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Pythagorean Triples and Fermat`s Last Theorem
... The Pythagorean theorem says that the sum of the squares of the sides of a right triangle equals the square of the hypotenuse. In symbols, a2 + b2 = c2 . Since we are interested in Number Theory, that is, the theory of whole numbers, we ask if there are any Pythagorean triangles whose sides are whol ...
... The Pythagorean theorem says that the sum of the squares of the sides of a right triangle equals the square of the hypotenuse. In symbols, a2 + b2 = c2 . Since we are interested in Number Theory, that is, the theory of whole numbers, we ask if there are any Pythagorean triangles whose sides are whol ...
Mathematical Logic and Foundations of
... undecidability of (N, +, ×, =) decidability of (R, +, ×, =) introduction to set theory axiomatic set theory the Axiom of Choice the Continuum Hypothesis ...
... undecidability of (N, +, ×, =) decidability of (R, +, ×, =) introduction to set theory axiomatic set theory the Axiom of Choice the Continuum Hypothesis ...
On the greatest prime factor of 22)—1 for a prime p
... is a large positive constant depending only on 8 27 since b—a> 2 = (2/logao)loga,> (2/logao )log a whenever a < ao . We shall assume tha t b —a < (log a) e2 and arrive at a contradiction . Recall the expressions (5) for a and b . Notic e that P i . . . p s
... is a large positive constant depending only on 8 27 since b—a> 2 = (2/logao)loga,> (2/logao )log a whenever a < ao . We shall assume tha t b —a < (log a) e2 and arrive at a contradiction . Recall the expressions (5) for a and b . Notic e that P i . . . p s
MIDTERM REVIEW FOR MATH 500 1. The limit Define limn→∞ an
... In class, we provide several examples and theorems to explain how we use the completeness Axiom establish some surprising properties of the real numbers, for instance, the Archimedean property of real numbers, and the “approximation” of any numbers in R by integers, and approximation of real numbers ...
... In class, we provide several examples and theorems to explain how we use the completeness Axiom establish some surprising properties of the real numbers, for instance, the Archimedean property of real numbers, and the “approximation” of any numbers in R by integers, and approximation of real numbers ...
Binomial Theorem
... The Binomial Theorem is a quick way (okay, it's a less slow way) of expanding (or multiplying out) a binomial expression that has been raised to some (generally inconveniently large) power. For instance, the expression (3x – 2)10 would be very painful to multiply out by hand. Thankfully, somebody fi ...
... The Binomial Theorem is a quick way (okay, it's a less slow way) of expanding (or multiplying out) a binomial expression that has been raised to some (generally inconveniently large) power. For instance, the expression (3x – 2)10 would be very painful to multiply out by hand. Thankfully, somebody fi ...
Exploring Mathematics Universe - KSU Web Home
... hanging were on Friday then it would not be a surprise, since he would know by Thursday night that he was to be hanged the following day, as it would be the only day left. Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday. He th ...
... hanging were on Friday then it would not be a surprise, since he would know by Thursday night that he was to be hanged the following day, as it would be the only day left. Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday. He th ...
Section 3 - Divisibility
... k = (n/d). So, if (n/d) is an integer, then d | n. • This leads to an easy way to test for nondivisibility: If (n/d) is not an integer, then d cannot divide n. • Examples: 3 | 12 since 12/3 = 4 ∈ Z. 5 | 12 since 12/5 = 2.4 ∉ Z. ...
... k = (n/d). So, if (n/d) is an integer, then d | n. • This leads to an easy way to test for nondivisibility: If (n/d) is not an integer, then d cannot divide n. • Examples: 3 | 12 since 12/3 = 4 ∈ Z. 5 | 12 since 12/5 = 2.4 ∉ Z. ...
Theorem
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.