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The question is still open as to whether there exist infinitely many Fermat primes or infinitely many composite Fermat numbers. The same question concerning Mersenne numbers is also unanswered. Extending some recent results of Megrelishvili and the author, we characterize the Fermat primes and the Mersenne primes in terms of the topological minimality of some matrix groups. This is achieved by showing, among other things, that if F is a subfield of a local field of characteristic ≠2, then the special upper triangular group ST+(n,F) is minimal precisely when the special linear group SL(n,F) is. We provide criteria for the minimality (and total minimality) of SL(n,F) and ST+(n,F), where F is a subfield of C. Let Fπ and Fc be the set of Fermat primes and the set of composite Fermat numbers, respectively. As our main result, we prove that the following conditions are equivalent for A∈Fπ,Fc: A is finite; ∏Fn∈ASL(Fn−1,Q(i)) is minimal, where Q(i) is the Gaussian rational field; and ∏Fn∈AST+(Fn−1,Q(i)) is minimal. Similarly, denote by Mπ and Mc the set of Mersenne primes and the set of composite Mersenne numbers, respectively, and let B∈Mπ,Mc. Then the following conditions are equivalent: B is finite; ∏Mp∈BSL(Mp+1,Q(i)) is minimal; and ∏Mp∈BST+(Mp+1,Q(i)) is minimal.

In this volume, an abstract theory of 'forms' is developed, thus providing a conceptually satisfying framework for the classification of forms of Fermat equations. The classical results on diagonal forms are extended to the broader class of all forms of Fermat varieties. The main topic is the study of forms of the Fermat equation over an arbitrary field K. Using Galois descent, all such forms are classified; particularly, a complete and explicit classification of all cubic binary equations is given. If K is a finite field containing the d-th roots of unity, the Galois representation on l-adic cohomology (and so in particular the zeta function) of the hypersurface associated with an arbitrary form of the Fermat equation of degree d is computed.

In this paper, we deal mainly with a class of periodic tridiagonal Toeplitz matrices with perturbed corners. By matrix decomposition with the Sherman–Morrison–Woodbury formula and constructing the corresponding displacement of matrices we derive the formulas on representation of the determinants and inverses of the periodic tridiagonal Toeplitz matrices with perturbed corners of type I in the form of products of Fermat numbers and some initial values. Furthermore, the properties of type II matrix can be also obtained, which benefits from the relation between type I and II matrices. Finally, we propose two algorithms for computing these properties and make some analysis about them to illustrate our theoretical results.

Leta, b, cbe relatively prime positive integers such thata 2+b 2=c 2. In 1956, Jeśmanowicz conjectured that for any positive integern, the only solution of (an) x + (bn) y = (cn) z in positive integers is (x, y, z) = (2, 2, 2). Letk≥ 1 be an integer andFk = 22 k + 1 bek-th Fermat number. In this paper, we show that Jeśmanowicz’ conjecture is true for Pythagorean triples (a, b, c) = (Fk − 2, 22 k−1+1,Fk ). 2010Mathematics Subject Classification: 11D61. Key words and phrases: Jeśmanowicz’ conjecture, Diophantine equation, Fermat numbers.

We define a Gauss factorial Nn!N_n! to be the product of all positive integers up to NN that are relatively prime to n∈Nn\\in \\mathbb N. In this paper we study particular aspects of the Gauss factorials ⌊n−1M⌋n!\\lfloor \\frac {n-1}{M}\\rfloor _n! for M=3M=3 and 6, where the case of nn having exactly one prime factor of the form p≡1(mod6)p\\equiv 1\\pmod {6} is of particular interest. A fundamental role is played by those primes p≡1(mod3)p\\equiv 1\\pmod {3} with the property that the order of p−13!\\frac {p-1}{3}! modulo pp is a power of 2 or 3 times a power of 2; we call them Jacobi primes. Our main results are characterizations of those n≡±1(modM)n\\equiv \\pm 1\\pmod {M} of the above form that satisfy ⌊n−1M⌋n!≡1(modn)\\lfloor \\frac {n-1}{M}\\rfloor _n!\\equiv 1\\pmod {n}, M=3M=3 or 6, in terms of Jacobi primes and certain prime factors of generalized Fermat numbers. We also describe the substantial and varied computations used for this paper.

We note that one more factor is missing from Table 1 in Bjorn-Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), 441 446, in addition to the three already reported upon in Bjorn-Riesel, Table errata to \"Factors of generalized Fermat numbers\", Math. Comp. 74 (2005), p. 2099.

The main purpose of this paper is using the elementary and combinational methods to obtain several new identities for Euler and Bernoulli polynomials. As some applications of these identities, we give several interesting relationships between these polynomials and some famous sequences (such as Fibonacci sequences and Lucas sequences) or numbers (such as Mersenne numbers and Fermat numbers).

We note that one more factor is missing from Table 1 in Björn–Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), 441–446, in addition to the three already reported upon in Björn–Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), p. 2099.

Have you ever daydreamed about digging a hole to the other side of the world? Robert Banks not only entertains such ideas but, better yet, he supplies the mathematical know-how to turn fantasies into problem-solving adventures. In this sequel to the popularTowing Icebergs, Falling Dominoes(Princeton, 1998), Banks presents another collection of puzzles for readers interested in sharpening their thinking and mathematical skills. The problems range from the wondrous to the eminently practical. In one chapter, the author helps us determine the total number of people who have lived on earth; in another, he shows how an understanding of mathematical curves can help a thrifty lover, armed with construction paper and scissors, keep expenses down on Valentine's Day. In twenty-six chapters, Banks chooses topics that are fairly easy to analyze using relatively simple mathematics. The phenomena he describes are ones that we encounter in our daily lives or can visualize without much trouble. For example, how do you get the most pizza slices with the least number of cuts? To go from point A to point B in a downpour of rain, should you walk slowly, jog moderately, or run as fast as possible to get least wet? What is the length of the seam on a baseball? If all the ice in the world melted, what would happen to Florida, the Mississippi River, and Niagara Falls? Why do snowflakes have six sides? Covering a broad range of fields, from geography and environmental studies to map- and flag-making, Banks uses basic algebra and geometry to solve problems. If famous scientists have also pondered these questions, the author shares the historical details with the reader. Designed to entertain and to stimulate thinking, this book can be read for sheer personal enjoyment.