36(4)
... Since Fn mdFn+l are coprime integers, and because any prime/? divides infinitely many Fibonacci numbers Fn_l = Fn+l-Fn, for every prime p the congruence Fn = Fn+l # 0 modp is satisfied for infinitely many pairs of Fibonacci numbers Fn and Fn+l. Proof of Theorem 1.2: Let 36 c: (0,1) be the subset of ...
... Since Fn mdFn+l are coprime integers, and because any prime/? divides infinitely many Fibonacci numbers Fn_l = Fn+l-Fn, for every prime p the congruence Fn = Fn+l # 0 modp is satisfied for infinitely many pairs of Fibonacci numbers Fn and Fn+l. Proof of Theorem 1.2: Let 36 c: (0,1) be the subset of ...
Sample Segment
... Lemma 1. Among any set of 5 points in the plane, no three collinear, we can find 4 that are the vertices of a convex quadrilateral. Proof. Consider three possible cases. case 1: the 5 points are the vertices of a convex pentagon. We are done. case 2: 4 points are the vertices of a convex quadrilater ...
... Lemma 1. Among any set of 5 points in the plane, no three collinear, we can find 4 that are the vertices of a convex quadrilateral. Proof. Consider three possible cases. case 1: the 5 points are the vertices of a convex pentagon. We are done. case 2: 4 points are the vertices of a convex quadrilater ...
MTH_63_3rd_Edition_Detailed_Solutions_Section_1.1_Reviewed
... steel that is 120” x 68”. If the part is 2.34” wide and 1.87” tall, calculate the number of rows and columns that will fit if there is a .6” space between each part. How many parts will this design produce? Note: the drawing is not to ...
... steel that is 120” x 68”. If the part is 2.34” wide and 1.87” tall, calculate the number of rows and columns that will fit if there is a .6” space between each part. How many parts will this design produce? Note: the drawing is not to ...
Introduction to Number Theory
... If we just want to find any two divisors, there may be many ways to do so. We’d like to be able to find a list of divisors in such a way that the same list is always found. Instead of looking for any divisors, let’s agree to find all prime divisors of a number. ...
... If we just want to find any two divisors, there may be many ways to do so. We’d like to be able to find a list of divisors in such a way that the same list is always found. Instead of looking for any divisors, let’s agree to find all prime divisors of a number. ...
Barwise: Infinitary Logic and Admissible Sets
... There are many natural examples of mathematical properties expressible in Lω1 ω . Let α be a countable ordinal. In the vocabulary L = {≤} of orderings, there is an Lω1 ω sentence whose models are just the orderings of type α, and there is an Lω1 ω formula saying, in a linear ordering, that the inter ...
... There are many natural examples of mathematical properties expressible in Lω1 ω . Let α be a countable ordinal. In the vocabulary L = {≤} of orderings, there is an Lω1 ω sentence whose models are just the orderings of type α, and there is an Lω1 ω formula saying, in a linear ordering, that the inter ...
Non-standard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson. He wrote:[...] the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection [...] that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latterRobinson argued that this law of continuity of Leibniz's is a precursor of the transfer principle. Robinson continued:However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits.Robinson continues:It is shown in this book that Leibniz's ideas can be fully vindicated and that they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory.In 1973, intuitionist Arend Heyting praised non-standard analysis as ""a standard model of important mathematical research"".