The Fibonacci zeta function - Department of Mathematics and Statistics
... 1998, [6]) F (2k) is transcendental for all k ≥ 1. As will be explained below, the essential ingredient in the proof of Theorem 1.2 is a deep theorem of Nesterenko regarding the transcendence of special values of Eisenstein series, proved in 1996. We will attempt to explain why the case of even expo ...
... 1998, [6]) F (2k) is transcendental for all k ≥ 1. As will be explained below, the essential ingredient in the proof of Theorem 1.2 is a deep theorem of Nesterenko regarding the transcendence of special values of Eisenstein series, proved in 1996. We will attempt to explain why the case of even expo ...
Polarizing Double-Negation Translations
... – In the absence of the cut rule, a formula is never active in different sides (both as an hypothesis and as a conclusion) of the turnstyle, having therefore a welldefined polarity. This last fact holds for all the rules except the axiom rule, which is easily dealt with, by an η-expansion-like argum ...
... – In the absence of the cut rule, a formula is never active in different sides (both as an hypothesis and as a conclusion) of the turnstyle, having therefore a welldefined polarity. This last fact holds for all the rules except the axiom rule, which is easily dealt with, by an η-expansion-like argum ...
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... The second half of the proof above, showing that C(n) = F2n+1 , can also be accomplished using a tiling argument of Anderson and Lewis [1] which allows tiles of any length. Think of a convex subset of k as a white tile on a 1 × k strip. Then a collection of mutually disjoint convex subsets of k may ...
... The second half of the proof above, showing that C(n) = F2n+1 , can also be accomplished using a tiling argument of Anderson and Lewis [1] which allows tiles of any length. Think of a convex subset of k as a white tile on a 1 × k strip. Then a collection of mutually disjoint convex subsets of k may ...
Full text
... Suppose first that 012K £ Q. By a theorem of Weyl (see [2], Theorem 445), the sequence of fractional parts of ndlln is uniformly distributed in the interval [0,1]. In particular, there is a sequence of integers nf such that nfi I 2K + {0 + 2y) I 4K -kt is very small for some integers kf, and the lim ...
... Suppose first that 012K £ Q. By a theorem of Weyl (see [2], Theorem 445), the sequence of fractional parts of ndlln is uniformly distributed in the interval [0,1]. In particular, there is a sequence of integers nf such that nfi I 2K + {0 + 2y) I 4K -kt is very small for some integers kf, and the lim ...
Propositional Logic
... To show that the query Q follows from the knowledge base KB, we can also add the negated query ¬Q to the knowledge base and derive a contradiction. Because of the equivalence A ∧ ¬A ⇔ f from Theorem 2.1 on page 18 we know that a contradiction is unsatisfiable. Therefore, Q has been proved. This proc ...
... To show that the query Q follows from the knowledge base KB, we can also add the negated query ¬Q to the knowledge base and derive a contradiction. Because of the equivalence A ∧ ¬A ⇔ f from Theorem 2.1 on page 18 we know that a contradiction is unsatisfiable. Therefore, Q has been proved. This proc ...
Tiling Proofs of Recent Sum Identities Involving Pell Numbers
... length and must therefore contain at least one square. Also note that k cannot equal 2n or 2n + 1 since X and Y both end in dominoes.) As before, to the right of the fault line at cell k, we have dominoes in staggered formation, and just a single square s on cell k + 1 in A or B (but not both). Here ...
... length and must therefore contain at least one square. Also note that k cannot equal 2n or 2n + 1 since X and Y both end in dominoes.) As before, to the right of the fault line at cell k, we have dominoes in staggered formation, and just a single square s on cell k + 1 in A or B (but not both). Here ...
Group action
... So, for example (a + bi)(a – bi) = 1234321 = 11112 = (101·11)2 = (10 + i)2(10 – i)2112 Factorization of a + bi is a unit times some factors from the right. Norms of a + bi and a – bi are the same, so they should get the same number of factors. Each of them should get 11 (since they are conjugate) an ...
... So, for example (a + bi)(a – bi) = 1234321 = 11112 = (101·11)2 = (10 + i)2(10 – i)2112 Factorization of a + bi is a unit times some factors from the right. Norms of a + bi and a – bi are the same, so they should get the same number of factors. Each of them should get 11 (since they are conjugate) an ...
Fibonacci numbers at most one away from a perfect power
... form G(u, v) = 1 where G is a binary form of degree p. We do not solve these Thue equations completely, but we compute explicit upper bounds for their solutions using classical methods (see for example [6]). This provides us with upper bounds for n in terms of p. To be precise, we prove that n < 109 ...
... form G(u, v) = 1 where G is a binary form of degree p. We do not solve these Thue equations completely, but we compute explicit upper bounds for their solutions using classical methods (see for example [6]). This provides us with upper bounds for n in terms of p. To be precise, we prove that n < 109 ...
Powerpoint
... • Induction proof Basis k 2; straightforward Induction Assume claim holds for k 1 states Suppose ...
... • Induction proof Basis k 2; straightforward Induction Assume claim holds for k 1 states Suppose ...
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.