Introduction to mathematical reasoning Chris Woodward Rutgers
... world series, is a compound proposition. It might even be true! The statement The author of this book is an American and the Devil Rays are a baseball team is a compound proposition, which happens to be true, because of the two simple propositions that make up the compound the proposition both happe ...
... world series, is a compound proposition. It might even be true! The statement The author of this book is an American and the Devil Rays are a baseball team is a compound proposition, which happens to be true, because of the two simple propositions that make up the compound the proposition both happe ...
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... These equations can always be solved by back-substitution; linear algebra is not required. Three cases arise, according to the number of zero roots of the characteristic equation ar 2 + br + c = 0. The values m = n, n + 1, n + 2 correspond to zero, one or two roots r = 0. Case 1: [No root r = 0]. Th ...
... These equations can always be solved by back-substitution; linear algebra is not required. Three cases arise, according to the number of zero roots of the characteristic equation ar 2 + br + c = 0. The values m = n, n + 1, n + 2 correspond to zero, one or two roots r = 0. Case 1: [No root r = 0]. Th ...
Numbers: Rational and Irrational
... listed in tables of logarithmic and trigonometric functions are ostensibly rational, actually they are only rational approximations of the true values,which are irrational with few exceptions. Clearly, then, irrational numbers occur in various natural ways in elementary mathematics. The real numbers ...
... listed in tables of logarithmic and trigonometric functions are ostensibly rational, actually they are only rational approximations of the true values,which are irrational with few exceptions. Clearly, then, irrational numbers occur in various natural ways in elementary mathematics. The real numbers ...
p-ADIC QUOTIENT SETS
... with a little heavy machinery in the form of the Brun-Titchmarsh and BombieriVinogradov theorems. The upshot of Theorem 8.3 is that there are infinitely many pairs of primes (p, q) so that the ratio set of {pj : j > 0} ∪ {q k : k > 0} is dense in Qp but not in Qq . A number of related questions are ...
... with a little heavy machinery in the form of the Brun-Titchmarsh and BombieriVinogradov theorems. The upshot of Theorem 8.3 is that there are infinitely many pairs of primes (p, q) so that the ratio set of {pj : j > 0} ∪ {q k : k > 0} is dense in Qp but not in Qq . A number of related questions are ...
Modeling Chebyshev`s Bias in the Gaussian Primes as a Random
... to find all primes less than or equal to a certain value. The Sieve of Eratosthenes starts by marking all multiples of 2 as composite, then proceeding to multiples of 3, 5, 7 and so on up to x. For example, after all even numbers up to (and including) 30 have been marked as composite, we have: ...
... to find all primes less than or equal to a certain value. The Sieve of Eratosthenes starts by marking all multiples of 2 as composite, then proceeding to multiples of 3, 5, 7 and so on up to x. For example, after all even numbers up to (and including) 30 have been marked as composite, we have: ...
Problem Shortlist with Solutions - International Mathematical Olympiad
... Answer. The sets A for which pA is maximal are the sets the form {d, 5d, 7d, 11d} and {d, 11d, 19d, 29d}, where d is any positive integer. For all these sets pA is 4. Solution. Firstly, we will prove that the maximum value of pA is at most 4. Without loss of generality, we may assume that a1 < a2 < ...
... Answer. The sets A for which pA is maximal are the sets the form {d, 5d, 7d, 11d} and {d, 11d, 19d, 29d}, where d is any positive integer. For all these sets pA is 4. Solution. Firstly, we will prove that the maximum value of pA is at most 4. Without loss of generality, we may assume that a1 < a2 < ...
Counting degenerate polynomials of fixed degree and bounded height
... constant was recently given in [12] (see [23, Example 266], [8] and [19] for some previous bounds in this problem). Many results and asymptotic formulas counting algebraic numbers of fixed degree and bounded heights (but other than naive height) have been obtained in [6] (Mahler measure), [20], [22] ...
... constant was recently given in [12] (see [23, Example 266], [8] and [19] for some previous bounds in this problem). Many results and asymptotic formulas counting algebraic numbers of fixed degree and bounded heights (but other than naive height) have been obtained in [6] (Mahler measure), [20], [22] ...
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. The cases n = 1 and n = 2 were known to have infinitely many solutions. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The theretofore unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof it was in the Guinness Book of World Records for ""most difficult mathematical problems"".