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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115
votes
11answers
9k views

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 ...
113
votes
15answers
17k views

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 ...
47
votes
10answers
9k views

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). ...
39
votes
9answers
7k views

Can sum of a rational number and its reciprocal be an integer?

Can sum of a rational number and its reciprocal be an integer? My brother asked me this question and I was unable to answer it. The only trivial solutions which I could think of are $1$ and $-1$. ...
35
votes
5answers
38k views

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{...
35
votes
1answer
319 views

"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 ...
34
votes
4answers
1k views

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 $n$...
33
votes
2answers
13k views

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 ...
32
votes
6answers
9k views

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. ...
30
votes
6answers
6k views

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 ...
30
votes
6answers
4k views

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 ...
28
votes
1answer
1k views

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, ...
24
votes
5answers
6k views

Induction proof. Explain in detail why it’s incorrect [duplicate]

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 ...
24
votes
2answers
570 views

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 ...
24
votes
2answers
8k views

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=...
22
votes
19answers
1k views

Special properties of the number $146$

I'm a math teacher. Next week I'll give a special lecture about number theory curiosities. It will treat special properties of numbers — the famous story with Ramanujan, taxicab numbers, later numbers ...
22
votes
7answers
2k views

How can I find integers which satisfy $\frac{150+n}{15+n}=m$?

Here are some facts about myself: In 2017, I was $15$ years old. Canada, my country, was $150$ years old. When will be the next time that my country's age will be a multiple of mine? I've toned ...
21
votes
6answers
8k views

Does topology apply to the integers? [closed]

What is the natural topology (or topologies) on the integers. Can we define a metric on the integers?
21
votes
3answers
2k views

Prove that the number 14641 is the fourth power of an integer in any base greater than 6?

Prove that the number $14641$ is the fourth power of an integer in any base greater than $6$? I understand how to work it out, because I think you do $$14641\ (\text{base }a > 6) = a^4+4a^3+6a^2+...
21
votes
1answer
697 views

The equation with binomial coefficient $\binom{n-m}{k+m}=\binom{n+m}{k-m}$

Find all positive integers $n,k$ such that $$\binom{n-m}{k+m}=\binom{n+m}{k-m}$$ 1) I solved problem if $m=1$. Its here: $k=1; n=3$ 2) $$\binom{n-m}{k+m}=\binom{n+m}{k-m}$$ $k=m, n=3m$ is root of ...
19
votes
3answers
2k views

proving that the area of a 2016 sided polygon is an even integer

Let $P$ be a $2016$ sided polygon with all its adjacent sides perpendicular to each other, i.e., all its internal angles are either $90$°or $270$°. If the lengths of its sides are odd integers, prove ...
19
votes
1answer
534 views

Is $\mathbb{Z}$ the only totally-ordered PID that is "special"?

(All my rings are commutative and unital.) Definition. Call a totally-ordered ring $R$ special iff for all non-zero $b \in R,$ every coset of $bR$ has a unique element in the interval $[0,|b|).$ ...
18
votes
1answer
912 views

How to prove a double sum is always an integer?

I have verified the following double sum is always an integer for $s$ up to $1000$ via Maple. But I can not prove it. Proofs, hints, or references are all welcome. Thanks! $$\sum_{m=s}^{2s}\sum_{k=0}^{...
17
votes
10answers
3k views

Square root of $2$ is irrational

I am studying the proof that $\sqrt 2$ is an irrational number. Now I understand most of the proof, but I lack an understanding of the main idea which is: We assume $\frac{m^2}{n^2} = 2$. Then ...
17
votes
3answers
352 views

What are all the integral solutions of $n!=m(m^2-1)$?

Observe that: $3!=2(2^2-1)$ $4!=3(3^2-1)$ $5!=5(5^2-1)$ $6!=9(9^2-1)$ Question: What are all the integral solutions of $n!=m(m^2-1)$? I guess it is just $(n,m) = (3,2),(4,3),(5,5),(6,9)$, but how to ...
16
votes
2answers
3k views

How to prove that the product of eight consecutive numbers can't be a number raised to exponent 4?

How to prove this? I tried something like $$P(n,8)=\frac{n!}{(n-8)!} = b^4$$ but I can't proceed to a solution.
15
votes
8answers
4k views

Is there a quick way to write say positive integers in an interval in mathematical notation?

For example I find myself wanting to write $x$ is an element of the integers from $1$ to $50$, Is this the quickest way? $x\in \left[ 1,50\right] \cap \mathbb{N} $ Also is this standard on here? $\...
14
votes
2answers
7k views

What is the remainder when $1! + 2! + 3! +\cdots+ 1000!$ is divided by $12$?

What is the remainder when $$1! + 2! + 3! +\cdots+ 1000!$$ is divided by $12$. I have tried to find the answer using the Binomial Theorem but that doesn't help. How will we do this? Please help.
14
votes
3answers
37k views

What is the difference between natural numbers and positive integers?

I was reading sets and came to some reserved letters for a few sets. Two of them really confused me. They were - $\mathbb N$ : For the set of natural numbers. $\mathbb Z^+$ : For the set if all ...
14
votes
1answer
282 views

Does $\operatorname{product-of-digits}(n)=n/3$ have at least one solution in any bases except for the power of $3$?

Define a function $P_b(x)$ as $$P_b(x)=\text{"the product of digits of x in base b"}$$ Is it true that for any $b\in\mathbb{N}$, if $b$ is not a power of $3$, then there exists at least one positive ...
13
votes
7answers
12k views

Proof of Lemma: Every integer can be written as a product of primes

I'm new to number theory. This might be kind of a silly question, so I'm sorry if it is. I encountered the classic lemma about every nonzero integer being the product of primes in Ireland and Rosen's ...
13
votes
6answers
1k views

Is every integer a unary operation?

I have been thinking about mathematical operations, and I was trying to ponder what an operation simpler than addition would be. I started thinking about what each operation did (exponents: repeated ...
13
votes
4answers
4k views

Proving that a number is an integer.

Prove that the following number is an integer: $$\left( \dfrac{76}{\dfrac{1}{\sqrt[\large{3}]{77}-\sqrt[\large{3}]{75}}-\sqrt[\large{3}]{5775}}+\dfrac{1}{\dfrac{76}{\sqrt[\large{3}]{77}+\sqrt[\large{3}...
13
votes
1answer
2k views

The word "integral" in calculus unrelated to "integral" / "integer" in algebra?

I think that the word integral in calculus is nothing to do with integer or integer numbers. But why is integral is chosen for integration? In algebra, integral means related to integers, and this is ...
13
votes
1answer
280 views

Zero joint subsums of integers

Fix integers $x_1,y_1,\ldots,x_{10},y_{10}\in [-2,2]$ such that $$ \sum x_i=\sum y_i=0\,. $$ Then, does it necessarily exist a nonempty proper subset $J$ of $\{1,\ldots,10\}$ such that $$ \sum_{j \...
13
votes
5answers
3k views

Intuitive/direct proof that a rectangle partitioned into rectangles each with at least one integer side must itself have an integer side

A challenge problem asked to show that if rectangle $R$ with axis-parallel sides is partitioned into finitely many subrectangles $R_1,R_2,\ldots,R_n$ (also with axis-parallel sides), such that each $...
13
votes
2answers
136 views

Are Transitions in a Hydrogen Atom Unique

So there was a question on a past exam paper of a test I have recently taken and despite the test being over I feel the need to know the answer. I am a physics major and the test was a generic test on ...
13
votes
2answers
493 views

Prove the equation $\left(2x^2+1\right)\left(2y^2+1\right)=4z^2+1$ has no solution in the positive integers

Prove the equation $$\left(2x^2+1\right)\left(2y^2+1\right)=4z^2+1$$ has no solution in the positive integers My work: 1) I have the usually problem $$\left(nx^2+1\right)\left(my^2+1\right)=(m+n)z^...
12
votes
3answers
290 views

Can $\sqrt{ab}$ and $\sqrt{a^2 + b^2}$ be both integers if $a$ and $b$ are natural numbers?

Does there exist an $a \in \mathbb{N}$ and $b \in \mathbb{N}$ such that $\sqrt{ab} \in \mathbb{Z}$ and $\sqrt{a^2 + b^2} \in \mathbb{Z}$?
12
votes
2answers
1k views

How to find all polynomials that map integers to integers?

How can one find all polynomials of degree k that map integers to integers? In other words, how to get all combinations of coefficients $a_0,...,a_k \in \Bbb R$ ...
12
votes
2answers
1k views

When are quadratic rings of integers unique factorization domains?

Let $D$ be a square free integer. Let $R_D$ be the integral closure of $\mathbb{Z}$ in the field $\mathbb{Q}(\sqrt{D})$. For some values $D$, the ring $R_D$ is a $UFD$, but not for all. For example, ...
12
votes
1answer
383 views

What's special about 323 and squared rectangles?

The minimal number of squares for rectangles up to longest side 380 is known. The data was calculated for the question "tiling a rectangle with the smallest number of squares". I took a look at hard ...
11
votes
5answers
9k views

Notation for every odd integer number

I have this equation: $$f(x)=\tan(x)$$ I found the vertical asymptotes to be: $$x=\frac{\pi}{2}k$$ What is the proper notation for that k is equal to every odd number integer(negative,...
11
votes
2answers
9k views

Why is the set of integers with the operation of addition considered a cyclic group?

The first sentence in the Wikipedia article entitled "Cyclic Groups" states that "In algebra, a cyclic group is a group that is generated by a single element". How is this consistent with addition on ...
11
votes
3answers
1k views

Proving the area is irrational for triangle with integer vertices

Question: $A,B,C$ are three non-collinear points lying on a plane whose normal vector is $(\hat{i}+\hat{j}+\hat{k})$. If all the three coordinates of every point is an integer, then prove that the ...
11
votes
4answers
1k views

Is every property of the integers provable?

I've been researching provability of properties, and I came across and interesting argument which states that every property of the integers is provable, yet doesn't the incompleteness theorem tell us ...
11
votes
1answer
406 views

Collatz divide by -2 instead

I've been toying around with the Collatz conjecture for a while, and in an effort to extend it to the negative integers I tried diving by $-2$ instead of by $2$. The new iteratively applied function ...
11
votes
2answers
656 views

For which values of $\theta$ does this equation $x^{\cos\theta} +y^{\sin\theta }=1$ have solutions in integers?

For which values of $\theta$ does this equation $$x^{\cos\theta} +y^{\sin\theta}=1$$ have solutions in integers ? Note : $x, y$ integers, $\theta$ is real number. Thank you for your help.
11
votes
4answers
273 views

Does there exist integers $a, n > 1$ such that $1 + \frac{1}{1 + a} + \frac{1}{1 + 2a} + ... + \frac{1}{1 + na}$ is an integer?

Does there exist integers $a, n > 1$ such that $1 + \frac{1}{1 + a} + \frac{1}{1 + 2a} + ... + \frac{1}{1 + na}$ is an integer? I have no clue how to begin. I've tried to simplify this somehow, but ...
11
votes
2answers
423 views

Is there a less-trivial integer function with described properties?

To be found are integer one-valued functions $f(n_1,n_2)$ with following properties: $f(n_1,n_2)=f(n_2,n_1)$, $f(f(n_1,n_2),n_3)=f(n_1,f(n_2,n_3))$, $f(n_1+n_2,n_1+n_3)=n_1+f(n_2,n_3)$. So far I ...

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