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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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 ...
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=...
4
votes
5answers
2k views

How to prove $\gcd(a^2,b^2) = (\gcd(a,b))^2$?

How to prove $\gcd(a^2, b^2) = (\gcd(a, b))^2$? My attempt: Let $\gcd(a, b) = d$. Then $d|a$ and $d|b$ then $d^2|a^2$ and $d^2|b^2$. i.e $d^2$ divides $a^2 ~~\&~~ b^2$.
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 $-1$. ...
9
votes
3answers
2k views

Closed form for $1^k + ... + n^k$ (generalized Harmonic number)

This question must have been asked, it's just very hard to search for such questions. I'm looking for the cleanest method I can find for getting a closed form formula for $\sum_{i=1}^n i^k$ ...
7
votes
5answers
4k views

What is a natural number? [duplicate]

According to page 25 of the book A First Course in Real Analysis, an inductive set is a set of real numbers such that $0$ is in the set and for every real number $x$ in the set, $x + 1$ is also in the ...
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 ...
10
votes
4answers
3k views

Difference between $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}_n$

In every Modern Algebra book I've read, I've seen that the groups $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}_n$ are isomorphic, but not equal. I understand the difference between "isomorphic" and "equal,...
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 ...
10
votes
5answers
1k views

Is $1234567891011121314151617181920212223......$ an integer?

This question came from that one and from that talk where it's noted that "integers have a finite count of digits", so that the "number" in the title is not at all a number (not integer nor rational ...
3
votes
3answers
2k views

If $a,b$ are positive integers such that $\gcd(a,b)=1$, then show that $\gcd(a+b, a-b)=1$ or $2$. and $\gcd(a^2+b^2, a^2-b^2)=1$ or $2 $

If $a,b$ are positive integers such that $\gcd(a,b)=1$, then show that $\gcd(a+b, a-b)=1$ or $2$ and $\gcd(a^2+b^2, a^2-b^2)=1$ or $2 $. Progress We have $\gcd(a,b)=1\implies \exists u,v\in \mathbb ...
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.
9
votes
3answers
2k views

Number of integer triplets $(a,b,c)$ such that $a<b<c$ and $a+b+c=n$

What is an equivalent combinatorial presentation for the problem? Can I use the stars and bars approach to find the number of integral solutions of $a+b+c=n$ where $a,b,c\geq 0$? If in addition $a+b&...
4
votes
5answers
224 views

Show that if $a \neq b$ and a and b are positive then $\frac{a}{b}+\frac{b}{a}$ is never an integer

Some observations I made is for $\frac{a}{b}+\frac{b}{a}$, is either: the denominator has to be one, the numerator has to be a multiple of the denominator or the numerator and denominator have to be ...
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 ...
2
votes
2answers
301 views

Is getting a random integer even possible?

On googling I got Random numbers are numbers that occur in a sequence such that two conditions are met: (1) the values are uniformly distributed over a defined interval or set, and (2) it is ...
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 ...
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 $...
1
vote
2answers
149 views

A polynomial with integer coefficients that attains the value $5$ at four distinct points

There is a polynomial $f$ of integer coefficients such that $\deg(f) \geq 4$. Let's assume that there are four integers $a,b,c,d$ for which $f(a)=f(b)=f(c)=f(d)=5$. Prove that there is no integer $k$ ...
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+...
1
vote
1answer
322 views

Prove $\text{lcm}(a,b,c) = \frac{a \cdot b \cdot c \cdot \gcd(a,b,c)}{\gcd(a,b)\gcd(b,c)\gcd(a,c)}$ [closed]

With $\{a,b,c\}\subset \mathbb{Z}$ given that $\text{lcm}(a,b) = \frac{a \cdot b}{\gcd(a,b)}$. Prove that: $$\text{lcm}(a,b,c) = \frac{a \cdot b \cdot c \cdot \gcd(a,b,c)}{\gcd(a,b)\gcd(b,c)\gcd(a,...
6
votes
1answer
713 views

Reducibility of $x^3+nx+1$ over $ \Bbb Z$

For what values of $n$, where $n$ is an integer, the polynomial $x^3+nx+1$ is reducible over $\Bbb Z$. My attempt: When $n= 0,-2 $, the given polynomial is reducible over $\Bbb Z$ as $x=-1$ and $x=1$ ...
3
votes
1answer
188k views

What three odd integers have a sum of 30? [duplicate]

I've been asked the following question: What three odd integers from the set {1,3,5,7,9,11,13,15} that when summed together equals to 30? Note that any integer can be used more than once. Is there ...
19
votes
1answer
533 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 $[0,|b|).$ ...
10
votes
1answer
769 views

Prove that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$

I am having difficulty proving the following inequality: $$ \sqrt[n]{n} < 1 + \sqrt{\frac{2}{n}} \quad \text{for all positive integers}\,\,\, n. $$ I am trying to use mathematical induction but I ...
2
votes
2answers
961 views

Determine all integral solutions of $a^2+b^2+c^2=a^2b^2$ [duplicate]

I started like this : $a^2+c^2=b^2(a^2-1)\\c^2 +1=(a^2-1)(b^2-1)$ but it's leads to nowhere. can you help please ?
4
votes
1answer
2k views

Calculating a Factorial Base Representation

My friend thought of a system in which each number $n$ (I will first restrict my question to positive integers $n$) is represented by a digit string $(d_l,...,d_1)$ as follows $\forall n \in \mathbb{N}...
0
votes
5answers
5k views

Prove that $n^3=n \text{ mod }6$ for every integer $n$. [duplicate]

Prove that for every integer $n$ , $n^3=n \text{ mod }6$ I was having no clue how to do this, then I thought of case-by-case analysis and obviously it worked. The problem is that there were six case ...
3
votes
6answers
129 views

For what integer values of $y$ is $\frac{3y-1}{y-3}$ an integer?

I have encountered this as part of a bigger problem but I really don't know how to go on about it. I would also appreciate it if you could specify a certain technique to follow when facing such a ...
1
vote
5answers
99 views

Prove that $a_1!\cdots a_k! < n!$ whenever $a_1+\cdots+a_k < n$

Prove that for positive integers $a_1,\dots,a_k$ (where $k\geq 1)$ are such that $a_1+\cdots+a_k < n$ , then $a_1!\cdots a_k! < n!$. So far, i've tried adding the condition that suppose they ...
1
vote
5answers
135 views

If $2x + 3y$ is multiple of $17$, then $9x + 5y$ is multiple of $17$

I want proof that: If $2x + 3y$ is multiple of $17$, then $9x + 5y$ is multiple of $17$. My attempt: By the Bezout's theorem, how $mdc(2, 3) = 1$, exists $r, s \in \mathbb{Z}$ such that $$2r + 3s ...
0
votes
1answer
82 views

Show $\sum_{k=1}^n \frac{1}{k^2} \le 2$ and $\ln(n!) \ge 1 -n+n\ln(n)$ for all positive integers n

I have been trying to show these two inequalities hold for all positive integers n, but I don't know how to proceed at all... I have tried playing around with them but I didn't find anything helpful. ...
0
votes
2answers
90 views

Division in $1$ variable

Let the sum of all integers $n$ such that $(2n^2+9/n+3)$ is an integer be $A$, what is $|A|$? I tried expressing it to: $k$ is an integer, $(n+3)k + 0 = 2n^2 + 9$, $n(2n-k) + 3(3-k) = 0$, from which ...
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. ...
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 ...
8
votes
3answers
479 views

Prove that $x^x+y^y=z^z$ doesn't have integer solutions

Prove that $x^x+y^y=z^z$ doesn't have integer solutions To be honest, I don't see any way to start this problem, I tried for hours but it's not as easy as I thought. Any hints? As you can see in ...
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{...
9
votes
3answers
683 views

Find the limit $\lim_{n\to\infty}\left(\sqrt{n^2+n+1}-\left\lfloor\sqrt{n^2+n+1}\right\rfloor\right)$ [duplicate]

$$\lim_{n\to\infty}\left(\sqrt{n^2+n+1}-\left\lfloor\sqrt{n^2+n+1}\right\rfloor\right)\;=\;?\quad(n\in I) \\ \text{where $\lfloor\cdot\rfloor$ is the greatest integer function.}$$ This is what I did: ...
8
votes
2answers
7k views

Convert a piecewise linear non-convex function into a linear optimisation problem.

Update: Problem and solution found here (p. 17, 61), although my prof's solution (formulation) is different. Convert $$\min z = f(x)$$ where $$f(x) = \left\{\begin{matrix} 1-x, & ...
2
votes
3answers
132 views

Diophantine equation power of 7 and 2

$$ 7^x = 2^y \cdot 3 + 1$$ Find all positive $(x,y) \in \mathbb{N}^2$ When I look at this equation $\mod 3$ or $\mod 7$ it does hold - but how can I continue from here? I know that $7^x -1$ is even so ...
10
votes
3answers
1k views

Find all functions of positive integers for $f(f(n))=n+2$

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with induction, which yields the shortest, simplest proofs for proving the ...
7
votes
2answers
137 views

Relation between primeness and co-primeness of integers

I wonder what this stunning formal analogy between the definitions of being co-prime (for two integers) and being prime (for one integer) might reveal – and how: $\alpha, \beta$ are co-prime ...
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, ...
6
votes
1answer
122 views

Compute the kernel of the group hom $\Omega : \Bbb{Q}^{\times} \to \Bbb{Z}^+$.

The $\Omega$ function is the counting function that returns precisely the number of primes $\Omega(n)$ (including multiplicity) that divide a natural number $n \in \Bbb{N}$. For example $\Omega(6) = ...
5
votes
4answers
474 views

Can we prove the existence of a gcd in $\mathbb Z$ without using division with remainder?

For $a,b\in\mathbb Z$ not both $0$, we say $d$ is a gcd of $a$ and $b$ if $d$ is a common divisor of $a$ and $b$ and if every common divisor of $a$ and $b$ divides $d$. With this definition, can we ...
5
votes
3answers
4k views

Find all ordered pairs $(a,b)$ such that $1/a + 1/b = 3/2018$ and $a,b$ are positive integers

I gave this problem my best attempt and am now trying to understand the solution for it. This is problem #1 to the 79th William Lowell Putnam Math Competition. This is the given solution by Kiran ...
4
votes
3answers
440 views

$a+b+c=x+y+z$ and $abc=xyz$ , in which each two of them unequal.

$a,b,c,x,y,z\in \mathbb Z$ ,they are all positive. And not equal to each other. Let $a>b>c>0,x>y>z>0$ $$ \begin{cases} a+b+c=x+y+z\\ abc=xyz \end{cases} $$ now I have try out $1+8+...
4
votes
3answers
2k views

Find three real orthogonal matrices of order $3$ having all integer entries. [closed]

Find three real orthogonal matrices of order $3$ having all integer entries. I have no idea to solve the problem. I don't know how to start. If $A$ be such matrix then $AA^T=A^TA=I_3$. Please help me....
4
votes
1answer
113 views

Finding integral points on an elliptic $y^2-3y=x^3+x^2$ curve using the LMFDB-database

I have the following elliptic curve that I want to look up in the LMFDB-database: $$\text{k}:\space\space\space y^2-3y=x^3+x^2$$ Using the Weierstrass form of my elliptic curve, I wrote my equation ...

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