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... this follows a = r by reason of definition of a = a(p). Thus/? is divisible by a = a(p). Should it happen that/? is divisible by a = a(p), then, due to the Vorobev's previous theorem, Fn is divisible by Fa(pj and so Fw is "divisible by/7, too. With this we proved the Lemma 1 and from this follows th ...
... this follows a = r by reason of definition of a = a(p). Thus/? is divisible by a = a(p). Should it happen that/? is divisible by a = a(p), then, due to the Vorobev's previous theorem, Fn is divisible by Fa(pj and so Fw is "divisible by/7, too. With this we proved the Lemma 1 and from this follows th ...
DUCCI SEQUENCES IN HIGHER DIMENSIONS Florian Breuer
... 5. If 2r ≡ −1 mod m for some r ∈ Z, then P (n) divides n(2r − 1). The above result is far from exhaustive, in particular, much more is known about the function P (n), see for example [13] and [14]. Some of these results have been rediscovered many times. As far as the author can tell, (1) is first p ...
... 5. If 2r ≡ −1 mod m for some r ∈ Z, then P (n) divides n(2r − 1). The above result is far from exhaustive, in particular, much more is known about the function P (n), see for example [13] and [14]. Some of these results have been rediscovered many times. As far as the author can tell, (1) is first p ...
Foundations of Mathematics I Set Theory (only a draft)
... which will be forbidden by an axiom that will be introduced in the next part, in other words such an object will not be a set (it may be something else!) The sets {x, x, y}, {x, y} and {y, x} are equal because they contain the same elements, namely x and y. This set has either one or two elements: I ...
... which will be forbidden by an axiom that will be introduced in the next part, in other words such an object will not be a set (it may be something else!) The sets {x, x, y}, {x, y} and {y, x} are equal because they contain the same elements, namely x and y. This set has either one or two elements: I ...
A Few New Facts about the EKG Sequence
... From Lemma 4.5, there are at most π( (a + ε)n) indices i smaller than zan , for which ai < (a + ε)n, and ai+1 ≥ (a + ε)n and ai is composite. In the other case, if ai < (a + ε)n and ai+1 ≥ (a + ε)n and ai is prime, the next term ai+2 will be smaller than (a + ε)n. Suppose there is an index i < zan f ...
... From Lemma 4.5, there are at most π( (a + ε)n) indices i smaller than zan , for which ai < (a + ε)n, and ai+1 ≥ (a + ε)n and ai is composite. In the other case, if ai < (a + ε)n and ai+1 ≥ (a + ε)n and ai is prime, the next term ai+2 will be smaller than (a + ε)n. Suppose there is an index i < zan f ...
20(3)
... smallest Fibonacci number used in the Zeckendorf representation occurred respectively with an even or with an odd subscript. Since the Zeckendorf representation is unique, sets A and B cover the set of positive integers and are disjoint. Notice thats if the smallest subscript for a Fibonacci number ...
... smallest Fibonacci number used in the Zeckendorf representation occurred respectively with an even or with an odd subscript. Since the Zeckendorf representation is unique, sets A and B cover the set of positive integers and are disjoint. Notice thats if the smallest subscript for a Fibonacci number ...
Some Aspects and Examples of In nity Notions T ZF
... The investigation of di erent de nitions of in nity constitutes a signi cant part of the development of axiomatic set theory. Tarski [15], Mostowski [11], Levy [9], and many other authors have devoted research papers to the theme of niteness de nitions (which is the most often used term for this s ...
... The investigation of di erent de nitions of in nity constitutes a signi cant part of the development of axiomatic set theory. Tarski [15], Mostowski [11], Levy [9], and many other authors have devoted research papers to the theme of niteness de nitions (which is the most often used term for this s ...
DISTRIBUTION OF RESIDUES MODULO p - Harish
... interest to Number Theorists for many decades. The set of all non-zero residues modulo p can be divided into two classes, namely, the set of all quadratic residues (or squares) and quadratic non-residues (or non-squares) modulo p. In natural numbers, there are no consecutive squares as the differenc ...
... interest to Number Theorists for many decades. The set of all non-zero residues modulo p can be divided into two classes, namely, the set of all quadratic residues (or squares) and quadratic non-residues (or non-squares) modulo p. In natural numbers, there are no consecutive squares as the differenc ...
Functional Dependencies in a Relational Database and
... slightly different from this set, but the two sets are equivalent, since it is easy to check that this set implies each axiom in the original set and that Armstrong’s original axioms imply each of these. It turns out that axioms ( A 1) - (A3) are more convenient for our purposes than Armstrong’s ori ...
... slightly different from this set, but the two sets are equivalent, since it is easy to check that this set implies each axiom in the original set and that Armstrong’s original axioms imply each of these. It turns out that axioms ( A 1) - (A3) are more convenient for our purposes than Armstrong’s ori ...
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.