ON A VARIATION OF PERFECT NUMBERS Douglas E. Iannucci
... Φ97 (3) has any prime divisors q > 103 . By Lemma 1, q ≡ 1 (mod 194), so it suffices merely to test all such primes q < 103 to see if 397 ≡ 1 (mod q). Thus we obtain the product, say R, of all primes less than 103 which divide Φ97 (3). If ln R < 96 ln 3, then Φ97 (3) must contain a prime divisor gre ...
... Φ97 (3) has any prime divisors q > 103 . By Lemma 1, q ≡ 1 (mod 194), so it suffices merely to test all such primes q < 103 to see if 397 ≡ 1 (mod q). Thus we obtain the product, say R, of all primes less than 103 which divide Φ97 (3). If ln R < 96 ln 3, then Φ97 (3) must contain a prime divisor gre ...
![The full Müntz Theorem in C[0,1]](http://s1.studyres.com/store/data/019844035_1-f7b6943c075c22a54f0d5d67f46bc0e0-300x300.png)
The full Müntz Theorem in C[0,1]
... The original Müntz Theorem proved by Müntz [18] in 1914, by Szász [25] in 1916, and anticipated by Bernstein [3] was only for sequences of exponents tending to infinity. Later works, see, for example, [22] and [19], include the above result, as well as a treatment of the case when {λi }∞ i=1 is a ...
... The original Müntz Theorem proved by Müntz [18] in 1914, by Szász [25] in 1916, and anticipated by Bernstein [3] was only for sequences of exponents tending to infinity. Later works, see, for example, [22] and [19], include the above result, as well as a treatment of the case when {λi }∞ i=1 is a ...
Sets, Logic, Relations, and Functions
... Because sets are only defined by their elements, {x , y} = {y, x }. Definition 1.5.1. 1. (x , y) is an ordered pair, and (x , y) = (y, x ) iff x = y.11 2. (x1 , . . . xk ) is a k-tuple, and (x1 , . . . xk ) = (y1 , . . . yk ) iff every xi = yi . Definition 1.5.2. 1. If S , T are sets, S × T = { s : ...
... Because sets are only defined by their elements, {x , y} = {y, x }. Definition 1.5.1. 1. (x , y) is an ordered pair, and (x , y) = (y, x ) iff x = y.11 2. (x1 , . . . xk ) is a k-tuple, and (x1 , . . . xk ) = (y1 , . . . yk ) iff every xi = yi . Definition 1.5.2. 1. If S , T are sets, S × T = { s : ...
Chapter 5 of my book
... put our arguments on solid mathematical ground. Russell’s Paradox: Assume for any property P the collection of all elements having property P is a set. Consider R = {x : x 6∈ x}; thus x ∈ R if and only if x 6∈ x. Most objects are not elements of themselves; for example, N 6∈ N because the set of nat ...
... put our arguments on solid mathematical ground. Russell’s Paradox: Assume for any property P the collection of all elements having property P is a set. Consider R = {x : x 6∈ x}; thus x ∈ R if and only if x 6∈ x. Most objects are not elements of themselves; for example, N 6∈ N because the set of nat ...
Did I get it right? COS 326 Andrew W. Appel Princeton University
... However, that is not how O’Caml evaluation works. O’Caml evaluates it’s arguments to a value first, and then calls the function. Don’t worry: if you know that the expression will evaluate to a value (and will not infinite loop or raise an exception) then you can substitute the symbolic expression fo ...
... However, that is not how O’Caml evaluation works. O’Caml evaluates it’s arguments to a value first, and then calls the function. Don’t worry: if you know that the expression will evaluate to a value (and will not infinite loop or raise an exception) then you can substitute the symbolic expression fo ...
Additive properties of even perfect numbers
... 1) is an even perfect number. Two millennia later, Euler proved the converse. Theorem 1. An even positive integer n is perfect if and only if it is of the form n = 2p−1 (2p − 1), where 2p − 1 is prime. For a proof, see [1, Theorem 1.3.3, p. 22]. So we have a characterization of even perfect numbers. ...
... 1) is an even perfect number. Two millennia later, Euler proved the converse. Theorem 1. An even positive integer n is perfect if and only if it is of the form n = 2p−1 (2p − 1), where 2p − 1 is prime. For a proof, see [1, Theorem 1.3.3, p. 22]. So we have a characterization of even perfect numbers. ...
Stable Kneser hypergraphs and ideals in N with the Nikodym
... if for any sequence P (xn ) of positive reals such that n=1 xn diverges, there exists I ∈ I for which n∈I xn diverges as well. It is not hard to show that the ideal Z of sets of density zero has (PSP). This ‘folklore’ fact can be traced back (at least) to a short note of Auerbach [2] from 1930, and ...
... if for any sequence P (xn ) of positive reals such that n=1 xn diverges, there exists I ∈ I for which n∈I xn diverges as well. It is not hard to show that the ideal Z of sets of density zero has (PSP). This ‘folklore’ fact can be traced back (at least) to a short note of Auerbach [2] from 1930, and ...
PLATONISM IN MODERN MATHEMATICS A University Thesis
... with physics. The formalist school holds the view that every mathematical problem is solvable by finite proof methods, thus a new field of studying the structure, or syntax and mathematical arguments and proof, was developed in this school known as mathematics. Thus, the meaning of mathematical sym ...
... with physics. The formalist school holds the view that every mathematical problem is solvable by finite proof methods, thus a new field of studying the structure, or syntax and mathematical arguments and proof, was developed in this school known as mathematics. Thus, the meaning of mathematical sym ...
Squarefree smooth numbers and Euclidean prime generators
... of a, and by multiplicativity it follows that d + n/d has a prime factor q satisfying pq = −1. ...
... of a, and by multiplicativity it follows that d + n/d has a prime factor q satisfying pq = −1. ...
Number Theory Learning Module 3 — The Greatest Common
... Classical Greek mathematics concerned itself mostly with geometry. The notion of measurement is fundamental to geometry, and the Greeks were the first to provide a formal foundation for this concept. Surprisingly, however, they never used fractions to express measurements (and never developed an ari ...
... Classical Greek mathematics concerned itself mostly with geometry. The notion of measurement is fundamental to geometry, and the Greeks were the first to provide a formal foundation for this concept. Surprisingly, however, they never used fractions to express measurements (and never developed an ari ...
Chapter 1: The Foundations: Logic and Proofs
... 1.3 Predicates and Quantifiers 1.4 Nested Quantifiers 1.5 Rules of Inference 1.6 Introduction to Proofs 1.7 Proof Methods and Strategy ...
... 1.3 Predicates and Quantifiers 1.4 Nested Quantifiers 1.5 Rules of Inference 1.6 Introduction to Proofs 1.7 Proof Methods and Strategy ...
![arXiv:1510.00735v3 [math.NT] 14 Oct 2015](http://s1.studyres.com/store/data/015647175_1-32c28f0e3a09596218fdb4a3c649b24f-300x300.png)
Beautifying Gödel - Department of Computer Science
... what an interpreter does: it turns passive data into active program. It is a familiar fact to programmers that we can write an interpreter for a language in that same language, and that is just what we are doing here. To finish defining we must decide the details of an entire theory. We shall not do ...
... what an interpreter does: it turns passive data into active program. It is a familiar fact to programmers that we can write an interpreter for a language in that same language, and that is just what we are doing here. To finish defining we must decide the details of an entire theory. We shall not do ...
Section 9.3: Mathematical Induction
... is true. Thus we need to show that a2 = a + (2 − 1)d. Since P (1) is true, we have a1 = a, and by the definition of an arithmetic sequence, a2 = a1 +d = a+d = a+(2−1)d. So P (2) is true. We now use the fact that P (2) is true to show that P (3) is true. Using the fact that a2 = a + (2 − 1)d, we show ...
... is true. Thus we need to show that a2 = a + (2 − 1)d. Since P (1) is true, we have a1 = a, and by the definition of an arithmetic sequence, a2 = a1 +d = a+d = a+(2−1)d. So P (2) is true. We now use the fact that P (2) is true to show that P (3) is true. Using the fact that a2 = a + (2 − 1)d, we show ...
Incompleteness Result
... The positive story of the development of classical logic has culminated in Godel’s completeness proof: “Whatever is true is provable” and this would encourage many working mathematicians to continue their pursuit to prove or disprove those historically well-known mathematical conjectures. However, t ...
... The positive story of the development of classical logic has culminated in Godel’s completeness proof: “Whatever is true is provable” and this would encourage many working mathematicians to continue their pursuit to prove or disprove those historically well-known mathematical conjectures. However, t ...
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.