Mathematics Stack Exchange
Mathematics Stack Exchange

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.

The integers are the whole numbers, positive, negative and zero. That is, the integers are the numbers that appear in the infinite list

$$.\quad .\quad .\quad -5\quad -4\quad -3\quad -2\quad -1\quad 0\quad 1\quad 2\quad 3\quad 4\quad 5\quad .\quad .\quad .\quad$$

The set of all integers is denoted by $\mathbb{Z}$. The letter Z comes from the German word "Zahlen" which means "numbers". The integers are related to many other familiar sets of numbers:

$$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}.$$

The set of integers are closed under addition, subtraction, and multiplication. Together with the additive identity $0$ and the multiplicative identity $1$, the integers form an example of a commutative ring with unity. In fact, it is a Euclidean domain.

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Is there a domain "larger" than (i.e., a supserset of) the complex number domain?

I've been teaching my 10yo son some (for me, anyway) pretty advanced mathematics recently and he stumped me with a question. The background is this. In the domain of natural numbers, addition and multiplication always generate natural numbers,…
user1324
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What is the smallest unknown natural number?

There are several unknown numbers in mathematics, such as optimal constants in some inequalities. Often it is enough to some estimates for these numbers from above and below, but finding the exact values is also interesting. There are situations…
Joonas Ilmavirta
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Why are integers subset of reals?

In most programming languages, integer and real (or float, rational, whatever) types are usually disjoint; 2 is not the same as 2.0 (although most languages do an automatic conversion when necessary). In addition to technical reasons, this…
Aivar
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The best symbol for non-negative integers?

I would like to specify the set $\{0, 1, 2, \dots\}$, i.e., non-negative integers in an engineering conference paper. Which symbol is more preferable? $\mathbb{N}_0$ $\mathbb{N}\cup\{0\}$ $\mathbb{Z}_{\ge 0}$ $\mathbb{Z}_{+}$
Ari
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Probability that two random numbers are coprime is $\frac{6}{\pi^2}$

This is a really natural question for which I know a stunning solution. So I admit I have a solution, however I would like to see if anybody will come up with something different. The question is What is the probability that two numbers randomly…
uforoboa
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"Rectangularity" of integers

We can sort of think of a number $n$ as "almost square" if $n = kl$ where $$\frac{k}{l} \approx 1.$$ More generally, we can talk about the "rectangularity" of an integer $n$ as $$ \max_{k \leq l | kl = n} \frac{k}{l}. $$ (We might also look at the…
Jeffrey Sun
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A sequence of coefficients of $x+(x+(x+(x+(x+(x+\dots)^6)^5)^4)^3)^2$

Let's consider a function (or a way to obtain a formal power series): $$f(x)=x+(x+(x+(x+(x+(x+\dots)^6)^5)^4)^3)^2$$ Where $\dots$ is replaced by an infinite sequence of nested brackets raised to $n$th power. The function is defined as the limit…
Yuriy S
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Sum of all integers

No, I'm not talking about $-\frac{1}{12}$. I was talking with someone the other day, and they said that the sum of all integers, positive and negative, is zero because they all cancel each other out.…
user3932000
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What is the smallest integer greater than 1 such that $\frac12$ of it is a perfect square and $\frac15$ of it is a perfect fifth power?

What is the smallest integer greater than 1 such that $\frac12$ of it is a perfect square and $\frac15$ of it is a perfect fifth power? I have tried multiplying every perfect square (up to 400 by two and checking if it is a perfect 5th power, but…
J. DOEE
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Is (a/b)/c equal to a/(b*c) for integer division?

Let $\div$ denote a binary operator that returns the integer quotient of two integers, i.e. (assuming that both integers are positive) $a \div b = \left\lfloor \frac ab \right\rfloor$. This corresponds to the integer division operator in many…
Nithin
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A curious sequence (Ascending and descending a staircase)

The following story is true, not just to make it sound mysterious or coincidental. I found a very curious sequence of integers, and searching it gave no result. I am trying to learn more about it, just for fun. We were taking a break with two…
Jonathan H
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Every natural number is covered by consecutive numbers that sum to a prime power.

Conjecture. For every natural number $n \in \Bbb{N}$, there exists a finite set of consecutive numbers $C\subset \Bbb{N}$ containing $n$ such that $\sum\limits_{c\in C} c$ is a prime power. A list of the first few numbers in $\Bbb{N}$ has several…
Abstract Space Crack
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A conjecture about numbers of the form $10^{m}(2^{k}−1)+2^{k-1}−1$, where $m$ is the number of decimal digits of $ 2^{k-1}$.

Question Numbers $n$ of the form $10^{m}(2^{k}−1)+2^{k-1}−1$, where $m$ is the number of decimal digits of $ 2^{k-1}$. For example: $k=1$ then $n=10$. $k=2$ then $n=31$. $k=3$ then $n=73$. $k=4$ then $n=157.$ Conjecture: the number…
Enzo Creti
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Induction proof. Explain in detail why it’s incorrect

Can somebody give a clear explanation why this is incorrect? thank you Theorem 1: All positive integers are equal. Proof: We show that any two positive integers are equal, from which the result follows. We do this by induction on the maximum of…
studymaths
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Does topology apply to the integers?

What is the natural topology (or topologies) on the integers. Can we define a metric on the integers?
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