On the Number of False Witnesses for a Composite Number
... Lagrange's theorem gives F(«)|<|>(«), whereis Euler's function.
There are composite numbers « for which F(n) = («),such as « = 561. Such
numbers are called Carmichael numbers and probably there are infinitely many of
them, but this has never been proved. It is known that Carmichael numbers ar ...
... Lagrange's theorem gives F(«)|<|>(«), where
A RIGOROUS TIME BOUND FOR FACTORING INTEGERS For real
... One such algorithm, the class group relations method, is due to Seysen [32]. Under the assumption of the generalized Riemann hypothesis (GRH) for Lfunctions of abelian characters of imaginary quadratic fields, Seysen showed that his method runs in expected time at most LnH, ..)5/4 + 0(1)]. A. K. Len ...
... One such algorithm, the class group relations method, is due to Seysen [32]. Under the assumption of the generalized Riemann hypothesis (GRH) for Lfunctions of abelian characters of imaginary quadratic fields, Seysen showed that his method runs in expected time at most LnH, ..)5/4 + 0(1)]. A. K. Len ...
Some properties of the space of fuzzy
... E-mail address: [email protected] (J.-X. Fang). 0165-0114/$ - see front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.fss.2008.07.014 ...
... E-mail address: [email protected] (J.-X. Fang). 0165-0114/$ - see front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.fss.2008.07.014 ...
What is "formal logic"?
... relativity, the theory of models, arithmetic, not in its restricted logical sense meaning a set of sentences, or a deductively closed set of sentences. Formalization was applied mainly to logic and mathematics one hundred years ago. The formalist program, especially promoted by Hilbert, was the atte ...
... relativity, the theory of models, arithmetic, not in its restricted logical sense meaning a set of sentences, or a deductively closed set of sentences. Formalization was applied mainly to logic and mathematics one hundred years ago. The formalist program, especially promoted by Hilbert, was the atte ...
was the congruence
... congruence classes mod m. (Thus, every integer belongs to exactly one congruence class mod m and no two congruence classes have any numbers in common.) There are exactly m congruence classes mod m and they are determined by the m possible remainders (or in Gauss’ terminology, residues) r = 0, 1, … , ...
... congruence classes mod m. (Thus, every integer belongs to exactly one congruence class mod m and no two congruence classes have any numbers in common.) There are exactly m congruence classes mod m and they are determined by the m possible remainders (or in Gauss’ terminology, residues) r = 0, 1, … , ...
Slide 1
... contradictions, equivalence, and logical proofs is no different for fuzzy sets; the results, however, can differ considerably from those in classical logic. If the truth values for the simple propositions of a fuzzy logic compound proposition are strictly true (1) or false (0), the results follow id ...
... contradictions, equivalence, and logical proofs is no different for fuzzy sets; the results, however, can differ considerably from those in classical logic. If the truth values for the simple propositions of a fuzzy logic compound proposition are strictly true (1) or false (0), the results follow id ...
40(4)
... Articles should be submitted using the format of articles in any current issues of THE FIBONACCI QUARTERLY. They should be typewritten or reproduced typewritten copies, that are clearly readable, double spaced with wide margins and on only one side of the paper. The full name and address of the auth ...
... Articles should be submitted using the format of articles in any current issues of THE FIBONACCI QUARTERLY. They should be typewritten or reproduced typewritten copies, that are clearly readable, double spaced with wide margins and on only one side of the paper. The full name and address of the auth ...
First-order possibility models and finitary
... Humberstone’s original motivations for considering possibility models. For many normal modal logics extending K with standard axioms (such as D, T, 4, B, and 5), Holliday [Hol14] is able to give such a completeness proof. On the other hand, there are normal modal logics which do not admit such a fin ...
... Humberstone’s original motivations for considering possibility models. For many normal modal logics extending K with standard axioms (such as D, T, 4, B, and 5), Holliday [Hol14] is able to give such a completeness proof. On the other hand, there are normal modal logics which do not admit such a fin ...
THE INSOLUBILITY OF CLASSES OF DIOPHANTINE EQUATIONS
... Proof, Using the proof of Theorem III and Lemma 2, we immediately infer that there exists no solution of X,m + X,m + X,m e 0 (mod p) and X,X .X, i~6 0 (mod p) . Hence, if there exists a rational solution X,11d + X2+ X,m - 0, then p I X,X.X, . If q denotes any prime factor of 9n, and (X,X2X,, in) =1 ...
... Proof, Using the proof of Theorem III and Lemma 2, we immediately infer that there exists no solution of X,m + X,m + X,m e 0 (mod p) and X,X .X, i~6 0 (mod p) . Hence, if there exists a rational solution X,11d + X2+ X,m - 0, then p I X,X.X, . If q denotes any prime factor of 9n, and (X,X2X,, in) =1 ...
A Few Basics of Probability
... with certainty. Probability 1 is the highest probability that any statements are supposed to get (see below) so it makes sense to give logical truths probability 1. The third axiom is a bit more confusing than the others but still makes intuitive sense. If two sentences cannot be true together, then ...
... with certainty. Probability 1 is the highest probability that any statements are supposed to get (see below) so it makes sense to give logical truths probability 1. The third axiom is a bit more confusing than the others but still makes intuitive sense. If two sentences cannot be true together, then ...
The binomial theorem
... So the answer is 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40 320 ways. The notation for this is 8!. This is read as eight factorial. Most calculators have a factorial key. The number of ways to arrange n different objects in a row is n!. It is important to note that n! = n(n − 1)!. For this statement to be t ...
... So the answer is 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40 320 ways. The notation for this is 8!. This is read as eight factorial. Most calculators have a factorial key. The number of ways to arrange n different objects in a row is n!. It is important to note that n! = n(n − 1)!. For this statement to be 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.