Notes for Number theory (Fall semester)
... 3. The well-ordering principle is easily seen to be false if we replace N with any of the other standard number system: for instance, the set of the strictly positive rational numbers has no least element. 4. The previous principle implies 0 element, the 1 element and the addition ` (as initial data ...
... 3. The well-ordering principle is easily seen to be false if we replace N with any of the other standard number system: for instance, the set of the strictly positive rational numbers has no least element. 4. The previous principle implies 0 element, the 1 element and the addition ` (as initial data ...
Math 780: Elementary Number Theory
... Notation. We write ajb if a divides b, and we write a b if a does not divide b. De nition. An integer p is prime (or is a prime) if it is > 1 and divisible by no other positive integer other than 1 and itself. (In Algebra, the condition that p be > 1 is replaced by jpj > 1.) The division algorithm. ...
... Notation. We write ajb if a divides b, and we write a b if a does not divide b. De nition. An integer p is prime (or is a prime) if it is > 1 and divisible by no other positive integer other than 1 and itself. (In Algebra, the condition that p be > 1 is replaced by jpj > 1.) The division algorithm. ...
Flat primes and thin primes
... Therefore, by the quoted result above, the number of thin primes less than or equal to x is bounded below by a constant times x/ log2 x, so must be infinite. However there are parts of the proof of [19, Theorem 3] that do not appear to work, even for the given case a = 2, and, in addition, the impli ...
... Therefore, by the quoted result above, the number of thin primes less than or equal to x is bounded below by a constant times x/ log2 x, so must be infinite. However there are parts of the proof of [19, Theorem 3] that do not appear to work, even for the given case a = 2, and, in addition, the impli ...
Congruences
... can add or multiply congruences. But how about adding an equation to a congruency or multiplying a congruency by an equation? Note that "adding an equation to a congruency" is a fancy way of saying adding the same integer to both sides of a congruency. Similarly the other fancy phrase means multiply ...
... can add or multiply congruences. But how about adding an equation to a congruency or multiplying a congruency by an equation? Note that "adding an equation to a congruency" is a fancy way of saying adding the same integer to both sides of a congruency. Similarly the other fancy phrase means multiply ...
Lectures on Number Theory
... latter case is excluded, we conclude that r − r0 = 0, that is r = r0 . Therefore a(q − q 0 ) = 0, which implies q − q 0 = 0, i.e. q = q 0 . More generally, we say that r0 is a remainder when b is divided by a whenever there is an integer q 0 such that b = aq 0 + r0 without any further restriction on ...
... latter case is excluded, we conclude that r − r0 = 0, that is r = r0 . Therefore a(q − q 0 ) = 0, which implies q − q 0 = 0, i.e. q = q 0 . More generally, we say that r0 is a remainder when b is divided by a whenever there is an integer q 0 such that b = aq 0 + r0 without any further restriction on ...
Color - Alex Kocurek
... Very often, these inexpressibility claims are justified in the literature only by example: all of the most straightforward attempts at formalizing these English sentences into first-order modal logic fail. While this style of argument may be convincing, it does not constitute a proof. One can someti ...
... Very often, these inexpressibility claims are justified in the literature only by example: all of the most straightforward attempts at formalizing these English sentences into first-order modal logic fail. While this style of argument may be convincing, it does not constitute a proof. One can someti ...
On Weak Ground
... characterization is intended to be a definition of weak ground in terms of strict ground. Nevertheless, the question of whether any of these characterizations can be used to specify the notion is worth exploring. The arguments that follow, if sound, provide reasons not only for rejecting the full ex ...
... characterization is intended to be a definition of weak ground in terms of strict ground. Nevertheless, the question of whether any of these characterizations can be used to specify the notion is worth exploring. The arguments that follow, if sound, provide reasons not only for rejecting the full ex ...
Number Theory for Mathematical Contests
... This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under the conditions s ...
... This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under the conditions s ...
Number Theory for Mathematical Contests
... Thus in the Principle of Mathematical Induction, we try to verify that some assertion P(n) concerning natural numbers is true for some base case k0 (usually k0 = 1, but one of the examples below shows that we may take, say k0 = 33.) Then we try to settle whether information on P(n − 1) leads to favo ...
... Thus in the Principle of Mathematical Induction, we try to verify that some assertion P(n) concerning natural numbers is true for some base case k0 (usually k0 = 1, but one of the examples below shows that we may take, say k0 = 33.) Then we try to settle whether information on P(n − 1) leads to favo ...
![arXiv:math/0604314v2 [math.NT] 7 Sep 2006 On](http://s1.studyres.com/store/data/016163041_1-f0b243339013698f62e5d1397917ed3b-300x300.png)
Distribution of Prime Numbers
... The result below is a simple deduction from Theorem 1B. THEOREM 1C. The arithmetic functions d : N → C and σ : N → C are both multiplicative. Natural numbers n ∈ N where σ(n) = 2n are of particular interest, and are known as perfect numbers. A perfect number is therefore a natural number which is eq ...
... The result below is a simple deduction from Theorem 1B. THEOREM 1C. The arithmetic functions d : N → C and σ : N → C are both multiplicative. Natural numbers n ∈ N where σ(n) = 2n are of particular interest, and are known as perfect numbers. A perfect number is therefore a natural number which is eq ...
QUASI-MV ALGEBRAS. PART III
... qMV algebras, in particular about the structure of clouds. For each cloud C we have a twin cloud C ′ = {c′ : c ∈ C} of cardinality equal to that of C, and ′ is a bijection between C and C ′ . Moreover, C and C ′ are disjoint, except possibly for a single cloud, for which C ′ = C. This unique cloud, ...
... qMV algebras, in particular about the structure of clouds. For each cloud C we have a twin cloud C ′ = {c′ : c ∈ C} of cardinality equal to that of C, and ′ is a bijection between C and C ′ . Moreover, C and C ′ are disjoint, except possibly for a single cloud, for which C ′ = C. This unique cloud, ...
Lectures on Sieve Methods - School of Mathematics, TIFR
... THESE LECTURES were given during a seven-week course at the Tata Institute of Fundamental Research. The aim was to provide an introduction to modern sieve methods, i.e. to various forms of both the large sieve (part I) and the small sieve (part II), as well as their interconnections and applications ...
... THESE LECTURES were given during a seven-week course at the Tata Institute of Fundamental Research. The aim was to provide an introduction to modern sieve methods, i.e. to various forms of both the large sieve (part I) and the small sieve (part II), as well as their interconnections and applications ...
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.