Let $f$ be a positive integer, and let $χ$ be a primitive quadratic character of conductor $f$. Let $k$ be a positive integer, and write $B_k(χ,X)$ for the $k$-th Bernoulli polynomial corresponding to $χ$. Suppose $B_k(χ,X)$ is irreducible and of degree at least $2$. Then for 100% of positive integers $m$ divisible by $f$, the Diophantine equation \[ χ(1) \cdot (x+1)^k+χ(2) \cdot (x+2)^k+\cdots+χ(m) \cdot (x+m)^k \, =\, y^n, \] has no solutions with $x$, $y$, $n$ integers, and $n \ge 2$.

Tian’s conjecture states that for any fixed distinct prime numbers p1,…,pm, the Diophantine equation n+12=p1α1·p2α2···pmαm in positive integers n,α1,…,αm has at most m solutions. In this paper, we develop a computational method to verify some special cases of this conjecture. We also give an alternative proof using the classical Zsigmondy theorem. For m=2 and 3, a sharp absolute upper bound for the number of solutions is given.

In this paper, we study the Diophantine equation bk+(a+ b)k+(2a+ b)k++(a(x-1) +b)k=y(y +c)(y +2c)(y +(& ell;-1) c) , where a, b, c, k, & ell; are given integers under natural conditions. We prove some effective results for special values for c, k and & ell;, and obtain a general ineffective result based on the Bilu-Tichy method.

In this short survey paper, some recent results on the zero-structure of shifted Bernoulli polynomials $B_k(X)+b$ and Euler polynomials $E_k(X)+b$ are presented. Further, using some previous results, we show that there are only finitely many pairs $(k,b)$ with $k\geq 3$, $b\in \mathbb{C}$ for which $B_k(X)+b$, resp., $E_k(X)+b$ has no three simple zeros, and we give explicitly all these pairs $(k,b)$.

Motivated by the work of David Singmaster, we study the number of times an integer can appear among the Stirling numbers of both kinds. We provide an upper bound for the occurrences of all the positive integers, and present certain questions for further study. Some numerical results and conjectures concerning the related diohantine equations are collected.

A divisor B of a nonzero polynomial A, defined over the prime field of two elements, is unitary (resp. bi-unitary) if gcd(B,A/B)=1 (resp. gcdu(B,A/B)=1), where gcdu(B,A/B) denotes the greatest common unitary divisor of B and A/B. We denote by σ**(A) the sum of all bi-unitary monic divisors of A. A polynomial A is called a bi-unitary superperfect polynomial over F2 if the sum of all bi-unitary monic divisors of σ**(A) equals A. In this paper, we give all bi-unitary superperfect polynomials divisible by one or two irreducible polynomials over F2. We prove the nonexistence of odd bi-unitary superperfect polynomials over F2.

An M/M/1//N retrial queueing system with two-way communication to the infinite source and impatient customers in the orbit is considered in the paper. There is a finite source in which the primary or regular customers are coming, while requests from the infinite source are the secondary customers. Because of not having waiting queues, the service of an arriving, primary customer begins immediately. Otherwise, in the case of a busy server, the primary customers are forwarded to the orbit waiting an exponentially distributed random time to try to reach the service unit. When the service unit is in an idle state, it may call a customer from the infinite source for service. All requests possess an impatience property resulting in an earlier departure from the system through the orbit if they wait too much for being served. Besides, the service unit is supposed to break down according to several distributions which have a specialty in removing all the customers located in the system. In the case of a faulty state, blocking is applied not allowing the customers into the system until the service unit fully recovers. This work concentrates on examining the effect of those distributions on several performance measures like the distribution of the number of customers in the system, and the probability of a primary customer departing because of catastrophe. The obtained results are graphically realized to show the differences and curiosities among the used parameter settings of the various distributions.

For $m \geq 3$, we define the $m$th order pyramidal number by \[ \mathrm{Pyr}_m(x) = \frac{1}{6} x(x+1)((m-2)x+5-m). \] In a previous paper, written by the first-, second-, and fourth-named authors, all solutions to the equation $\mathrm{Pyr}_m(x) = y^2$ are found in positive integers $x$ and $y$, for $6 \leq m \leq 100$. In this paper, we consider the question of higher powers, and find all solutions to the equation $\mathrm{Pyr}_m(x) = y^n$ in positive integers $x$, $y$, and $n$, with $n \geq 3$, and $5 \leq m \leq 50$. We reduce the problem to a study of systems of binomial Thue equations, and use a combination of local arguments, the modular method via Frey curves, and bounds arising from linear forms in logarithms to solve the problem.

In this paper a retrial queuing system is considered with the help of two-way communication where the server is subject to random breakdowns. This is a M/M/1//N type of system so the population of the source is finite. The server becoming idle enables calls the customers in the orbit (outgoing call or secondary customers). The service time of the primary and secondary customers follows exponential distribution with different rates μ 1 and μ 2 respectively. All the random variables included in the model construction are assumed to be totally independent of each other. The novelty of this paper is to show the effect of the different distributions of failure time on the main performance measures such as the mean waiting time of an arbitrary customer or the utilization of the service unit. In order to achieve a valid comparison a fitting process is done; thus, in case of every distribution the mean value and dispersion is the same. Graphical illustrations are given with the help of the self-developed simulation program.

In this paper, a finite-source retrial queueing system is considered with impatient customers and catastrophic breakdowns. The characteristic of the system includes collision which occurs when a new job arrives in the system and the service facility is occupied with a job, they will collide. Both jobs will be forwarded to the virtual waiting room the so-called orbit. Here, the customers initiate other attempts to reach the server after a random time. But they give up retrying after staying in the orbit a while and leave the system which is the impatient attribute of the customers. In case of a negative event, a catastrophic breakdown takes place meaning that all the customers at the server and in the orbit depart from the system. The novelty of this paper is to investigate that feature in a collision environment with impatient customers using different distributions of the service time.

The purpose of the present paper is to consider the extensibility of the Diophantine triple {a, b, c}, where a < b < c with b = 3a, and to prove that such a set cannot be extended to an irregular Diophantine quadruple. We succeed in that for some families of c's (depending on a). As corollary, we prove that any Diophantine quadruple which contains the pair {a, 3a} is regular. Finally in this paper, we will see that by considering the case b = 8a we obviously obtain similar results. (C) 2021 Elsevier Inc. All rights reserved.

The receptorial responsiveness method (RRM) is a procedure that is based on a simple nonlinear regression while using a model with two variables (X, Y) and (at least) one parameter to be determined (cx). The model of RRM describes the co-action of two agonists that consume the same response capacity (due to the use of the same postreceptorial signaling in a biological system). While using RRM, uniquely, an acute increase in the concentration of an agonist (near the receptors) can be quantified (as cx), via evaluating E/c curves that were constructed with the same or another agonist in the same system. As this measurement is sensitive to the implementation of the curve fitting, the goal of the present study was to test RRM by combining different ways and setting options, namely: individual vs. global fitting, ordinary vs. robust fitting, and three weighting options (no weighting vs. weighting by 1/Y2 vs. weighting by 1/SD2). During the testing, RRM was used to estimate the known concentrations of stable synthetic A1 adenosine receptor agonists in isolated, paced guinea pig left atria. The estimates were then compared to the known agonist concentrations (to assess the accuracy of RRM); furthermore, the 95% confidence limits of the best-fit values were also considered (to evaluate the precision of RRM). It was found that, although the global fitting offered the most convenient way to perform RRM, the best estimates were provided by the individual fitting without any weighting, almost irrespective of the fact whether ordinary or robust fitting was chosen.

In earlier studies, we generated concentration-response (E/c) curves with CPA (N6-cyclopentyladenosine; a selective A1 adenosine receptor agonist) or adenosine, in the presence or absence of S-(2-hydroxy-5-nitrobenzyl)-6-thioinosine (NBTI, a selective nucleoside transport inhibitor), and with or without a pretreatment with 8-cyclopentyl-N3-[3-(4-(fluorosulfonyl)-benzoyloxy)propyl]-N1-propylxanthine (FSCPX, a chemical known as a selective, irreversible A1 adenosine receptor antagonist), in isolated, paced guinea pig left atria. Meanwhile, we observed a paradoxical phenomenon, i.e., the co-treatment with FSCPX and NBTI appeared to enhance the direct negative inotropic response to adenosine. In the present in silico study, we aimed to reproduce eight of these E/c curves. Four models (and two additional variants of the last model) were constructed, each one representing a set of assumptions, in order to find the model exhibiting the best fit to the ex vivo data, and to gain insight into the paradoxical phenomenon in question. We have obtained in silico evidence for an interference between effects of FSCPX and NBTI upon our ex vivo experimental setting. Regarding the mechanism of this interference, in silico evidence has been gained for the assumption that FSCPX inhibits the effect of NBTI on the level of endogenous (but not exogenous) adenosine. As an explanation, it may be hypothesized that FSCPX inhibits an enzyme participating in the interstitial adenosine formation. In addition, our results suggest that NBTI does not stop the inward adenosine flux in the guinea pig atrium completely.

In the present paper we prove that under certain conditions the linear combination of two Euler polynomials with odd degrees P n , m ( x ) = E n ( x ) + c E m ( x ) is always indecomposable over C , where c denotes a rational number.

Based on in silico results, recently we have assumed that FSCPX, an irreversible A(1) adenosine receptor antagonist, inhibits the action of NBTI that is apparent on E/c curves of adenosine receptor agonists. As a mechanism for this unexpected effect, we hypothesized that FSCPX might modify the equilibrative and NBTI-sensitive nucleoside transporter (ENT1) in a way that allows ENT1 to transport adenosine but impedes NBTI to inhibit this transport. This assumption implies that our method developed to estimate receptor reserve for agonists with short half-life such as adenosine, in its original form, overestimates the receptor reserve. In this study, therefore, our goals were to experimentally test our assumption on this effect of FSCPX, to improve our receptor reserve-estimating method and then to compare the original and improved forms of this method. Thus, we improved our method and assessed the receptor reserve for the direct negative inotropic effect of adenosine with both forms of this method in guinea pig atria. We have found that FSCPX inhibits the effects of NBTI that are mediated by increasing the interstitial concentration of adenosine of endogenous (but not exogenous) origin. As a mechanism for this action of FSCPX, inhibition of enzymes participating in the interstitial adenosine production can be hypothesized, while modification of ENT1 can be excluded. Furthermore, we have shown that, in comparison with the improved form, the original version of our method overestimates receptor reserve but only to a small extent. Nevertheless, use of the improved form is recommended in the future.

Dujella and Pethő, generalizing a result of Baker and Davenport, proved that the set {1,3} cannot be extended to a Diophantine quintuple. As a consequence of our main result, we show that the Diophantine pair {1,b} is regular if b−1 is a prime power.

The term receptor reserve, first introduced and used in the traditional receptor theory, is an integrative measure of response-inducing ability of the interaction between an agonist and a receptor system (consisting of a receptor and its downstream signaling). The underlying phenomenon, i.e., stimulation of a submaximal fraction of receptors can apparently elicit the maximal effect (in certain cases), provides an opportunity to assess the receptor reserve. However, determining receptor reserve is challenging for agonists with short half-lives, such as adenosine. Although adenosine metabolism can be inhibited several ways (in order to prevent the rapid elimination of adenosine administered to construct concentration–effect (E/c) curves for the determination), the consequent accumulation of endogenous adenosine biases the results. To address this problem, we previously proposed a method, by means of which this bias can be mathematically corrected (utilizing a traditional receptor theory-independent approach). In the present investigation, we have offered in silico validation of this method by simulating E/c curves with the use of the operational model of agonism and then by evaluating them using our method. We have found that our method is suitable to reliably assess the receptor reserve for adenosine in our recently published experimental setting, suggesting that it may be capable for a qualitative determination of receptor reserve for rapidly eliminating agonists in general. In addition, we have disclosed a possible interference between FSCPX (8-cyclopentyl-N3-[3-(4-(fluorosulfonyl)benzoyloxy)propyl]-N1-propylxanthine), an irreversible A1 adenosine receptor antagonist, and NBTI (S-(2-hydroxy-5-nitrobenzyl)-6-thioinosine), a nucleoside transport inhibitor, i.e., FSCPX may blunt the effect of NBTI.

In the present paper, we prove that the general integer linear combination of several Bernoulli and Euler polynomials with odd degree is always indecomposable over the field of complex numbers.

We derive sharp upper bounds for the size of the intersection of certain linear recurrence sequences. As a consequence of these, we partially resolve a conjecture of Yuan on simultaneous Pellian equations, under the condition that one of the parameters involved is suitably large.