All Questions
Tagged with well-orders elementary-number-theory
15
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No positive integers $(a,b)$ satisfying $2^b -1\mid 2^a +1$ (AoPS Vol 2)
I posted this proof on AoPS but I'm having trouble understanding what's wrong with it. I was hoping this community could offer a different perspective?
Suppose the claim is false. That is to say, ...
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Well-Ordering Principle "proof"
Theorem. Well-Ordering Principle.
Every non-empty subset of natural numbers has a least element.
I have seen some proofs for the theorem, but is very "complex"proof really needed here?
My ...
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Confusion in well ordering principle
Well-ordering principle states that every non-empty set of positive integers contains a least element.
I have a set S which is a subset of natural numbers. Now by well-ordering principle I can ...
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Using the WOP to check divisibility
The well-ordering principle states that every non-empty set of positive integers contains a least element.
I need to prove that 9|$n^3+(n+1)^3+(n+2)^3$ , $n∈N$ using this principle.
Regards
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Proof related with Well ordering principle
In each of the following scenarios use the Well Ordering Principle to answer the questions:
a) Is there a process that is capable of generating an infinitely long sequence of numbers $a_1,a_2,a_3\dots$...
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Help solving this proof using Well Ordering Principle
Let $a_1,a_2, ... , a_n ∈ \Bbb{N} $ . Prove that there exists $ l ∈ \Bbb{N} $ such that $a_i | l$ for all $i ∈ \{1,2,...,n\}$ and if $x ∈ \Bbb{N} $ is such that each $a_i$ divides $x$, then $l | x$.
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proof of Well Ordering Principle over positive integers
Theorem If $A$ is any nonempty subset of $\mathbb{Z}^+$, there is some element $m \in A$ such that $m \leq a$, for all $a \in A$ where such $m$ is the minimal element of $A$. (AA: Dummit and Foote)
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Well ordering principle
I am trying to understand why is the well ordering principle stated as an axiom of integers.
In the process, I found a "proof" of the principle (which is obviously wrong) and want to ...
- 259
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Application of the Well-Ordering Principle
I am suppose to show that there is no infinite sequence of strictly decreasing non-negative integers and know that the Well-ordering principle will have to come into play. Also, will induction need to ...
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well-ordering principle for natural numbers from the definition of real numbers.
Define the set of real numbers $\mathbb{R}$ by means of the following axiom: There exists a totally ordered field $(\mathbb{R},+,\cdot,\leq)$ which is Dedekind complete. We also assume that $a\leq b$ ...
user178826
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Show that the set {1/6, 1/7 , 1/8,.....} does not have a least element
Show that the set $\{\frac 16,\frac 17 ,\frac 18,\dots\}$ does not have a least element and conclude that no set containing this set is well ordered.
I am not sure how can I show this ... The set ...
- 619
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115
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Is this proof using well ordering principle correct? Are all parts necessary?
Example. Suppose the Royal Canadian Mint was to introduce a 3 cent coin like the British three pence to replace the 1 cent penny. Prove that 7 cents is now the largest quantity unable to be made with ...
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Question about choice of set used in proof of Division Algorithm with the Well Ordering Principle
When we prove the existence part of the division algorithm, namely
"Given natural numbers $a, n$, there exist integers $q$ and $r$ such that $a = nq + r$ with $0\leq r < n$" ,
I constructed the ...
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Prove that all integers can be represented as powers of 2 multiplied by an odd number [closed]
I would like to prove that $\forall n \in \mathbb{N},n\geq1$ $, n = 2^k\times m$ with $k \in \mathbb{Z}$ and $ m $ odd and $m \in \mathbb{N}$.
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Proof of Lehman's Lemma with the well ordering principle
I was reading an article on the well ordering principle and there was a problem that asked to use the well ordering principle to solve Lehman's Lemma:
That there are no positive integer solutions to ...
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