Comments on the Equation ±ra
نویسنده
چکیده
in nonnegative integers x, y and integers u, v ∈ {0, 1}. Note that the choice of x and y uniquely determines the choice of u and v, so we will usually refer to a solution (x, y). The Case (ra, sb) = 1 There are only a finite number of cases with N > 3 solutions to Equation (P) [23]. There are at least five infinite families of cases with N = 3 solutions to (P), as well as a number of anomalous cases with N = 3 (by ‘anomalous case’ we mean a case not a member of a known infinite family). Some of these anomalous cases are quite high, e.g., (a, b, c, r, s) = (56744, 1477, 83810889, 1478, 56743), [23]. We have not been able to give a complete finite list of such anomalous solutions, so the question arises: what additional restrictions on the variables would make possible a proof which gives a complete list of anomalous solutions, thus improving the result to N = 2 except for completely designated exceptions? This question has been essentially answered with the additional restriction x > 0 and y > 0 (see [22], in which the problem is reduced to a finite search). But even if only one of the exponents (x1, x2, . . . , xN , y1, y2, . . . , yN) is equal to zero, the problem becomes more difficult: even with the further restriction rs = 1 the methods of [3] and [21] do not suffice without additional heavy restrictions such as placing an upper bound on one of b, c, or min(x1, x2, . . . , xN) (when min(y1, y2, . . . , yN) = 0). But if one adds the yet further restriction that a and b be prime, it is possible to give a complete list of infinite families and a complete list of anomalous solutions, thus obtaining N = 2 with completely designated exceptions (see [26]). The restriction that a and b be prime is perhaps not as artificial as it may seem: computer searches in [3] and [21] (supplemented with calculations on the second author’s website) suggest that the list of exceptions in Theorem 3 would remain unchanged even if p and q were allowed to be any relatively prime integers (here of course we would be redefining F and M to allow composite Fermat and Mersenne numbers). The General Case In what follows we will refer to a set of solutions to (P) which we will write as
منابع مشابه
روش های تربیت عقلانی بر مبنای اندیشه امام خمینی(ره)
This paper reviews training methods based on rational thoughts of Imam Khomeini, "RAh" is. It can be intellectual as the main point of theoretical challenges, the comments cited in the field. Such is the wisdom with multiple meanings and its different divisions, subject always dialogue was thinkers East and West. Research methods - descriptive and analytical. Hence the author has tried to ana...
متن کاملNatural Convection Heat Transfer within Octagonal Enclosure
The problem of steady, laminar and incompressible natural convection flow in an octagonalenclosure was studied. In this investigation, two horizontal walls were maintained at a constant hightemperature, two vertical walls were kept at a constant low temperature and all inclined walls wereconsidered adiabatic. The enclosure was assumed to be filled with a Bousinessq fluid. The studyincludes comp...
متن کاملAntithetical Gendered Stances in Readers’ Comments on Domestic Violence against Men
Domestic violence against women (DVAW) has received much attention from scholars across disciplines, leading to a circumvention of studies on domestic violence against men (DVAM). This paper, therefore, engages in a qualitative dialogic analysis of readers’ comments on cases of DVAM reported in select blogs in order to elicit opposing gendered stances on DVAM in the selected readers’ comments; ...
متن کاملThe Elephant Will Kick the Donkey: An Attitudinal Analysis of Online Comments on the GOP Letter to the Iranian Supreme Leader
With the growth in sociality and interaction around touching national and international topics, news sites are increasingly becoming places for communities to discuss and address issues spurred by news articles. Proponents of cyberspace promise that online discourse will increase political participation and pave the road for a democratic utopia (Papacharissi, 2004). Comment writers, according t...
متن کاملA Typology of Supervisor Written Feedback on L2 Students’ Theses/Dissertations
The present study aimed at providing a typology of Iranian supervisors’ written feedback on L2 graduate students’ theses/dissertations and examining the way different speech functions are employed to put the supervisors’ thoughts and feelings into words. In so doing, a corpus of comments, including 15,198 comments provided on 87 TEFL theses and dissertations by 30 supervisors were analyzed. We ...
متن کاملHow to reply to referees' comments when submitting manuscripts for publication
Background: The publication of articles in peer-reviewed scientific journals is a fairly complex and stepwise process that involves responding to referees’ comments. Little guidance is available in the biomedical literature on how to deal with such comments. Objective: The objective of this article is to provide guidance to notice writers on dealing with peer review comments in a way that maxim...
متن کامل