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Selected Chapters from Number Theory and Algebra
... 10 Problem 1.9. As a warm-up, find the three smallest primes larger than 41 which are not in the list of 40 primes just mentioned. Answer. These are the first three primes not in the range: 59, 67, 73 6= n2 − n + 41 for any natural number n. We now address the question which other primes besides 41 ...
... 10 Problem 1.9. As a warm-up, find the three smallest primes larger than 41 which are not in the list of 40 primes just mentioned. Answer. These are the first three primes not in the range: 59, 67, 73 6= n2 − n + 41 for any natural number n. We now address the question which other primes besides 41 ...
Zeros of Polynomial Functions:
... Every polynomial of the nth degree with real coefficients has precisely n zeros in C (complex number system). The fundamental theorem of algebra states that every non-zero single-variable polynomial, with complex coefficients, has exactly as many complex roots as its degree, if repeated roots are co ...
... Every polynomial of the nth degree with real coefficients has precisely n zeros in C (complex number system). The fundamental theorem of algebra states that every non-zero single-variable polynomial, with complex coefficients, has exactly as many complex roots as its degree, if repeated roots are co ...
Congruent Numbers Via the Pell Equation and its Analogous
... From this we define a Pythagorean triple as follows: a = s2 −t2 , b = 2st, and c = s2 +t2 , where s = (mk+r+1), and t = (mk+r). Now, the results are immediate from the above corollary. Remark 4.8. From (A1), we see that the product of any three consecutive numbers is a congruent number. In particula ...
... From this we define a Pythagorean triple as follows: a = s2 −t2 , b = 2st, and c = s2 +t2 , where s = (mk+r+1), and t = (mk+r). Now, the results are immediate from the above corollary. Remark 4.8. From (A1), we see that the product of any three consecutive numbers is a congruent number. In particula ...
On the Product of Divisors of $n$ and of $sigma (n)
... some of its properties. In particular, he proved several results pertaining to multiplicative perfect numbers, which, by analogy, are numbers n for which the relation T (n) = nk holds with some positive integer k. In this paper, we compare T (n) with T (σ(n)). Our first result is: Theorem 1. The ine ...
... some of its properties. In particular, he proved several results pertaining to multiplicative perfect numbers, which, by analogy, are numbers n for which the relation T (n) = nk holds with some positive integer k. In this paper, we compare T (n) with T (σ(n)). Our first result is: Theorem 1. The ine ...
A conjecture of Erdos on graph Ramsey numbers
... that r(Kn ) > 2n/2 for n > 2. Over the last sixty years, there have been several improvements on these bounds (see, e.g., [5]). However, despite efforts by various researchers, the constant factors in the above exponents remain the same. Unsurprisingly then, the field has stretched in different dire ...
... that r(Kn ) > 2n/2 for n > 2. Over the last sixty years, there have been several improvements on these bounds (see, e.g., [5]). However, despite efforts by various researchers, the constant factors in the above exponents remain the same. Unsurprisingly then, the field has stretched in different dire ...
4 slides/page
... The equation ax = b for a, b ∈ R is uniquely solvable if a 6= 0: x = ba−1. • Can we also (uniquely) solve ax ≡ b (mod m)? • If x0 is a solution, then so is x0 + km ∀k ∈ Z ◦ . . . since km ≡ 0 (mod m). So, uniqueness can only be mod m. But even mod m, there can be more than one solution: • Consider 2 ...
... The equation ax = b for a, b ∈ R is uniquely solvable if a 6= 0: x = ba−1. • Can we also (uniquely) solve ax ≡ b (mod m)? • If x0 is a solution, then so is x0 + km ∀k ∈ Z ◦ . . . since km ≡ 0 (mod m). So, uniqueness can only be mod m. But even mod m, there can be more than one solution: • Consider 2 ...
www.fq.math.ca
... Among all polygonal chains, the hexagonal chains were studied the most extensively, since they are of great importance in chemistry, namely, benzenoid hydrocarbon chains. Each perfect matching of a hexagonal chain corresponds to a Kekule structure of the corresponding benzenoid hydrocarbon. The stab ...
... Among all polygonal chains, the hexagonal chains were studied the most extensively, since they are of great importance in chemistry, namely, benzenoid hydrocarbon chains. Each perfect matching of a hexagonal chain corresponds to a Kekule structure of the corresponding benzenoid hydrocarbon. The stab ...
Rédei symbols and arithmetical mild pro-2-groups
... i.e. if S ∩ Sp = ∅, the structure of these groups has remained rather mysterious for a long time. Some of the few known results are still conjectural such as the Fontaine-Mazur conjecture which predicts that these groups are either finite or not p-adic analytic. In his fundamental paper [7], J. Labu ...
... i.e. if S ∩ Sp = ∅, the structure of these groups has remained rather mysterious for a long time. Some of the few known results are still conjectural such as the Fontaine-Mazur conjecture which predicts that these groups are either finite or not p-adic analytic. In his fundamental paper [7], J. Labu ...
25(4)
... The THIRD INTERNATIONAL CONFERENCE ON FIBONACCI NUMBERS AND THEIR APPLICATIONS will take place at The University of Pisa, Pisa, Italy, from July 25-29, 1988. This conference is sponsored jointly by The Fibonacci Association and The University of Pisa. Papers on all branches of mathematics and scienc ...
... The THIRD INTERNATIONAL CONFERENCE ON FIBONACCI NUMBERS AND THEIR APPLICATIONS will take place at The University of Pisa, Pisa, Italy, from July 25-29, 1988. This conference is sponsored jointly by The Fibonacci Association and The University of Pisa. Papers on all branches of mathematics and scienc ...
Midpoints and Exact Points of Some Algebraic
... (assuming rounded-to-nearest arithmetic), or a floating-point number (assuming rounded toward #1 or toward 0 arithmetic). In the first case, we say that fðxÞ is a midpoint, and in the second case, we say that fðxÞ is an exact point. For some usual algebraic functions and various floating-point forma ...
... (assuming rounded-to-nearest arithmetic), or a floating-point number (assuming rounded toward #1 or toward 0 arithmetic). In the first case, we say that fðxÞ is a midpoint, and in the second case, we say that fðxÞ is an exact point. For some usual algebraic functions and various floating-point forma ...
INTEGER FACTORIZATION ALGORITHMS
... In the Trial division and Fermat factorization, we know that both will be checked for every possible factor of number, n. These factorization algorithms always exhaust search the number to be factored. However, the algorithms are suitable for factoring small numbers like two or three digit numbers. ...
... In the Trial division and Fermat factorization, we know that both will be checked for every possible factor of number, n. These factorization algorithms always exhaust search the number to be factored. However, the algorithms are suitable for factoring small numbers like two or three digit numbers. ...
Theory of L-functions - Institut für Mathematik
... There is another quite remarkable line of investigation, namely the impact of Random Matrix Theory, i.e., the recent idea to model L-functions by large unitary random matrices; this approach is motivated by Montgomery’s celebrated pair correlation conjecture and computations observing that the neare ...
... There is another quite remarkable line of investigation, namely the impact of Random Matrix Theory, i.e., the recent idea to model L-functions by large unitary random matrices; this approach is motivated by Montgomery’s celebrated pair correlation conjecture and computations observing that the neare ...
35(1)
... K. T. Atanassov and others, in [3], [1], and [2], introduced (2, F) and (3, F) sequences which were pairs and triples of sequences defined by two or three simultaneous Fibonacci-like recurrences, respectively, for which the exact definition will be given at the end of this section. There are four (2 ...
... K. T. Atanassov and others, in [3], [1], and [2], introduced (2, F) and (3, F) sequences which were pairs and triples of sequences defined by two or three simultaneous Fibonacci-like recurrences, respectively, for which the exact definition will be given at the end of this section. There are four (2 ...
lesson-4modular-arithmetric1
... Definition 6a: Additive Identity Element and Additive Inverse In the table above for +5, we see that any element in 5 , say a, a +5 0 = a and 0 +5 a = a. We say that 0 is the additive identity element in 5. We also notice that 1 +5 4 = 0 , 4 +5 1 = 0, 2 +5 3 = 0 and 3 +5 2 = 0. We say that 1 is th ...
... Definition 6a: Additive Identity Element and Additive Inverse In the table above for +5, we see that any element in 5 , say a, a +5 0 = a and 0 +5 a = a. We say that 0 is the additive identity element in 5. We also notice that 1 +5 4 = 0 , 4 +5 1 = 0, 2 +5 3 = 0 and 3 +5 2 = 0. We say that 1 is th ...
Let m be a positive integer. Show that a mod m = b mod m if a ≡ b
... p, then we would have p|ja − ia, or p|(j − i)a. By Lemma 1, since a is not divisible by p, p must divide j − i. But this is impossible, since j − i is a positive integer less than p. Therefore no two of these integers are congruent modulo p. (b) By part (a), since no two of a, 2a, · · · , (p − 1)a a ...
... p, then we would have p|ja − ia, or p|(j − i)a. By Lemma 1, since a is not divisible by p, p must divide j − i. But this is impossible, since j − i is a positive integer less than p. Therefore no two of these integers are congruent modulo p. (b) By part (a), since no two of a, 2a, · · · , (p − 1)a a ...
Elementary Number Theory with the TI-89/92
... In fact the function pwm(a,b,n) we have written (where a, b, n are natural numbers with n > 0), computes ab mod n by cleverly combining the reduction modulo n and the binary representation of the exponent b (bibliography [4]). It achieves both the results of reducing the number of multiplications an ...
... In fact the function pwm(a,b,n) we have written (where a, b, n are natural numbers with n > 0), computes ab mod n by cleverly combining the reduction modulo n and the binary representation of the exponent b (bibliography [4]). It achieves both the results of reducing the number of multiplications an ...
A proof of GMP square root
... 2. We compute the square root and remainder for n0 = 27; it yields s0 = 5 and r0 = 2. We place 2 below 27. 3. We lower n1 , i.e. 0, and divide the number obtained, 20, by twice s0 (2s0 = 10). This yields q = 2 and r00 = 0. Hence the estimation of the square root is 10s0 + q = 52. 4. We determine the ...
... 2. We compute the square root and remainder for n0 = 27; it yields s0 = 5 and r0 = 2. We place 2 below 27. 3. We lower n1 , i.e. 0, and divide the number obtained, 20, by twice s0 (2s0 = 10). This yields q = 2 and r00 = 0. Hence the estimation of the square root is 10s0 + q = 52. 4. We determine the ...
The secret life of 1/n: A journey far beyond the decimal point
... to discuss those values of n for which ` is as large as possible; this brings us face-to-face with the curious idea of a primitive root in number theory, and with an enduring mystery. We return in §1.4 to the general question of how we can determine ` in terms of n, by making a significant improveme ...
... to discuss those values of n for which ` is as large as possible; this brings us face-to-face with the curious idea of a primitive root in number theory, and with an enduring mystery. We return in §1.4 to the general question of how we can determine ` in terms of n, by making a significant improveme ...