Solutions
... p | m such that r | m/p. Indeed, take p to be any prime dividing m/r. Then am/p = (ar )m/(pr) ≡ 1 (mod n) getting a contradiction. Thus m = r is the order. 2. Suppose p is an odd prime and n ≥ 1. If d | ϕ(pn ) show that there are exactly ϕ(d) numbers in Z× pn of multiplicative order d. [Hint: Use pr ...
... p | m such that r | m/p. Indeed, take p to be any prime dividing m/r. Then am/p = (ar )m/(pr) ≡ 1 (mod n) getting a contradiction. Thus m = r is the order. 2. Suppose p is an odd prime and n ≥ 1. If d | ϕ(pn ) show that there are exactly ϕ(d) numbers in Z× pn of multiplicative order d. [Hint: Use pr ...
Full text
... Proof: Since x0 is minimal of level k with A:>2 and x- > 3 , we have x0 £(16m + 3) 1. We prove that x- £(16w + 3) by contradiction. If x, e(l&m-3), pick J satisfying 2y + l = x / . Clearly a(y) = o-(x/), hence y Ghk. Since j>
... Proof: Since x0 is minimal of level k with A:>2 and x- > 3 , we have x0 £(16m + 3) 1. We prove that x- £(16w + 3) by contradiction. If x, e(l&m-3), pick J satisfying 2y + l = x / . Clearly a(y) = o-(x/), hence y Ghk. Since j>
Characterizing the number of coloured $ m $
... modulo m; again, they were able to obtain a (more complicated) explicit expression, and again this expression depended only on the coefficients in the base m representation of n. See also Edgar [E16] and Ekhad and Zeilberger [EZ15] for more on these results. The study of congruences for integer part ...
... modulo m; again, they were able to obtain a (more complicated) explicit expression, and again this expression depended only on the coefficients in the base m representation of n. See also Edgar [E16] and Ekhad and Zeilberger [EZ15] for more on these results. The study of congruences for integer part ...
Continuum Hypothesis, Axiom of Choice, and Non-Cantorian Theory
... The failure to identify the Continuum Hypothesis with any of the Axioms of set theory, led Godel in 1938 to confirm the consistency of the Hypothesis with the other Axioms of set theory, and led Cohen in 1963 to confirm the consistency of the HypothesisNegation. Since the Continuum Hypothesis is equ ...
... The failure to identify the Continuum Hypothesis with any of the Axioms of set theory, led Godel in 1938 to confirm the consistency of the Hypothesis with the other Axioms of set theory, and led Cohen in 1963 to confirm the consistency of the HypothesisNegation. Since the Continuum Hypothesis is equ ...
The Connectedness of Arithmetic Progressions in
... arithmetic progressions in Golomb’s topology D on N. Theorem 3.3. Let a, b ∈ N. The arithmetic progression {an + b} is connected in (N, D) if and only if Θ(a) ⊆ Θ(b). In particular, i) the progression {an} is D-connected, and ii) if the progression {an + b} is an element of the basis B, then it is D ...
... arithmetic progressions in Golomb’s topology D on N. Theorem 3.3. Let a, b ∈ N. The arithmetic progression {an + b} is connected in (N, D) if and only if Θ(a) ⊆ Θ(b). In particular, i) the progression {an} is D-connected, and ii) if the progression {an + b} is an element of the basis B, then it is D ...
Propositional Statements Direct Proof
... , because a/2 = ab . But we said definition, so b is even. Since a and b are both even, a/2 and b/2 are integers. and 2 = a/2 b/2 b/2 a a before b is in its simplest form and cannot be reduced. We just reduced b by a factor of 2, so this is a contradiction. X ...
... , because a/2 = ab . But we said definition, so b is even. Since a and b are both even, a/2 and b/2 are integers. and 2 = a/2 b/2 b/2 a a before b is in its simplest form and cannot be reduced. We just reduced b by a factor of 2, so this is a contradiction. X ...
MAA245 NUMBERS 1 Natural Numbers, N
... If ∗ is a binary operation on a set A, we define its inverse ∗ as follows. Given s, t ∈ A, if ∃ r ∈ A such that r ∗ s = t, then t ∗ s = r. If ∗ is commutative, then t ∗ s = r ⇔ t ∗ r = s. If ∃ r ∈ A with r ∗ s = t for every s, t ∈ A, then ∗ is closed on A: t ∗ s ∈ A ∀s, t ∈ A. The need to find sets ...
... If ∗ is a binary operation on a set A, we define its inverse ∗ as follows. Given s, t ∈ A, if ∃ r ∈ A such that r ∗ s = t, then t ∗ s = r. If ∗ is commutative, then t ∗ s = r ⇔ t ∗ r = s. If ∃ r ∈ A with r ∗ s = t for every s, t ∈ A, then ∗ is closed on A: t ∗ s ∈ A ∀s, t ∈ A. The need to find sets ...
Math 3000 Section 003 Intro to Abstract Math Homework 4
... Solution: (a) Direct Proof. (b) The proofs starts out with an integer and assumes that its fourth power is even. (c) The proof must conclude that the third multiple of that integer increased by 1 is an odd number. (d) i.ii. We know (already have proven) that an integer and its square have same parit ...
... Solution: (a) Direct Proof. (b) The proofs starts out with an integer and assumes that its fourth power is even. (c) The proof must conclude that the third multiple of that integer increased by 1 is an odd number. (d) i.ii. We know (already have proven) that an integer and its square have same parit ...
Proof
... We could try indirect proof also, but in this case, it is a little simpler to just use proof by contradiction (very similar to indirect). So, what are we trying to show? Just that x + y is irrational. That is, :9i, j: (x + y ) = ji . What happens if we hypothesize the negation of this statement? ...
... We could try indirect proof also, but in this case, it is a little simpler to just use proof by contradiction (very similar to indirect). So, what are we trying to show? Just that x + y is irrational. That is, :9i, j: (x + y ) = ji . What happens if we hypothesize the negation of this statement? ...
Explicit formulas for Hecke Gauss sums in quadratic
... for the ordinary Gauss sums and ordinary quadratic reciprocity, which, however, follows from (2). It is remarkable, that hence, as a consequence, the quadratic reciprocity law for quadratic number fields is not a genuine new reciprocity law. This is in contrast to what is suggested by Hecke’s proof ...
... for the ordinary Gauss sums and ordinary quadratic reciprocity, which, however, follows from (2). It is remarkable, that hence, as a consequence, the quadratic reciprocity law for quadratic number fields is not a genuine new reciprocity law. This is in contrast to what is suggested by Hecke’s proof ...
Reverse Mathematics and the Coloring Number of Graphs
... M.Sc. Mathematics, University of Connecticut, Storrs, Connecticut, 2006 ...
... M.Sc. Mathematics, University of Connecticut, Storrs, Connecticut, 2006 ...
Logical Inference and Mathematical Proof
... Quote from “A Study in Scarlet” “And now we come to the great question as to the reason why. Robbery has not been the object of the murder, for nothing was taken. Was it politics, then, or was it a woman? That is the question which confronted me. I was inclined from the first to the latter suppositi ...
... Quote from “A Study in Scarlet” “And now we come to the great question as to the reason why. Robbery has not been the object of the murder, for nothing was taken. Was it politics, then, or was it a woman? That is the question which confronted me. I was inclined from the first to the latter suppositi ...
Methods of Proof
... a pretty obvious thing to do, at least when you see someone else do it, this step, in which you bring your knowledge to the problem, may seem like a big one to take, and you may find yourself stalling out at this point. One possible reason this may happen is that you may be trying to do too much at ...
... a pretty obvious thing to do, at least when you see someone else do it, this step, in which you bring your knowledge to the problem, may seem like a big one to take, and you may find yourself stalling out at this point. One possible reason this may happen is that you may be trying to do too much at ...
Discrete Mathematics and Logic II. Formal Logic
... The system E (see chapter 1 of the textbook) has 15 axioms, starting with Associativity of ==, (3.1). Its inference rules are Leibniz (1.5), Transitivity of equality (1.4), and Substitution (1.1). Its theorems are the formulas that can be shown to be equal to an axiom using these inference rules. ...
... The system E (see chapter 1 of the textbook) has 15 axioms, starting with Associativity of ==, (3.1). Its inference rules are Leibniz (1.5), Transitivity of equality (1.4), and Substitution (1.1). Its theorems are the formulas that can be shown to be equal to an axiom using these inference rules. ...