Solutions - Math.utah.edu
... • Between two rational numbers there is an irrational number. • Between two irrational numbers there is an rational number. Proof. The proof of the second part was already done in Extra Problems #3, Exercise 0.4 (in fact, we showed there were infinitely many rational numbers between any two numbers) ...
... • Between two rational numbers there is an irrational number. • Between two irrational numbers there is an rational number. Proof. The proof of the second part was already done in Extra Problems #3, Exercise 0.4 (in fact, we showed there were infinitely many rational numbers between any two numbers) ...
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... many other computational problems could be done quickly. For example, the Euler phi function can obviously be computed quickly given the prime factorization of II. As a of the work on tests for primality we show that in fact the converse is true, assuming the ERH. Thus, computing the Euler phi funct ...
... many other computational problems could be done quickly. For example, the Euler phi function can obviously be computed quickly given the prime factorization of II. As a of the work on tests for primality we show that in fact the converse is true, assuming the ERH. Thus, computing the Euler phi funct ...
Max Lewis Dept. of Mathematics, University of Queensland, St Lucia
... for a given k 2 N. We have used it to show that there exists an n 2 N such that k(n) = k for every k 106 . For example, taking k = 106 , we find that q = 106 · (23 1) + 1 = 22000001 is prime and so k(23 · 22000001) = 106 . Unfortunately Lemma 4 does not lead to a proof that k is surjective. Let fk ...
... for a given k 2 N. We have used it to show that there exists an n 2 N such that k(n) = k for every k 106 . For example, taking k = 106 , we find that q = 106 · (23 1) + 1 = 22000001 is prime and so k(23 · 22000001) = 106 . Unfortunately Lemma 4 does not lead to a proof that k is surjective. Let fk ...
On normal numbers - Mathematical Sciences Publishers
... A number f is called simply normal in the scale of r if (1) holds for k = 1. A number is said to be absolutely normal if it is normal to every base r. It is well-known (see, for example, [6], Theorem 8.11) that almost every number ξ is absolutely normal. We write r ^ s, if there exist integers n, m ...
... A number f is called simply normal in the scale of r if (1) holds for k = 1. A number is said to be absolutely normal if it is normal to every base r. It is well-known (see, for example, [6], Theorem 8.11) that almost every number ξ is absolutely normal. We write r ^ s, if there exist integers n, m ...
Full text
... Lemma: Let C,n denote the primitive rfi1 root of unity cos-^f + Zsin^f. Suppose Then, for somey and k, £Jn + £kn - -1 if and only if n = 3 j and k = 2j. ...
... Lemma: Let C,n denote the primitive rfi1 root of unity cos-^f + Zsin^f. Suppose Then, for somey and k, £Jn + £kn - -1 if and only if n = 3 j and k = 2j. ...
On the Number of Prime Numbers less than a Given Quantity
... Prime numbers are probably one of the most beautiful objects in all of mathematics. It is remarkable, that they have such a simple definition: “p is prime iff p has no other divisors, besides 1 and p”, and at the same time their properties are so hard to explore. The importance of the primes was rea ...
... Prime numbers are probably one of the most beautiful objects in all of mathematics. It is remarkable, that they have such a simple definition: “p is prime iff p has no other divisors, besides 1 and p”, and at the same time their properties are so hard to explore. The importance of the primes was rea ...
Notes8
... However, the table for multiplication is a bit more interesting. There is obviously a row with all zeroes. But in each of the other rows, every value is there and there is no repeated value. This does not always happen; for example, if we wrote down the table for modulus 4, then we would see only ev ...
... However, the table for multiplication is a bit more interesting. There is obviously a row with all zeroes. But in each of the other rows, every value is there and there is no repeated value. This does not always happen; for example, if we wrote down the table for modulus 4, then we would see only ev ...
The Congruent Number Problem -RE-S-O-N-A-N-C-E--I-A-U-9-U
... Here we have yet another instance of a problem in number theory which is simple to state yet has hidden depths. There have been instances when the solutions of such problems have emerged only centuries after being posed. In such instances, a lot of deep and beautiful mathematics gets generated as a ...
... Here we have yet another instance of a problem in number theory which is simple to state yet has hidden depths. There have been instances when the solutions of such problems have emerged only centuries after being posed. In such instances, a lot of deep and beautiful mathematics gets generated as a ...
Week 1 - UCR Math Dept.
... Say, for example, we want to place 4 balls in 2 identical bins so that each bin gets at least one ball. There are two ways of doing it: Either split the balls 2, 2 or split them 3, 1. These correspond to the ”partitions” 4 = 2 + 2 and 4 = 3 + 1. Let pk (n) denote the number of partitions of n into k ...
... Say, for example, we want to place 4 balls in 2 identical bins so that each bin gets at least one ball. There are two ways of doing it: Either split the balls 2, 2 or split them 3, 1. These correspond to the ”partitions” 4 = 2 + 2 and 4 = 3 + 1. Let pk (n) denote the number of partitions of n into k ...
On a strong law of large numbers for monotone measures
... Received date: 30 July 2012 Revised date: 15 January 2013 Accepted date: 15 January 2013 Please cite this article as: Agahi, H., Mohammadpour, A., Mesiar, R., Ouyang, Y., On a strong law of large numbers for monotone measures. Statistics and Probability Letters (2013), doi:10.1016/j.spl.2013.01.021 ...
... Received date: 30 July 2012 Revised date: 15 January 2013 Accepted date: 15 January 2013 Please cite this article as: Agahi, H., Mohammadpour, A., Mesiar, R., Ouyang, Y., On a strong law of large numbers for monotone measures. Statistics and Probability Letters (2013), doi:10.1016/j.spl.2013.01.021 ...
Squarefree smooth numbers and Euclidean prime generators
... In [14], Mullin considered the sequence {pk }∞ k=1 defined so that, for every k ≥ 0, pk+1 is the smallest prime factor of 1 + p1 · · · pk . From the argument employed by Euclid to prove the infinitude of prime numbers, it follows that the pk are pairwise distinct, and Mullin’s sequence can thus be v ...
... In [14], Mullin considered the sequence {pk }∞ k=1 defined so that, for every k ≥ 0, pk+1 is the smallest prime factor of 1 + p1 · · · pk . From the argument employed by Euclid to prove the infinitude of prime numbers, it follows that the pk are pairwise distinct, and Mullin’s sequence can thus be v ...
How many ways can you make change
... gmu.edu/~geir/SylvDen2.pdf. [2] J. R. Alfonsin. The diophantine Frobenius problem. Oxford University Press, 2006. [3] G. Alon and P. Clark. On the number of represenations of an integer by a linear form. Journal of Integer Sequences, 8:Article 05.5.2, 2005. [4] V. Baldoni, N. Berline, J. D. Loera, B ...
... gmu.edu/~geir/SylvDen2.pdf. [2] J. R. Alfonsin. The diophantine Frobenius problem. Oxford University Press, 2006. [3] G. Alon and P. Clark. On the number of represenations of an integer by a linear form. Journal of Integer Sequences, 8:Article 05.5.2, 2005. [4] V. Baldoni, N. Berline, J. D. Loera, B ...
CHAPTER 2 NUMBER THEORY, NUMBER SYSTEM & COMPUTER
... • If you can factor a and b into primes, do so. For each prime number, look at the powers that it appears in the factorizations of a and b. Take the smaller of the two. Put these prime powers together to get the gcd. This is easiest understand by examples: i) 576 2632 , ...
... • If you can factor a and b into primes, do so. For each prime number, look at the powers that it appears in the factorizations of a and b. Take the smaller of the two. Put these prime powers together to get the gcd. This is easiest understand by examples: i) 576 2632 , ...
Multiplicities and Enumeration of Semidualizing Modules
... to computing the Hilbert-Samuel multiplicity eR (C), thus justifying our interest in the following question, motivated by the well-known equality eR (J; D) = eR (J; R): Question 1.4. Let C be a semidualzing R-module. For each m-primary ideal J, must we have an equality of Hilbert-Samuel multipliciti ...
... to computing the Hilbert-Samuel multiplicity eR (C), thus justifying our interest in the following question, motivated by the well-known equality eR (J; D) = eR (J; R): Question 1.4. Let C be a semidualzing R-module. For each m-primary ideal J, must we have an equality of Hilbert-Samuel multipliciti ...
1 Unique Factorization of Integers
... again gives us an analogue to the Fundamental Theorem of Arithmetic. Q[x]: the space of all polynomials with rational coefficients. In this setting, we also have unique factorization in the following sense. Given a polynomial f (x) ∈ Q[x] then f (x) = c f1 (x) . . . fr (x) where c ∈ Q and the fi are ...
... again gives us an analogue to the Fundamental Theorem of Arithmetic. Q[x]: the space of all polynomials with rational coefficients. In this setting, we also have unique factorization in the following sense. Given a polynomial f (x) ∈ Q[x] then f (x) = c f1 (x) . . . fr (x) where c ∈ Q and the fi are ...
Full text
... Note that, for n>m>r, p(n, m) = H%}m+l(a0,...,ar_x), where {H%}m+l(a^...,a^)}^ is the sequence of multivariate Fibonacci polynomials of order r of Philippou (cf. [1]). Let n and j be two states such that 0
... Note that, for n>m>r, p(n, m) = H%}m+l(a0,...,ar_x), where {H%}m+l(a^...,a^)}^ is the sequence of multivariate Fibonacci polynomials of order r of Philippou (cf. [1]). Let n and j be two states such that 0
Cardinality, countable and uncountable sets
... looks very much like “an equivalence relation in the class of all sets”, and indeed this can be formalized in axiomatic set theory, but we’ll leave that for the advanced course. The notion of “cardinality” of a set was develop in the late 19th/early 20th centuries by the German mathematician Georg C ...
... looks very much like “an equivalence relation in the class of all sets”, and indeed this can be formalized in axiomatic set theory, but we’ll leave that for the advanced course. The notion of “cardinality” of a set was develop in the late 19th/early 20th centuries by the German mathematician Georg C ...
Discrete Mathematics Lecture 3 Elementary Number Theory and
... • To show that the statement in the form “∀x ∈ D, P(x) à Q(x)” is not true one needs to show that the negation, which has a form “∃x ∈ D, P(x) ∧ ~Q(x)” is true. x is called a counterexample. • Famous conjectures: – Fermat big theorem: there are no non-zero integers x, y, z such that xn + yn = zn, fo ...
... • To show that the statement in the form “∀x ∈ D, P(x) à Q(x)” is not true one needs to show that the negation, which has a form “∃x ∈ D, P(x) ∧ ~Q(x)” is true. x is called a counterexample. • Famous conjectures: – Fermat big theorem: there are no non-zero integers x, y, z such that xn + yn = zn, fo ...
Solutions
... Due Wednesday, 2015-09-23, in class Do 5 of the following 7 problems. Please only attempt 5 because I will only grade 5. 1. Suppose n ≥ 2 and a ∈ Z× n . Show that a has multiplicative order m modulo n if and only if the following two conditions are satisfied: (a) am ≡ 1 (mod n) and (b) for every pri ...
... Due Wednesday, 2015-09-23, in class Do 5 of the following 7 problems. Please only attempt 5 because I will only grade 5. 1. Suppose n ≥ 2 and a ∈ Z× n . Show that a has multiplicative order m modulo n if and only if the following two conditions are satisfied: (a) am ≡ 1 (mod n) and (b) for every pri ...
