always maps Sturmian words to infinite words of full complexity.

A nonempty subset A ⊆ {1, 2, . . . , n } is relatively prime if gcd( A ) = 1. Let ƒ( n ) denote the number of relatively prime subsets of {1, 2, . . . , n }. The sequence given by the values of ƒ( n ) is sequence A085945 in Sloane's On-Line Encyclopedia of Integer Sequences. In this article we show that ƒ( n ) is never a square if n ≥ 2. Moreover, we show that reducing the terms of this sequence modulo any prime l ≠ 3 leads to a sequence which is not periodic modulo l .

The Levinson–Durbin recursion is used to construct the coefficients which define the minimum mean square error predictor of a new observation for a discrete time, second-order stationary stochastic process. As the sample size varies, the coefficients determine what is called a Levinson–Durbin sequence. A generalized Levinson–Durbin sequence is also defined, and we note that binomial coefficients constitute a special case of such a sequence. Generalized Levinson–Durbin sequences obey formulas which generalize relations satisfied by binomial coefficients. Some of these results are extended to vector stationary processes.

In the present paper, by employing Cilleruelo's method, we prove that neither (4 k 2 + 1) nor (2 k ( k – 1) + 1) is a perfect square for all n > 1, which confirms a conjecture of Amdeberhan, Medina, and Moll.

Recently, Lovejoy introduced the construct of overpartition pairs which are a natural generalization of overpartitions. Here we generalize that idea to overpartition k -tuples and prove several congruences related to them. We denote the number of overpartition k -tuples of a positive integer n by k ( n ) and prove, for example, that for all n ≥ 0, t –1 ( tn + r ) ≡ 0 (mod t ) where t is prime and r is a quadratic nonresidue mod t .

In 1836 E. Midy published in France an article where he showed that if p is a prime number, such that the smallest repeating sequence of digits in the decimal expansion of has an even length, when this sequence is broken into two halves of equal length if these parts are added then the result is a string of 9′s. Later, J. Lewittes and H.W.Martin generalized this statement when the length of the smallest repeating sequence of digits is e = kd and the sequence is broken into d blocks of equal length and the expansion is over any number base; that fact was named Midy's property. We will give necessary and sufficient conditions (that are easy to check) for the integer N to satisfy Midy's property.