# Questions tagged [integers]

For questions about the structure, definition, and basic properties of the set of integers, or positive and negative whole numbers, commonly denoted $\mathbb{Z}$. Do not use this tag just because your question involves integers. Consider using (elementary-number-theory) or (number-theory) instead of or in addition to this tag.

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### Big Matrix Small Determinant

From a programming competition: Construct a square matrix with $N$ rows and $N$ columns consisting of non-negative integers from $0$ to $10^{18}$, such that its determinant is equal to $1$, and there ...
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### Set notation using integers [duplicate]

I am confused with the set notation used with integers, such as $\Bbb Z/n$, $\Bbb Z_n$,and $\Bbb Z/n\Bbb Z$. Please explain the difference between these notations.
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### What is the sum of all integers a such that $a^2-7a-7$ divided by $a-4$ yields an integer?

I sat down for several minutes over the span of a few days to try to solve this problem. I tried different methods, however, the only method I could devise was about making a well educated guess! I ...
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### Find the set of natural numbers $n \in \mathbb{N}$ for which $\lfloor\sqrt{n+1}\rfloor \neq\lfloor\sqrt{n}\rfloor$

I have to find the set of natural numbers $n \in \mathbb{N}$ for which $\lfloor\sqrt{n+1}\rfloor \neq\lfloor\sqrt{n}\rfloor$. I have tried writing the formal definition of the floor function and tried ...
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### From where this theorem about integer valued polynomials is coming from?

In his article "Polynomials with Integer values" (Resonance), Sury proved an interesting lemma: If $P$ is a non constant, integral valued, polynomial, then the number of prime divisors of ...
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### Proving well ordering principle with well ordering principle

Hypothesis: "Every nonempty subset S of the positive integers has a least element." Goal: "Every nonempty subset S of non-negative integers has a least element." Is this proof ...
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### Can $a^2 - 2^b$ be factored if $b$ is odd?

Let $a$ and $b$ be positive integers. If $b$ is even (i.e. $b=2c$ for some positive integer $c$), then $a^2 - 2^b$ can be factored as $$a^2 - 2^b = a^2 - 2^{2c} = (a + 2^c)(a - 2^c).$$ Edited: (...
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### Integer Values of $\sum_{k=1}^n k^r . \sum_{q=1}^n \frac{1}{q^r}$

For harmonic numbers $H_n = \sum_{k=1}^n \frac{1}{k}$ we know that this sum is never integer for any $n$. The same is true for generalized Harmonic numbers: the sum $\sum_{k=1}^n\frac{1}{k^r}$ is ...
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### Integer part problem

If $m,n$ are natural non-zero numbers show that $$[x]+[x+1/n]+[x+2/n]+...+[x+m/n]=[nx]$$ for any real $x$ if and only if $m=n-1$. $[x]$ is the integer part of $x$. I know from the Hermite Identity ...
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### Can I say the integer solution of this equation is unique?

I want to solve the equation $\frac{y!}{(x-2)!(y-x)!}=\frac{1340!}{659!\times679!}$ where $x,y \in \mathbb{Z}$. I think it is natural to say $(x,y)=(681,1340)$ and $(x,y)=(661,1340)$ are desired ...
### If Alice gives Bob $m$ candies, then he'll have $n$ times her candies; if Bob gives Alice $n$ candies, then she'll have $m$ times his candies.
I came up with a seemingly innocent problem of recreational mathematics by myself. It goes likes this. Alice and Bob have some different amount of candies ($>1$ each). If Alice gives Bob $m$ ...