My ORCID ID is 0000-0002-5619-3513. My Erdos Number is 3.

**Supercongruences related to 3F2(1) involving harmonic numbers**

*accepted for publication in the International Journal of Number Theory.*

We show various supercongruences for truncated series which involve central binomial coefficients and harmonic numbers. The corresponding infinite series are also evaluated.*- download a copy from arXiv:*

**From generating series to polynomial congruences**(with Sandro Mattarei).

*Journal of Number Theory, 182 (2018), 179-205.*

Consider an ordinary generating function $\sum_{k=0}^{\infty}c_kx^k$, of an integer sequence of some combinatorial relevance, and assume that it admits a closed form $C(x)$. Various instances are known where the corresponding truncated sum $\sum_{k=0}^{q-1}c_kx^k$, with $q$ a power of a prime $p$, also admits a closed form representation when viewed modulo $p$. Such a representation for the truncated sum modulo $p$ frequently bears a resemblance with the shape of $C(x)$, despite being typically proved through independent arguments. We develop a method which allows one to directly infer the closed-form representation of the truncated sum from the closed form of the series for a significant class of series involving central binomial coefficients.*- download a copy:*

**Revitalized automatic proofs: demonstrations**(with Tewodros Amdeberhan, David Callan, and Hideyuki Ohtsuka).

*INTEGERS, 17 (2017), A16.*

We consider three problems from the recent issues of the American Mathematical Monthly involving different versions of Catalan triangle. Our main results offer generalizations of these identities and demonstrate automated proofs with additional twists, and on occasion we furnish a combinatorial proof.*- download a copy:*

**Two Triple binomial sum supercongruences**(with Tewodros Amdeberhan)

*Journal of Number Theory, 175 (2017), 140-157.*

In a recent article, Apagodu and Zeilberger discuss some applications of an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequence. At the end, they propose some supercongruences as conjectures. Here we prove one of them, including a new companion enumerating abelian squares, and we leave some remarks for the others. This joint work extends this previous draft.*- download a copy from arXiv:*

**Multiple harmonic sums and multiple harmonic star sums are (nearly) never integers**(with Khodabakhsh Hessami Pilehrood and Tatiana Hessami Pilehrood).

*INTEGERS, 17 (2017), A10.*

A well known result of elementary number theory is that even though the partial sum H_n of the harmonic series increases to infinity, it is never an integer for n>1. Apparently the first published proof goes back to Leopold Theisinger in 1915, and, since then, it has been proposed as a challenging problem in several textbooks. In 1946, Erdos and Niven proved a stronger statement: there is only a finite number of integers n for which there is a positive integer r<=n such that the r-th elementary symmetric function of 1,1/2,...,1/n is an integer. In 2012, Chen and Tang refined this result and succeeded to show that the above sum is not an integer with the only two exceptions: either n=r=1 or n=3 and r=2. In this paper, we consider the integrality problem for sums which are not necessarily symmetric with respect to their variables: the multiple harmonic and multiple harmonic star sums.*- download a copy:*

**Some q-congruences for homogeneous and quasi-homogeneous multiple q-harmonic sums**(with Khodabakhsh Hessami Pilehrood and Tatiana Hessami Pilehrood).

*Ramanujan Journal, 43 (2017), 113-139.*

We show some new Wolstenholme type q-congruences for some classes of multiple q-harmonic sums of arbitrary depth with strings of indices composed of ones, twos and threes. Most of these results are q-extensions of the corresponding congruences for ordinary multiple harmonic sums obtained by the authors in a previous paper. Finally, we pose a conjecture concerning two kinds of cyclic sums of multiple q-harmonic sums.*- download a copy:*

**Restricted linear congruences**(with Khodakhast Bibak, Bruce M. Kapron, Venkatesh Srinivasan, and Laszlo Toth).

*Journal of Number Theory, 171 (2017), 128-144.*

In this paper, using properties of Ramanujan sums and of the finite Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence a_1*x_1+...+a_k*x_k=b (mod n), with (x_i,n)=t_i for 1<=i<=k. Some special cases of this problem have been studied in many papers, and have found very interesting applications in number theory, combinatorics, and cryptography, among other areas. We also propose an authenticated encryption scheme, and using our explicit formula, analyze the integrity of this scheme.*- download a copy:*

**Large Peg-Army Maneuvers**(with Luciano Guala', Stefano Leucci, and Emanuele Natale).

8th International Conference on Fun with Algorithms (FUN 2016), Volume 49, FUN 2016, June 8-10, 2016 - La Maddalena, Italy.

Despite its long history, the classical game of peg solitaire continues to attract the attention of the scientific community. In this paper, we consider two problems with an algorithmic flavour which are related with this game, namely Solitaire-Reachability and Solitaire-Army. In the first one, we show that deciding whether there is a sequence of jumps which allows a given initial configuration of pegs to reach a target position is NP-complete. Regarding Solitaire-Army, the aim is to successfully deploy an army of pegs in a given region of the board in order to reach a target position. By solving an auxiliary problem with relaxed constraints, we are able to answer some open questions raised by Csakany and Juhasz (Mathematics Magazine, 2000). To appreciate the combinatorial beauty of our solutions, we recommend to visit the gallery of animations provided HERE.*- download a copy:*

**Supercongruences for the Almkvist-Zudilin numbers**(with Tewodros Amdeberhan).

Acta Arithmetica, 173 (2016), 255-268.

Given a prime number p, the study of divisibility properties of a sequence c(n) has two contending approaches: p-adic valuations and superconcongruences. The former searches for the highest power of p dividing c(n), for each n; while the latter (essentially) focuses on the maximal powers r and t such that c(p^rn) is congruent to c(p^{r-1}n) modulo p^t. This is called supercongruence. In this paper, we prove a conjecture on supercongruences for sequences that have come to be known as the Almkvist-Zudilin numbers. Some other (naturally) related family of sequences will be considered in a similar vain.*- download a copy:*

**Some congruences for central binomial sums involving Fibonacci and Lucas numbers**.

*Journal of Integer Sequences, 19 (2016), 16.5.4.*

We present several polynomial congruences about sums with central binomial coefficients and harmonic numbers. In the final section we collectsome new congruences involving Fibonacci and Lucas numbers.*- download a copy:*

**New multiple harmonic sum identities**(with Helmut Prodinger).

*Electronic Journal of Combinatorics, 21 (2014), P2.43.*

We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series and congruences are given.*- download a copy:*

**New properties of multiple harmonic sums modulo p and p-analogues of Leshchiner's series**(with Khodabakhsh Hessami Pilehrood and Tatiana Hessami Pilehrood).*Transactions of the American Mathematical Society, 366 (2014), 3131-3159.*

In this paper we present some new identities of hypergeometric type for multiple harmonic sums whose indices are the sequences (\{1\}^a,c,\{1\}^b), (\{2\}^a,c,\{2\}^b) and prove a number of congruences for these sums modulo a prime p. The congruences obtained allow us to find nice p-analogues of Leshchiner's series for zeta values and to refine a result due to M. Hoffman and J. Zhao about the set of generators of the multiple harmonic sums of weight 7 and 9 modulo p. Moreover, we are also able to provide a new proof of Zagier's formula for \zeta^{*}(\{2\}^a,3,\{2\}^b) based on a finite identity for partial sums of the zeta-star series.*- download a copy:*

**Determinants of grids, tori, cylinders and Mobius ladders**(with Khodakhast Bibak).*Discrete Mathematics, 313 (2013), 1436-1440.*

Recently, Bien [A. Bien, The problem of singularity for planar grids, Discrete Math. 311 (2011), 921--931] obtained a recursive formula for the determinant of a grid. Also, recently, Pragel [D. Pragel, Determinants of box products of paths, Discrete Math. 312 (2012), 1844--1847], independently, obtained an explicit formula for this determinant. In this paper, we give a short proof for this problem. Furthermore, applying the same technique, we get explicit formulas for the determinant of a torus, a cylinder, and a Mobius ladder.*- download a copy:*

**Some q-analogs of congruences for central binomial sums**.

*Colloquium Mathematicum, 133 (2013), 133-143.*

We establish q-analogs for four congruences involving central binomial coefficients. The q-identities necessary for this purpose are shown via the q-WZ method.*- download a copy:*

**Congruences for central binomial sums and finite polylogarithms**(with Sandro Mattarei).

*Journal of Number Theory, 133 (2013), 131-157.*

We prove congruences, modulo a power of a prime p, for certain finite sums involving central binomial coefficients \binom{2k}{k} in the denominators. Specifically, we evaluate the sums p\sum_{k=1}^{p-1} {t^k k^{-d}\binom{2k}{k}^{-1}}\pmod{p^3} and p\sum_{k=1}^{p-1} {t^k H_{k-1}(2)\,k^{-d}\binom{2k}{k}^{-1}}\pmod{p}, for t\in\{1,-1,2,3,4,-1/2\} and d=0,1,2.*- download a copy:*

**Congruences concerning Jacobi polynomials and Apery-like formulae**(with Khodabakhsh Hessami Pilehrood and Tatiana Hessami Pilehrood).*International Journal of Number Theory, 8 (2012), 1789-1811.*

Let p>5 be a prime. We prove congruences modulo p^{3-d} for sums of the general form \sum_{k=0}^{(p-3)/2}\binom{2k}{k}t^k/(2k+1)^{d+1} and \sum_{k=1}^{(p-1)/2}\binom{2k}{k}t^k/k^d with d=0,1. We also consider the special case t=(-1)^{d}/16 of the former sum, where the congruences hold modulo p^{5-d}. Part of this paper follows from arXiv:1110.4013.*- download a copy:*

**Supercongruences for a truncated hypergeometric series**.

*INTEGERS, 12 (2012), A45.*

The purpose of this note is to obtain some congruences modulo a power of a prime p involving the truncated hypergeometric series \sum_{k=1}^{p-1} {(x)_k(1-x)_k\over (1)_k^2}\cdot{1\over k^a} for a=1 and a=2. In the last section, the special case x=1/2 is considered.*- download a copy:*

**q-Analogs of some congruences involving Catalan numbers**.

*Advances in Applied Mathematics, 48 (2012), 603-614.*

We provide some variations on the Greene-Krammer's identity which involve q-Catalan numbers. Our method reveals a curious analogy between these new identities and some congruences modulo a prime.*- download a copy:*

**On some new congruences for binomial coefficients**(with Zhi-Wei Sun).

*International Journal of Number Theory, 7 (2011), 645-662.*

In this paper we establish some new congruences involving Catalan numbers as well as central binomial coefficients. Let p>3 be a prime. We show that sum_{k=0}^{p-1}C_{pn+k}/C_n=1-3(n+1)((p-1)/3) (mod p^2) for every n=0,1,2,..., where C_m is the Catalan number binom(2m,m)/(m+1), and (*/3) is the Legendre symbol. We also determine sum_{k=0}^{p^a-1}binom(2k,k+d) and sum_{k=0}^{p^a-1}k*binom(2k,k+d) modulo p^2 for all a=1,2,3,... and d=0,1,...,p.*- download a copy:*

**Constraining spacetime torsion with the Moon and Mercury**(with Riccardo March, Giovanni Bellettini and Simone Dell'Agnello ).

*Phys. Rev. D 83, 104008 (2011).*

We report a search for new gravitational physics phenomena based on Einstein-Cartan theory of General Relativity including spacetime torsion. Starting from the parametrized torsion framework of Mao, Tegmark, Guth and Cabi, we analyze the motion of test bodies in the presence of torsion, and in particular we compute the corrections to the perihelion advance and to the orbital geodetic precession of a satellite.*- download a copy:*

**Constraining spacetime torsion with LAGEOS**(with Riccardo March, Giovanni Bellettini and Simone Dell'Agnello ).

*General Relativity and Gravitation, 43 (2011), Number 11, 3099-3126.*

We compute the corrections to the orbital Lense-Thirring effect (or frame-dragging) in the presence of spacetime torsion. We derive the equations of motion of a test body in the gravitational field of a rotating axisymmetric massive body, using the parametrized framework of Mao, Tegmark, Guth and Cabi. We calculate the secular variations of the longitudes of the node and of the pericenter.*- download a copy:*

**More congruences for central binomial coefficients**.

*Journal of Number Theory, 130 (2010), 2639-2649.*

We present several congruences modulo a power of prime p concerning sums of the following type \sum_{k=1}^{p-1}{m^k\over k^r}{2k\choose k}^{-1} which reveal some interesting connections with the analogous infinite series.*- download a copy:*

**Congruences of multiple sums involving sequences invariant under the binomial transform**(with Sandro Mattarei).

*Journal of Integer Sequences, 13 (2010), 10.5.1.*

We will prove several congruences modulo a power of a prime such as sum_{0< k_1<...< k_{n}<p} leg{p-k_{n}}{3} {(-1)^{k_{n}}\over k_1... k_{n}} equiv -{2^{n+1}+2\over 6^{n+1}} p B_{p-n-1}({1\over 3}) mod p^2 if n is an odd positive integer and p is prime such that p>\max(n+1,3).*- download a copy:*

**New harmonic number identities with applications**.

*Seminaire Lotharingien de Combinatoire, 63 (2010), B63g.*

We determine the explicit formulas for the sum of products of homogeneous multiple harmonic sums \sum_{k=1}^n \prod_{j=1}^r H_k(\{1\}^{\lambda_j}) when \sum_{j=1}^r \lambda_j\leq 5. We apply these identities to the study of two congruences modulo a power of a prime.*- download a copy:*

**Congruences of alternating multiple harmonic sums**(with Jianqiang Zhao).

*Journal of Combinatorics and Number Theory, 2 (2010), 129-159.*

In this sequel to arXiv:0905.3327, we continue to study the congruence properties of the alternating version of multiple harmonic sums. As contrast to the study of multiple harmonic sums where Bernoulli numbers and Bernoulli polynomials play the key roles, in the alternating setting the Euler numbers and the Euler polynomials are also essential.*- download a copy from arXiv:*

**Congruences involving alternating multiple harmonic sum**.

*Electronic Journal of Combinatorics, 17 (2010), R16.*

We show that for any prime prime p\not=2 \sum_{k=1}^{p-1} {(-1)^k\over k}{-{1\over 2} \choose k} \equiv -\sum_{k=1}^{(p-1)/2}{1\over k} \pmod{p^3} by expressing the l.h.s. as a combination of alternating multiple harmonic sums.*- download a copy:*

**New congruences for central binomial coefficients**(with Zhi-Wei Sun).

*Advances in Applied Mathematics, 45 (2010), 125-148.*

Let p be a prime, and let d\in\{0,...,p^a\} with a\in\Z^+. In this paper we determine \sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k and \sum_{k=1}^{p-1}\binom{2k}{k+d}/(km^{k-1}) modulo p where m is an integer not divisible by p. For example, we show that if p\not=2,5 then \sum_{k=1}^{p-1}(-1)^k\frac{\binom{2k}k}k=-5\frac{F_{p-(\frac p5)}}p (mod p), where F_n denotes the nth Fibonacci number. We also prove that if p>3 then \sum_{k=1}^{p-1}\frac{\binom{2k}k}k={8/9} p^2B_{p-3} (mod p^3), where B_n is the nth Bernoulli number.*- download a copy:*

**An elementary proof of a Rodriguez-Villegas supercongruence**.

*arXiv:0911.4261**, (22/11/2009).*

We give a short proof of the following known congruence: for every odd prime p \sum_{k=0}^{p-1}{2k\choose k}^2 16^{-k}\equiv (-1)^{{p-1\over 2}}\pmod{p^2}. Moreover, we provide some new results connected with the above congruence.*- download a copy from arXiv:*

**Edge Cover Time for Regular Graphs**.

*Journal of Integer Sequences, 11 (2008), 8.4.4.*

Consider the following stochastic process on a graph: initially all vertices are uncovered and at each step cover the two vertices of a random edge. What is the expected number of steps required to cover all vertices of the graph? In this note we show that the mean cover time for a regular graph of N vertices is asymptotically (N log N)/2. Moreover, we compute the exact mean cover time for some regular graphs via generating functions.*- download a copy:*

**The operad Lie is free**(with Paolo Salvatore).

*Journal of Pure and Applied Algebra, 213 (2009), 224-230.*

We show that the operad Lie is free as a non-symmetric operad. Then we study the generating series counting the operadic generators, finding a recursive formula for its coefficients, and showing that the asymptotic density of the operadic generators is 1/e (see the corresponding new sequence A134988 in Neil Sloane's Online Encyclopedia of Integer Sequences.*- download a copy:*

**Rigidity of holomorphic generators and one-parameter semigroups**(with Mark Elin, Marina Levenshtein, and David Shoikhet).

*Dynamic Systems and Applications, 16 (2007), 251-266.*

In this paper we establish a rigidity property of holomorphic generators by using their local behavior at a boundary point \tau of the open unit disk \Delta. Namely, if f\in\mathrm{Hol}(\Delta,\mathbb{C}) is the generator of a one-parameter continuous semigroup \{F_{t}\}_{t\geq0}, we state that the equality f(z)=o\left(|z-\tau|^{3}\right) when z\rightarrow\tau in each non-tangential approach region at \tau implies that f vanishes identically on \Delta. Note, that if F is a self-mapping of \Delta then f=I-F is a generator, so our result extends the boundary version of the Schwarz Lemma obtained by D. Burns and S. Krantz.*- download a copy:*

**Congruences for Sums of Binomial Coefficients**(with Zhi-Wei Sun).

*Journal of Number Theory, 126 (2007), 287-296.*

We find an explicit integer nu_m(q) such that [n+ nu_m(q) ,r]_m=[n,r ]_m (mod q) where [n,r]_m=sum_{k=r mod m} binomial(n,k). This is a further extension of a congruence of Glaisher.*- download a copy:*

**The dinner table problem: the rectangular case**.

*INTEGERS, 6 (2006), A11.*

n people are seated randomly at a rectangular table with floor(n/2) and ceil(n/2) seats along the two opposite sides for two dinners. What's the probability that neighbors at the first dinner are no more neighbors at the second one? We give an explicit formula and we show that its asymptotic behavior as n goes to infinity is e^{-2}(1+4/n) (it is known that it is e^{-2}(1-4/n) for a round table). See the corresponding new sequence A110128 in Neil Sloane's Online Encyclopedia of Integer Sequences.

ERRATA. In Proposition 1, the term 736/(15n^5), taken from "The dinner table problem" by B. Aspvall and F. M. Liang, should be replaced by 796/(15*n^5) (see sequence A078630). I thank Vaclav Kotesovec for pointing out the error.*- download a copy:*

**A Common Fixed Point Theorem for Commuting Expanding Maps on Nilmanifolds**.

*Electronic Journal of Differential Equations, Conf. 12 (2005), 181-188.*

A self-map f of a compact connected manifold M is expanding if it locally expands distances with respect to some metric. We consider the case when M is a nilmanifold and we discuss a new common fixed point theorem for two expanding maps which commute.*- download a copy:*

**La matematica del domino.**

A divulgative paper about domino tiling problems that I've written in memory of Professor Franco Conti.

It has been published in the book*Ricordando Franco Conti*, Scuola Normale Superiore, Pisa, 2005.

- download a copy: |

**A New Domino Tiling Sequence**.

*Journal of Integer Sequences, 7 (2004), 2.3.*

In this short note, we prove that the sequence A061646 in Neil Sloane's Online Encyclopedia of Integer Sequences is connected with the number of domino tilings of a holey square.*- download a copy:*

**Commuting Holomorphic Self-Maps of the Unit Disc**.

*Ergodic Theory and Dynamical Systems, 24 (2004), 945-953.*

Let f,g be two commuting holomorphic self-maps of the unit disc. If f and g are fixed point free and they agree at the common Wolff point up to third order of derivatives then f and g are identically equal. Using the linear model of Baker and Pommerenke we improve a previous result for the parabolic case achieving a more ''natural'' smoothness conditions.*- download a copy:*

**The Lindelof principle and angular derivatives in convex domains of finite type**(with Marco Abate).

*Journal of the Australian Mathematical Society, 73 (2002), 1-30.*

We describe a generalization of the classical Julia-Wolff-Caratheodory theorem to a large class of bounded convex domains of finite type, including convex circular domains and convex domains with real analytic boundary. The main tools used in the proofs are several explicit estimates on the boundary behaviour of Kobayashi distance and metric, and a new Lindelof principle.*- download a copy:*

**Identity Principles for Commuting Holomorphic Self-Maps of the Unit Disc**(with Filippo Bracci and Fabio Vlacci).

*Journal of Mathematical Analysis and Applications, 270 (2002), 451-473.*

Let f,g be two commuting holomorphic self-maps of the unit disc. If f and g agree at the common Wolff point up to a certain order of derivatives (no more than 3 if the Wolff point is on the unit circle), then f and g are identically equal.*- download a copy:*

**Rigidity at the Boundary for Holomorphic Self-Maps of the Unit Disc**(with Fabio Vlacci).

*Complex Variables, Theory and Application, 45 (2001), 151-165.*

We prove a rigidity theorem which generalizes a result due to Burns and Krantz for holomorphic self-maps in the unit disc of the complex plane. The authors found that some conditions on the (boundary) Schwarzian derivative of a holomorphic self-map at specific points of the boundary of the disc may be sufficient to conclude that the map is a completely determined rational map.*- download a copy:*

**The Julia-Wolff-Caratheodory theorem(s)**(with Marco Abate).

*Kim, Kang-Tae (ed.) et al., Complex geometric analysis in Pohang. POSTECH-BSRI SNU-GARC international conference on several complex variables, Pohang, Korea, June 23-27, 1997. Providence, RI: American Mathematical Society. Contemporary Mathematics, 222 (1999), 161-172.*

It is described a general framework allowing generalization of the Julia-Wolff-Caratheodory theorem to several classes of bounded domain in C^n. As an example, the authors discuss the case of bounded convex circular domains.*- download a copy:*

**Sets of Periods for Expanding Maps on Flat Manifolds**.

*Monatshefte fur Mathematik, 128 (1999), 151-157.*

It is proven that the sets of periods for expanding maps on n-dimensional flat manifolds are uniformly cofinite, i.e. there is a positive integer m_0, which depends only on n,such that for any integer m>=m_0, for any n-dimensional flat manifold M and for any expanding map F on M, there exists a periodic point of F whose least period is exactly m.*- download a copy:*

**Common fixed points of commuting holomorphic maps of the polydisc which are expanding on the torus**.

*Advances in Mathematics, 138 (1998), 92-104.*

The following result is proved. Let F and G be two holomorphic maps of open polydisc in C^n which are continuous on the closure, map the torus in itself and are expanding on the torus. If F and G commute on the torus then they have a unique fixed point in the open polydisc.*- download a copy:*

**Centralizers of polynomials**.

*Rendiconti dell'Istituto di Matematica dell'Universita' di Trieste, 28 (1996), 63-69.*

The centralizer of a nonlinear polynomial P in C [z] is the set of nonlinear polynomials Q such that P(Q(z))=Q(P(z)). The centralizer is trivial if it consists only of the iterates of P. The author shows that there is an open dense subset S of the nonlinear polynomials whose elements have trivial centralizers. In fact if U is the set of polynomials of degree d>1 which have (d+1) different fixed points (including infinity) with different eigenvalues, then S=U\{T_2,-T_2\}, T_2(z) =2z^2-1.*- download a copy:*

**On fixed points of C^1 extensions of expanding maps in the unit disc.**

*Atti dell'Accademia Nazionale dei Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rendiconti Lincei, Mat. Appl., 5 (1994), 303-308.*

Using a result due to M. Shub, a theorem about the existence of fixed points inside the unit disc for C^1 extensions of expanding maps defined on the boundary is established.*- download a copy:*

**On fixed points of holomorphic maps of simply connected proper domains in C.**

*Atti dell'Accademia Nazionale dei Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rendiconti Lincei, Mat. Appl., 5 (1994), 197-202.*

A sufficient condition of the existence of fixed point of one-dimensional holomorphic map is given. If f has at least three zeros or a zero with multiplicity >=3 in {z: |z|<1/2}, then f satisfies the sufficient condition.*- download a copy:*

My Erdos number is 3 in two ways.

- Z.-W. Sun and R. Tauraso,
*Congruences for sums of binomial coefficients*, J. Number Theory 126 (2007), no. 2, 287-296.

- A. Granville and Z.-W. Sun,
*Values of Bernoulli polynomials*, Pacific J. Math. 172 (1996), no. 1, 117-137.

- T. Agoh, P. Erdös and A. Granville,
*Primes at a (somewhat lengthy) glance*, Amer. Math. Monthly 104 (1997), no. 10, 943-945.

- H. Prodinger and R. Tauraso,
*New multiple harmonic sum identities*, Electron. J. Comb. 21 (2014), no. 2, P2.43, 14 p.

- K. Hare, H. Prodinger, and J. Shallit,
*Three series for the generalized golden mean*, Fibonacci Q. 52 (1996), no. 4, 117-137.

- P. Erdös and J. Shallit,
*New bounds on the length of finite Pierce and Engel series*, Semin. Theor. Nombres Bordx. Ser. II 3 (1991), no. 1, 43-53.