On Two Problems Related to Divisibility Properties of z(n)

The order of appearance (in the Fibonacci sequence) function z:Z≥1→Z≥1 is an arithmetic function defined for a positive integer n as z(n)=min{k≥1:Fk≡0(modn)}. A topic of great interest is to study the Diophantine properties of this function. In 1992, Sun and Sun showed that Fermat’s Last Theorem is related to the solubility of the functional equation z(n)=z(n2), where n is a prime number. In addition, in 2014, Luca and Pomerance proved that z(n)=z(n+1) has infinitely many solutions. In this paper, we provide some results related to these facts. In particular, we prove that limsupn→∞(z(n+1)−z(n))/(logn)2−ϵ=∞, for all ϵ∈(0,2).

The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of z(n)

Let (Fn)n be the sequence of Fibonacci numbers. The order of appearance (in the Fibonacci sequence) of a positive integer n is defined as z(n)=min{k≥1:n∣Fk}. Very recently, Trojovská and Venkatachalam proved that, for any k≥1, the number z(n) is divisible by 2k, for almost all integers n≥1 (in the sense of natural density). Moreover, they posed a conjecture that implies that the same is true upon replacing 2k by any integer m≥1. In this paper, in particular, we prove this conjecture.

On the infinite Borwein product raised to a positive real power

In this paper, we study properties of the coefficients appearing in the q-series expansion of $$\prod _{n\ge 1}[(1-q^n)/(1-q^{pn})]^\delta $$ ∏ n ≥ 1 [ ( 1 - q n ) / ( 1 - q pn ) ] δ , the infinite Borwein product for an arbitrary prime p, raised to an arbitrary positive real power $$\delta $$ δ . We use the Hardy–Ramanujan–Rademacher circle method to give an asymptotic formula for the coefficients. For $$p=3$$ p = 3 we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent $$\delta $$ δ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the $$p=3$$ p = 3 case.

The Proof of a Conjecture Related to Divisibility Properties of z(n)

The order of appearance of n (in the Fibonacci sequence) z(n) is defined as the smallest positive integer k for which n divides the k—the Fibonacci number Fk. Very recently, Trojovský proved that z(n) is an even number for almost all positive integers n (in the natural density sense). Moreover, he conjectured that the same is valid for the set of integers n ≥ 1 for which the integer 4 divides z(n). In this paper, among other things, we prove that for any k ≥ 1, the number z(n) is divisible by 2k for almost all positive integers n (in particular, we confirm Trojovský’s conjecture).

Ideal structure of rings of analytic functions with non-Archimedean metrics

The work of Helmer [Divisibility properties of integral functions, Duke Math. J. 6(2) (1940) 345–356] applied algebraic methods to the field of complex analysis when he proved the ring of entire functions on the complex plane is a Bezout domain (i.e. all finitely generated ideals are principal). This inspired the work of Henriksen [On the ideal structure of the ring of entire functions, Pacific J. Math. 2(2) (1952) 179–184. On the prime ideals of the ring of entire functions, Pacific J. Math. 3(4) (1953) 711–720] who proved a correspondence between the maximal ideals within the ring of entire functions and ultrafilters on sets of zeroes as well as a correspondence between the prime ideals and growth rates on the multiplicities of zeroes. We prove analogous results on rings of analytic functions in the non-Archimedean context: all finitely generated ideals in the ring of analytic functions on an annulus of a characteristic zero non-Archimedean field are two-generated but not guaranteed to be principal. We also prove the maximal and prime ideal structure in the non-Archimedean context is similar to that of the ordinary complex numbers; however, the methodology has to be significantly altered to account for the failure of Weierstrass factorization on balls of finite radius in fields which are not spherically complete, which was proven by Lazard [Les zeros d’une function analytique d’une variable sur un corps value complet, Publ. Math. l’IHES 14(1) (1942) 47–75].

A note on ratios of Fibonacci hybrid and Lucas hybrid numbers

Irmak recently asked an open question related to divisibility properties of Fibonacci and Lucas quaternions [4, p. 374]. In this paper, we give an answer to Fibonacci and Lucas hybrid number version of this question.

The $ q $-WZ pairs and divisibility properties of certain polynomials

<abstract><p>Using the $ q $-WZ (Wilf-Zeilberger) pairs we give divisibility properties of certain polynomials. These results may deemed generalizations of some $ q $-congruences obtained by Guo earlier, or $ q $-analogues of some congruences of Sun. For example, we prove that, for $ n\geqslant 1 $ and $ 0\leqslant j\leqslant n $, the following two polynomials</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} &amp;\sum\limits_{k = j}^{n} (-1)^{k}[3k-2j+1]{2k-2j\brack k}\frac{(q;q^2)_k(q;q^2)_{k-j}(-q;q)_n^3}{(q;q)_k(q^2;q^2)_{k-j}},\\ &amp;\sum\limits_{k = j}^{n} (-1)^{n-k}q^{(k-j)^2}[4k+1]\frac{(q;q^2)_k^2(q;q^2)_{k+j}(-q;q)_n^6 }{(q^2;q^2)_k^2(q^2;q^2)_{k-j}(q;q^2)_j^2}. \end{align*} $\end{document} </tex-math></disp-formula></p> <p>are divisible by $ (1+q^n)^2[2n+1]{2n\brack n} $. Here $ [m] = 1+q+\cdots+q^{m-1}, (a; q)_m = (1-a)(1-aq)\cdots (1-aq^{m-1}) $, and $ {m\brack k} = (q^{m-k+1};q)_k/(q; q)_k $.</p></abstract>