# Tagged Questions

For question about natural numbers $\Bbb N$, their properties and applications

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### Gallian: is it true that the well-ordering principle can't be proven from properties of arithmetic?

I just started reading Gallian's Abstract algebra and on page 3 it says "An important property of the integers...is the so-called Well Ordering Principle. Since this property cannot be proved from the ...
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### Induction proof for $x \le y$

So these are $x \in \{1 ... i\}$ and $y \in \{ i + 1 ... n\}$, for $i, n \in \mathbb N$. I want to prove it for every $x \le y$. I know it`s easy but the solution is escaping me. I have tried with ...
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### Solve for x and y in z=x*2^y with a known z [closed]

I'm trying to figure out register values for a program I'm writing. I have a spreadsheet where I'm attempting to reverse engineer mantissa and exponent values so I can get the necessary register ...
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### Is there a unique specific description of the set $A=\{5, 7, 11, 29, 41\}$? [closed]

This question is what inspired mine. Let $A=\{5, 7, 11, 29, 41\}$. If we only refer to $A$ itself, and not to general properties (e.g. $7$ is the only prime not being a Sophie Germain), can we point ...
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### Solving system of inequalities, with solution in only natural numbers, with priority on variables

If I have the equations $27a+30b+33c+36c \geq x$ $a+b+c+d=4$ and want to solve them using only natural numbers (including 0) for both $x=131$ and $x=142$ preferably but not necessarily with ...
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### Show natural numbers ordered by divisibility is a distributive lattice.

I need a proof that the set of natural numbers with the the relationship of divisibility form a distributive lattice with gcd as AND and lcm as OR. I know it can be shown that a AND (b OR c) >= (a ...
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### Goldbach's conjecture can't be proved to be undecidable?

Conjectures concerning natural numbers which could be settled by a counterexample can, as far as I understand, not be proved to be undecidable without being proved not having a counterexample at the ...
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### showing the natural numbers exist from axioms (help with making sense of book)

I'm now on page 40 of a set theory book and I've hit the natural numbers. I think the book has oversimplified some things. The successor of a set $x$ is defined to be $S(x)=x\cup\{x\}$ A set $I$ is ...
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### Showing there is no natural number between two consecutive natural numbers

I want to show that: $x\subset S(x)$ where $S$ is the Successor function and $\not\exists z:x\subset z\subset S(x)$ These are obvious results, but the relation of $m<n\iff m\in n$ is given as a ...
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### Conjecture: three or more decompositions into powers with a base differing by 1 means its a perfect power

If $$(i_1)^{a_1}(i_1+1)^{b_1}=n$$ $$(i_2)^{a_2}(i_2+1)^{b_2}=n$$ $$(i_3)^{a_3}(i_3+1)^{b_3}=n$$ where all the terms are positive integers and the groups ...
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### Are these two definitions of the natural numbers equivalent?

If we consider two definitions of the natural numbers: Definition 1 $N$ is the set that satisfies all of: There is an element $0$ in $N$. For each element $n$ in $N$, there is the successor of ...
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### An isomorphic map from natural numbers to positive rational numbers that preserves addition, multiplication and order

Since $\mathbb{Q}^{+}$ is countable, there is a bijection between $\mathbb{Q}^{+}$ and $\mathbb{N}$ (0 included). Then the question now is, can we go further by constructing an isomorphic map between ...
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### Let $\gamma$ be the Euler-Mascheroni constant. Can there be natural numbers $a,b,c$ such that $\log a - \log b - \log \log \log c =\gamma$?

Can there be integers satisfying $$\ \log a - \log b - \log \log \log c = \gamma \ \ \ ?$$
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### Is this version of Lagrange's four-square theorem true?

Lagrange's four-square theorem states that any natural number $n$ can be represented as the sum of four integer squares.i.e. $n = a_1\times a_1 + a_2\times a_2 + a_3\times a_3 + a_4\times a_4$ ...
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### What is an example of a bijective function f: Z to N that isn't piecewise?

Like without using if even or odd. Like how you can define a bijection $f\colon\mathbb{N}\to\mathbb{Z}$ by is $f(n)=\lfloor n/2\rfloor\cdot(-1)^n$.