Infinity 1. Introduction
... useful class of infinite sets. Hence there is no separate theory of finite sets in topology: it is simply subsumed in the theory of compact sets. An advocate of actual infinity would draw a general moral from these examples: that the distinction between finite and infinite is less important than it ...
... useful class of infinite sets. Hence there is no separate theory of finite sets in topology: it is simply subsumed in the theory of compact sets. An advocate of actual infinity would draw a general moral from these examples: that the distinction between finite and infinite is less important than it ...
Chapter 1 Ways to Choose
... We begin with fundamental definitions and customary notation. A set is an assemblage of elements listed in any order without repetitions. The notation {3, 1, 2} describes a set with 1, 2, and 3 as elements; on the other hand {1, 2, 2, 3} is not a set. The number of elements in a set could be finite ...
... We begin with fundamental definitions and customary notation. A set is an assemblage of elements listed in any order without repetitions. The notation {3, 1, 2} describes a set with 1, 2, and 3 as elements; on the other hand {1, 2, 2, 3} is not a set. The number of elements in a set could be finite ...
MODULE 5 Fermat`s Theorem INTRODUCTION
... times, and the result divided by p, we get a remainder of one. For example, if we use a = 7 and p =3, the rule says that 72 divided by 3 will have a remainder of one. In fact 49/3 does not have remainder of one. This theorem has since become known as “Fermat’s Little Theorem,” or just “Fermat’s Theo ...
... times, and the result divided by p, we get a remainder of one. For example, if we use a = 7 and p =3, the rule says that 72 divided by 3 will have a remainder of one. In fact 49/3 does not have remainder of one. This theorem has since become known as “Fermat’s Little Theorem,” or just “Fermat’s Theo ...
MATH 521, WEEK 2: Rational and Real Numbers, Ordered Sets
... E. And yet our definition of being countably infinite implies that every element of E may be put in unique correspondence with exactly one element of N. Since we can clearly do this, it follows that, in this sense, E and N are exactly the same size. So strange things can happen when we consider infi ...
... E. And yet our definition of being countably infinite implies that every element of E may be put in unique correspondence with exactly one element of N. Since we can clearly do this, it follows that, in this sense, E and N are exactly the same size. So strange things can happen when we consider infi ...
FIELDS ON THE BOTTOM
... We say that a field F lies on the bottom if F contains no field E with 1 < [F : E] < ∞. By definition, each of the prime fields Q and Fp lie on the bottom. By a theorem of Artin, every separably closed field of positive characteristic lies on the bottom (see for example the proof of [Lan93, Cor. 9.3]). I ...
... We say that a field F lies on the bottom if F contains no field E with 1 < [F : E] < ∞. By definition, each of the prime fields Q and Fp lie on the bottom. By a theorem of Artin, every separably closed field of positive characteristic lies on the bottom (see for example the proof of [Lan93, Cor. 9.3]). I ...
Primes and Greatest Common Divisors
... There are infinitely many primes Lemma Every natural number greater than one is either prime or it has a prime divisor Follows from fundamental theorem Proof Suppose towards a contradiction that there are only finitely many primes p1 , p2 , p3 , . . . , pk . Consider the number q = p1 p2 p3 . . . p ...
... There are infinitely many primes Lemma Every natural number greater than one is either prime or it has a prime divisor Follows from fundamental theorem Proof Suppose towards a contradiction that there are only finitely many primes p1 , p2 , p3 , . . . , pk . Consider the number q = p1 p2 p3 . . . p ...
Full text
... that the representations agree with the first Fk+1 from k. Now, for the compositions of k, tack on the right side a 2 , on the far right of the nested greatest integer functions, and suppress all the excess right a's. This yields, with the new addition, representation for the numbers F ...
... that the representations agree with the first Fk+1 from k. Now, for the compositions of k, tack on the right side a 2 , on the far right of the nested greatest integer functions, and suppress all the excess right a's. This yields, with the new addition, representation for the numbers F ...
Fermat Numbers - William Stein
... because it seems to be more commonly used and it gives the properties that we have proved earlier. Notice that Theorem6 is false if we had chosen the other definition. A counterexample is 21 + 1 = 3 and 23 + 1 = 9 have a common factor 3. However, if we are only interested in Fermat numbers that are ...
... because it seems to be more commonly used and it gives the properties that we have proved earlier. Notice that Theorem6 is false if we had chosen the other definition. A counterexample is 21 + 1 = 3 and 23 + 1 = 9 have a common factor 3. However, if we are only interested in Fermat numbers that are ...
Solved and unsolved problems in elementary number theory
... Did Pythagoras invent arithmetic dynamics? Consider the map s : N ∪ {0} → N ∪ {0}, extended to have s(0) = 0. A perfect number is nothing other than a positive integer fixed point. We say n is amicable if n generates a two-cycle: in other words, s(n) 6= n and s(s(n)) = n. For example, s(220) = 284, ...
... Did Pythagoras invent arithmetic dynamics? Consider the map s : N ∪ {0} → N ∪ {0}, extended to have s(0) = 0. A perfect number is nothing other than a positive integer fixed point. We say n is amicable if n generates a two-cycle: in other words, s(n) 6= n and s(s(n)) = n. For example, s(220) = 284, ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.