The Experts below are selected from a list of 1548 Experts worldwide ranked by ideXlab platform

Carl Pomerance - One of the best experts on this subject based on the ideXlab platform.

  • Paul Erdős and the Rise of Statistical Thinking in Elementary Number Theory
    Bolyai Society Mathematical Studies, 2013
    Co-Authors: Paul Pollack, Carl Pomerance
    Abstract:

    It might be argued that Elementary Number Theory began with Pythagoras who noted two-and-a-half millennia ago that 220 and 284 form an amicable pair. That is, if s(n) denotes the sum of the proper divisors of n (“proper divisor” means d │ n and 1 ≤ d < n), then $$s(220) = 284\quad and\quad s(284) = 220.$$ When faced with remarkable examples such as this it is natural to wonder how special they are. Through the centuries mathematicians tried to find other examples of amicable pairs, and they did indeed succeed. But is there a formula? Are there infinitely many? In the first millennium of the common era, Thâbit ibn Qurra came close with a formula for a subfamily of amicable pairs, but it is far from clear that his formula gives infinitely many examples and probably it does not.

Paul Pollack - One of the best experts on this subject based on the ideXlab platform.

  • Paul Erdős and the Rise of Statistical Thinking in Elementary Number Theory
    Bolyai Society Mathematical Studies, 2013
    Co-Authors: Paul Pollack, Carl Pomerance
    Abstract:

    It might be argued that Elementary Number Theory began with Pythagoras who noted two-and-a-half millennia ago that 220 and 284 form an amicable pair. That is, if s(n) denotes the sum of the proper divisors of n (“proper divisor” means d │ n and 1 ≤ d < n), then $$s(220) = 284\quad and\quad s(284) = 220.$$ When faced with remarkable examples such as this it is natural to wonder how special they are. Through the centuries mathematicians tried to find other examples of amicable pairs, and they did indeed succeed. But is there a formula? Are there infinitely many? In the first millennium of the common era, Thâbit ibn Qurra came close with a formula for a subfamily of amicable pairs, but it is far from clear that his formula gives infinitely many examples and probably it does not.

  • Not Always Buried Deep: A Second Course in Elementary Number Theory
    2009
    Co-Authors: Paul Pollack
    Abstract:

    Number Theory is one of the few areas of mathematics where problems of substantial interest can be fully described to someone with minimal mathematical background. Solving such problems sometimes requires difficult and deep methods. But this is not a universal phenomenon; many engaging problems can be successfully attacked with little more than one's mathematical bare hands. In this case one says that the problem can be solved in an Elementary way. Such Elementary methods and the problems to which they apply are the subject of this book. ""Not Always Buried Deep"" is designed to be read and enjoyed by those who wish to explore Elementary methods in modern Number Theory. The heart of the book is a thorough introduction to Elementary prime Number Theory, including Dirichlet's theorem on primes in arithmetic progressions, the Brun sieve, and the Erdos-Selberg proof of the prime Number theorem. Rather than trying to present a comprehensive treatise, Pollack focuses on topics that are particularly attractive and accessible. Other topics covered include Gauss' Theory of cyclotomy and its applications to rational reciprocity laws, Hilbert's solution to Waring's problem, and modern work on perfect Numbers. The nature of the material means that little is required in terms of prerequisites: the reader is expected to have prior familiarity with Number Theory at the level of an undergraduate course and a first course in modern algebra (covering groups, rings, and fields). The exposition is complemented by over 200 exercises and 400 references.

Le Mao-hua - One of the best experts on this subject based on the ideXlab platform.

Adam Naumowicz - One of the best experts on this subject based on the ideXlab platform.

  • dataset description formalization of Elementary Number Theory in mizar
    International Conference on Intelligent Computer Mathematics, 2020
    Co-Authors: Adam Naumowicz
    Abstract:

    In this paper we present a dataset based on the Mizar formalization of selected problems related to Elementary Number Theory. The dataset comprises proofs of problems on several levels of difficulty. They are available in the form of full proofs, proof sketches, as well as bare statements equipped with suitable environments importing necessary notions from the Mizar Mathematical Library. The Elementary character of the underlying Theory makes the data particularly suitable as a starting point for developing courses in interactive theorem proving based mathematics education and recreational mathematics activities.

  • CICM - Dataset Description: Formalization of Elementary Number Theory in Mizar.
    Lecture Notes in Computer Science, 2020
    Co-Authors: Adam Naumowicz
    Abstract:

    In this paper we present a dataset based on the Mizar formalization of selected problems related to Elementary Number Theory. The dataset comprises proofs of problems on several levels of difficulty. They are available in the form of full proofs, proof sketches, as well as bare statements equipped with suitable environments importing necessary notions from the Mizar Mathematical Library. The Elementary character of the underlying Theory makes the data particularly suitable as a starting point for developing courses in interactive theorem proving based mathematics education and recreational mathematics activities.

Tommy Dreyfus - One of the best experts on this subject based on the ideXlab platform.

  • Secondary teachers’ knowledge of Elementary Number Theory proofs: the case of general-cover proofs
    Journal of Mathematics Teacher Education, 2011
    Co-Authors: Michal Tabach, Esther Levenson, Ruthi Barkai, Pessia Tsamir, Dina Tirosh, Tommy Dreyfus
    Abstract:

    In light of recent reform recommendations, teachers are expected to turn proofs and proving into an ongoing component of their classroom practice. Two questions emerging from this requirement are: Is the mathematical knowledge of high school teachers sufficient to prove various kinds of statements? Does teachers’ knowledge allow them to determine the validity of an argument made by their students? The results of this study, in which 50 secondary school teachers participated, point to a positive answer to the first question in the framework of Elementary Number Theory (ENT). However, the picture is more complex with respect to the second one. Results indicated that some teachers may over-value the generality of symbolic mode of representation and under-value the generality of verbal ones. Possibly, the verbal representation of an argument is less transparent and more difficult to understand.

  • VERBAL JUSTIFICATION—IS IT A PROOF? SECONDARY SCHOOL TEACHERS’ PERCEPTIONS
    International Journal of Science and Mathematics Education, 2010
    Co-Authors: Michal Tabach, Ruthi Barkai, Pessia Tsamir, Dina Tirosh, Tommy Dreyfus, Esther Levenson
    Abstract:

    According to reform documents, teachers are expected to teach proofs and proving in school mathematics. Research results indicate that high school students prefer verbal proofs to other formats. We found it interesting and important to examine the position of secondary school teachers with regard to verbal proofs. Fifty high school teachers were asked to prove various Elementary Number Theory statements, to write correct and incorrect proofs that students may use, and to evaluate given justifications to statements from Elementary Number Theory. While all the participants provided correct proofs to the statements, our findings indicate that teachers are not aware of students’ preference for verbal justifications. Also, about half of the teachers rejected correct verbal justifications. They claimed that these justifications lacked generality and are mere examples.