Tau Numbers: A Partial Proof of a Conjecture and Other Results
... Corollary 16. There are at most a finite number of integers n such that T (n) < .5π(n). Proof. Let b = .5 in the above corollary. Theorem 14 also implies that T (n) > π(n) for all sufficiently large n. Colton gave a table of T (n) showing that T (107 ) is about .59π(n). So T (n) must not drastically ...
... Corollary 16. There are at most a finite number of integers n such that T (n) < .5π(n). Proof. Let b = .5 in the above corollary. Theorem 14 also implies that T (n) > π(n) for all sufficiently large n. Colton gave a table of T (n) showing that T (107 ) is about .59π(n). So T (n) must not drastically ...
Math 319 Problem Set #6 – Solution 5 April 2002
... Next we consider all possible factorizations of 48 that might occur as products of the form ϕ(pα1 1 )ϕ(pα2 1 ) · · · ϕ(pαk k ). Write the factors in non-decreasing order. Factorizations with smallest factor 2: Factorizations starting 2 × 2: Since we can’t have more than two 2’s, any such factorizati ...
... Next we consider all possible factorizations of 48 that might occur as products of the form ϕ(pα1 1 )ϕ(pα2 1 ) · · · ϕ(pαk k ). Write the factors in non-decreasing order. Factorizations with smallest factor 2: Factorizations starting 2 × 2: Since we can’t have more than two 2’s, any such factorizati ...
A conjecture of Erdos on graph Ramsey numbers
... pairs of distinct vertices of G that are edges. Our next lemma extends the result of Erdős and Szemerédi to monochromatic pairs. Lemma 2.3. Let 0 < 1/7 and let t and N be positive integers satisfying t −1 and N t −14t . Then any red–blue edge-coloring of KN in which red has edge density ...
... pairs of distinct vertices of G that are edges. Our next lemma extends the result of Erdős and Szemerédi to monochromatic pairs. Lemma 2.3. Let 0 < 1/7 and let t and N be positive integers satisfying t −1 and N t −14t . Then any red–blue edge-coloring of KN in which red has edge density ...
Mathematical Logic
... To save parentheses in quantified formulas, we use a mild form of the dot notation: a dot immediately after ∀x or ∃x makes the scope of that quantifier as large as possible, given the parentheses around. So ∀x.A → B means ∀x(A → B), not (∀xA) → B. We also save on parentheses by writing e.g. Rxyz, Rt ...
... To save parentheses in quantified formulas, we use a mild form of the dot notation: a dot immediately after ∀x or ∃x makes the scope of that quantifier as large as possible, given the parentheses around. So ∀x.A → B means ∀x(A → B), not (∀xA) → B. We also save on parentheses by writing e.g. Rxyz, Rt ...
Normal numbers and the Borel hierarchy
... Now suppose ϕ is false. Let x be such that there are infinitely many y such that C(x, y). Let z be any positive integer. Each time an appending tuple of the form hx, yi with z < y is processed, x + z is appended to the output. Since we assumed there are infinitely many such tuples, x + z is appended ...
... Now suppose ϕ is false. Let x be such that there are infinitely many y such that C(x, y). Let z be any positive integer. Each time an appending tuple of the form hx, yi with z < y is processed, x + z is appended to the output. Since we assumed there are infinitely many such tuples, x + z is appended ...
CS1231 - Lecture 09
... – DO NOT BRING this restricted idea of ‘number’ in here. – A Cardinal Number is a descriptive numerical object, generalized to include ALL SORTS OF SETS – FINITE OR INFINITE. In particular, cardinals describe ‘how many elements’ in INFINITE sets as well. As such it describes how infinite the set is. ...
... – DO NOT BRING this restricted idea of ‘number’ in here. – A Cardinal Number is a descriptive numerical object, generalized to include ALL SORTS OF SETS – FINITE OR INFINITE. In particular, cardinals describe ‘how many elements’ in INFINITE sets as well. As such it describes how infinite the set is. ...
Variant of a theorem of Erdős on the sum-of-proper
... algorithm that achieves the more modest goal of enumerating s(N) (or equivalently, U) to x. Our algorithm has running time of the shape x1+o(1) . The algorithm of te Riele is based on an earlier one of Alanen [Ala72]. Alanen was able to count U to 5,000, while with te Riele’s improvements, he got th ...
... algorithm that achieves the more modest goal of enumerating s(N) (or equivalently, U) to x. Our algorithm has running time of the shape x1+o(1) . The algorithm of te Riele is based on an earlier one of Alanen [Ala72]. Alanen was able to count U to 5,000, while with te Riele’s improvements, he got th ...
Document
... computers do computation on binary numbers. Question: How do we know that every natural number can be written in binary? ...
... computers do computation on binary numbers. Question: How do we know that every natural number can be written in binary? ...
Curious and Exotic Identities for Bernoulli Numbers
... look at products of Bernoulli numbers and Bernoulli polynomials in more detail. In particular, we prove the result (discovered by Nielsen) that when a product of two Bernoulli polynomials is expressed as a linear combination of Bernoulli polynomials, then the coefficients are themselves multiples of ...
... look at products of Bernoulli numbers and Bernoulli polynomials in more detail. In particular, we prove the result (discovered by Nielsen) that when a product of two Bernoulli polynomials is expressed as a linear combination of Bernoulli polynomials, then the coefficients are themselves multiples of ...
A Computationally-Discovered Simplification of the Ontological
... Of course, this might be simplified so that the consequent of the main conditional reads Is the(x,F) -> Ex1(F,x), but we thought that this alternative wouldn’t capture the fact that the antecedent of Lemma 1 is an identity claim. We should note here that the representation of Lemma 1 must be derive ...
... Of course, this might be simplified so that the consequent of the main conditional reads Is the(x,F) -> Ex1(F,x), but we thought that this alternative wouldn’t capture the fact that the antecedent of Lemma 1 is an identity claim. We should note here that the representation of Lemma 1 must be derive ...
Theorem
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In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.Although they can be written in a completely symbolic form, for example, within the propositional calculus, theorems are often expressed in a natural language such as English. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed. Such arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being ""trivial"", or ""difficult"", or ""deep"", or even ""beautiful"". These subjective judgments vary not only from person to person, but also with time: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.