# Questions tagged [integers]

For questions about the set of integers, or positive and negative whole numbers, commonly denoted $\mathbb{Z}$.

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### How to prove that $4x^3+6x^2+4x+1$ is not a fourth power of an integer, for any $x\in\mathbb N$?

How do I prove that for all positive integers $x$ it's true that, $4x^3+6x^2+4x+1$ is not a fourth power of an integer? I've tried doing modulo 3 and 5 checks, and it didn't really go far from there
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### Induction and Maximum Principle

I wish to show that the following two assertions are equivalent: (Principle of Mathematical Induction) Let $S$ be a nonempty subset of the set of non-negative integers satisfying the following two ...
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### Prove that if $2^x,3^x, 5^x, 7^x, 11^x …$ are all integers then $x$ is an integer as well

How easy is it to prove that if $2^x,3^x, 5^x, 7^x, 11^x ...$ are integers then $x$ is an integer as well? I have read the definition of the exponent functions as given in my calculus text, and the ...
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### Prove that if a and b are positive integers, then there exists integers x and y such that 1/lcm(a,b)=x/a+y/b

My professor has not taught us the technique of writing proofs, he just continues to do them for us in class. So I am really stumped on this proof. Any help is greatly appreciated!
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### $\mathbb{Q}$ can not be embedded in $\mathbb{Z}$

Show that $\mathbb{Q}$ can not be embedded in $\mathbb{Z}$ (where both has the subspace topology of $\mathbb{R}$) My attempt at a solution Since Z is discrete, {k} is open in $\mathbb{Z}$ with ...
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### Add integers to a set number of lists, so that the sum of each completed list is as closely matching to the other lists as possible?

I am trying to figure out how to solve this problem in computer science. I won't go into the programming side of things, but basically what I need is this: I have a list of integers ranging from ...
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### Diophantine equation: $2(x^3+xy+y^3)=3(x+y)$

Here is a nice equation: $2(x^3+xy+y^3)=3(x+y)$ over $\mathbb{Z}$ x $\mathbb{Z}$. Any nice way to approach this?
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### Polygons inscribed in circles, with integer sides and integer radius

Is there a simple characterization for an integer partition $(s_1,\dots,s_k)$, such that a polygon with these sides is inscribed in a circle with integer radius? This is what I got so far: All ...
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### Encyclopedia of integers

Many years ago I read something that mentioned a book I would like to find. Apparently this book is sort of an encyclopedia for integers; each entry lists interesting mathematical facts about that ...
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### Equation that can easily be changed to output the digit in 1's, 10's ,100's etc?

I need an equation that can be easily changed to output the digit which is held in the 1's slot, 10's slot, 100's slot, etc. EX. I want the 100's digit in 6810 EX2. I want the 1's digit in 29115 ...
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### Find a function that maps x,y to $[0, n ( n + 1) / 2)$

Can you find me a bijective function that maps positive integers $x, y$ such that $0 \leq x < y \leq n$ to integers in $[0, n(n+1)/2)$ to use as a hash function?
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### Integers and place values?

Suppose the tens digit of a whole number between 80 and 90 is greater than the ones digit,but less than twice the ones digit. If the integer is even, what is it's value?