PERFECT NUMBERS WITH IDENTICAL DIGITS Paul Pollack1
... The next two results depend on Baker’s theory of linear forms in logarithms. We begin with a special case of [12, Theorem 9.6]. If S is a finite (possibly empty) set of rational primes, we call a natural number A an S-number if A is supported on the primes in S. Lemma 5. Let g ≥ 2. Let S be a finite ...
... The next two results depend on Baker’s theory of linear forms in logarithms. We begin with a special case of [12, Theorem 9.6]. If S is a finite (possibly empty) set of rational primes, we call a natural number A an S-number if A is supported on the primes in S. Lemma 5. Let g ≥ 2. Let S be a finite ...
Some sufficient conditions of a given series with rational terms
... on. In Diophantine approximation theory, we are in totally different situation that we already have necessary and sufficient condition to determine if a given real number is an irrational number or a transcendental number such as well known Roth theorem but seems to be lack of practical test just as ...
... on. In Diophantine approximation theory, we are in totally different situation that we already have necessary and sufficient condition to determine if a given real number is an irrational number or a transcendental number such as well known Roth theorem but seems to be lack of practical test just as ...
God, the Devil, and Gödel
... a mind: that the mind can outstrip any machine in deductive process. We must take care here, for there are some trivial ways in which the thesis could be false, and Lucas is well aware of many of these and does not intend it in these ways. For example, it is clear that present digital computers can ...
... a mind: that the mind can outstrip any machine in deductive process. We must take care here, for there are some trivial ways in which the thesis could be false, and Lucas is well aware of many of these and does not intend it in these ways. For example, it is clear that present digital computers can ...
Chowla`s conjecture
... paper [C-F]. Just as in the case of Yokoi‘s conjecture, Siegel‘s theorem implies ineffectively that for large p the class number is greater than 1, hence the problem is in fact to find an effective upper bound for p in the class number 1 case.We achieve this goal by proving the following theorem. TH ...
... paper [C-F]. Just as in the case of Yokoi‘s conjecture, Siegel‘s theorem implies ineffectively that for large p the class number is greater than 1, hence the problem is in fact to find an effective upper bound for p in the class number 1 case.We achieve this goal by proving the following theorem. TH ...
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... Then I maps all the predicate symbols in S to relations over U . What remains to be shown is ∀Y ∈ S. U,I|=Y. We prove this by structural induction on formulas, keeping in mind that the cases for γ and δ are straightforward generalizations of those for α and β. base case: If Y is an atomic formula th ...
... Then I maps all the predicate symbols in S to relations over U . What remains to be shown is ∀Y ∈ S. U,I|=Y. We prove this by structural induction on formulas, keeping in mind that the cases for γ and δ are straightforward generalizations of those for α and β. base case: If Y is an atomic formula th ...
Introduction to Logic
... only lies at its origin, ca. 500 BC, but has been the main force motivating its development since that time until the last century. There was a medieval tradition according to which the Greek philosopher Parmenides (5th century BC) invented logic while living on a rock in Egypt. The story is pure le ...
... only lies at its origin, ca. 500 BC, but has been the main force motivating its development since that time until the last century. There was a medieval tradition according to which the Greek philosopher Parmenides (5th century BC) invented logic while living on a rock in Egypt. The story is pure le ...
Pell`s equation and units in real quadratic fields
... This is not hard to see: Proposition 2.2 says that an integral ideal I is reduced if N (I ) < 2D , and it is easy to see that (a) follows this exactly. Although slightly less obvious, (b) follows from part (ii) of Prop. 2.2. Condition (b) makes it necessary that either ei or fi is equal to 0, for an ...
... This is not hard to see: Proposition 2.2 says that an integral ideal I is reduced if N (I ) < 2D , and it is easy to see that (a) follows this exactly. Although slightly less obvious, (b) follows from part (ii) of Prop. 2.2. Condition (b) makes it necessary that either ei or fi is equal to 0, for an ...
(pdf)
... which contradicts positive-definiteness (as we said earlier, j=1 ξj∗ aj +an 6= 0). 8. Extensions of (Normalized) Valuations The theorems in this section will not be proven. “Algebraic Number Theory” by Cassels and Fröhlich [1] (Chapter 2) is recommended, but they cite “Theory of Algebraic Numbers ...
... which contradicts positive-definiteness (as we said earlier, j=1 ξj∗ aj +an 6= 0). 8. Extensions of (Normalized) Valuations The theorems in this section will not be proven. “Algebraic Number Theory” by Cassels and Fröhlich [1] (Chapter 2) is recommended, but they cite “Theory of Algebraic Numbers ...
![[Michel Waldschmidt] Continued fractions](http://s1.studyres.com/store/data/018214733_1-ffc3ca0fc19190c3043e567af0ae13ba-300x300.png)
[Michel Waldschmidt] Continued fractions
... Example 5. Integer rectangle triangles having sides of the right angle as consecutive integers a and a + 1 have an hypothenuse c which satisfies a2 + (a + 1)2 = c2 . The admissible values for the hypothenuse is the set of positive integer solutions y to Fermat–Pell’s equation x2 −2y 2 = −1. The list ...
... Example 5. Integer rectangle triangles having sides of the right angle as consecutive integers a and a + 1 have an hypothenuse c which satisfies a2 + (a + 1)2 = c2 . The admissible values for the hypothenuse is the set of positive integer solutions y to Fermat–Pell’s equation x2 −2y 2 = −1. The list ...
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.