Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [integers]

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

3
votes
2answers
49 views

Reaching higher even numbers in Goldbach's conjecture, using lower even numbers.

Let $n \in \Bbb{N}, n \gt 1$. Let $\Bbb{P} = $ the prime numbers in $\Bbb{N}$. Define \begin{align*} A_n &= \{ (p,q) \in \Bbb{P}^2 : p + q = 2n\}, \\ B_n &= \{ (p, q) : p - q = 2n \}. \end{...
3
votes
2answers
32 views

$p=a^2+ab+41b^2$ iff $-163$ is a quadratic residue

Prove that a prime $p$ can be written as $p=a^2+ab+41b^2$ iff $-163$ is a quadratic residue modulo $p$. What I have in mind is something like this: look at $\mathbb{Q}[\sqrt{-163}]$ which has the ...
0
votes
3answers
42 views

Integers divisible by 5 but not divisible by 7 are countably infinite

I have searched for the answer to this question online and have come across some answers, however, I am struggling to understand it. How would I come up with a function for which the integers are ...
0
votes
0answers
23 views

Determining if a variable or function is an integer

Fermat's last theorem states that No three positive integers $a, b$, and $c$ satisfy the equation $a^n + b^n = c^n$ for any integer value of $n$ greater than $2$. The cases $n = 1$ and $n = 2$ ...
0
votes
1answer
41 views

Relations between $k, t$ and modulo $n$

Suppose we have $n$ people in a circle $\{0, 1, ..., n-1\}$ Also, suppose we have another person who goes around said circle and gives each of the $n$ people a gift, one each $k$ people, so: \begin{...
1
vote
0answers
26 views

$3n = 8x+4$ control flag inside of a function

I want to be able to evaluate wether the result of a function satisfies an expression. Is this possible to do? Example I have an expression, a linear polynomial: $$8x+4$$ for $x\in\mathbb{N}$ If I ...
-1
votes
2answers
36 views

Integer solutions of $ X+Y+Z=X\cdot Y\cdot Z $ [on hold]

An integer solution of above equation is $(X,Y,Z)=(1,2,3)$. But I am wondering: are there other natural solutions? And what about rational or irrational solutions, where $X,Y,Z$ are different ...
-2
votes
1answer
27 views

Highest power of 4 that divides n

Let $n\in\mathbb{N}$ or $\mathbb{Z^+}$. I know there are some ways to compute/find the highest power of $2$ that divides $n$, there's the ruler sequence, and some ways to do this with boolean algebra ...
0
votes
0answers
25 views

Calculate the floor-function of the expression

Let n be a positive integer and x a real number with x $\ge$ $\frac {3n^2+1}{3}$. Calculate $\lfloor \sqrt{x^2-nx}+\sqrt{x^2-n^2}+\sqrt{x^2+n^2}-3x \rfloor$, where $\lfloor t \rfloor$ is the usual ...
-2
votes
1answer
39 views

If $m\neq n$ what is $\mathrm{gcd}(a^{2n}+1,a^{2m}+1)?$

If $m\neq n,$ compute $\mathrm{gcd}(a^{2n}+1,a^{2m}+1).$ In my question, $m$ , $n$ , and $a$ are positive integers.
0
votes
2answers
31 views

Proof: sum of even and odd integer is odd

Statement: Sum of even and odd integer is odd $$ \forall(a,b) \in \mathbb{Z} : a \text{ mod } 2 = 0 \wedge b \text{ mod } 2 \neq 0 \implies a + b \text{ mod } 2 \neq 0 $$ Proof: $$ a \text{ mod } 2 ...
0
votes
1answer
20 views

Name of a kind of sequence

Each line count the numbers in previous line: 1 1 1 2 1 1 1, 1 2 3 1, 1 2 2 1, 1 2, 1 3 3 1, 2 2, 1 3 2 1, 2 2, 2 3 1 1, 4 2, 1 3 What is the name of a sequence obtained with this method? Does ...
0
votes
0answers
28 views

Question about primes in $\mathbb{Z}_{n} $ [duplicate]

For example, I kind of expect that in $\mathbb{Z}_{6}$ $\overline{2}$ is prime but I don't know how to prove it (considering the fact that I can test all the different products, but I want a more ...
1
vote
0answers
58 views

Proving that $\frac{\sum_{n=1}^{t} n^{2r+1}}{\sum_{n=1}^{t} n}$ is an integer

I am trying to prove that $$\frac{\sum_{n=1}^{t} n^{2r+1}}{\sum_{n=1}^{t} n}$$ is an integer for $r,t\geq 1$. I have managed to show this for $r=1$ and $r=2$ but I don't know where to go from here. ...
0
votes
1answer
50 views

Question from an IMC(International mathematics competition) key statge III selection exam

A positive integer n does not have any 9 digits, it has four 8 digits, three 7 digits, two 6 digits and some other digits. If the sum of the digits of the numbeer n is 104 andthe sum of the digits of ...
1
vote
2answers
50 views

Finding integer solutions out of a a | b

Determine all positive integer values of (n) such that $$ { n \choose 0 } + { n \choose 1 } + { n \choose 2 } + { n \choose 3 } \ \bigg| \ 2 ^ { 2008 } $$ What is the sum of all these values? ...
0
votes
1answer
21 views

Approximate rounding to nearest integer with a continuous function

I am trying to come up with a family of continuous functions, which will approximate a rounding to nearest integer function. I came up with the following solution: $f(x)=x-\beta*\frac{\sin(2 \pi x)}{...
5
votes
2answers
74 views

Calculate the sum of fractionals

Let $n \gt 1$ an integer. Calculate the sum: $$\sum_{1 \le p \lt q \le n} \frac 1 {pq} $$ where $p, q$ are co-prime such that $p + q > n$. Calculating the sum for several small $n$ value I found ...
0
votes
0answers
31 views

Is there utility in a boolean function that flags the set of integers?

I'm working on a math that might allow defining a differentiable function using elementary expressions which yields a 1 if a real number is an integer, and a 0 otherwise. I'm just curious if there ...
7
votes
4answers
887 views

How is this property called for mod?

We have a name for the property of integers to be $0$ or $1$ $\mathrm{mod}\ 2$ - parity. Is there any similar name for the remainder for any other base? Like a generalization of parity? Could I use ...
1
vote
1answer
35 views

prove that $x^2+7y^2=4p$ has an integer solution if and only if $u^2+7v^2=p$ has an integer solution

I need to prove this for $p$ prime. Clearly $x^2+7y^2=4p$ implies $x$ and $y$ are odd, but I'm not sure where to go from there. I'm far too tired to think I've been working for 16 hours straight (not ...
0
votes
2answers
50 views

kangaroo maths competition [closed]

How many three-digit positive integers $ABC$ exist, such that $(A + B)^c$ is a three-digit integer and an integer power of $2$? Note: An integer power of $2$ is a number in the form $2^k$ , where $k$ ...
4
votes
2answers
94 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 ...
1
vote
0answers
28 views

Predicate logic: What are the differences between ∀ and ∃ when it comes to comparing two variables?

Say I have the four following logical statements, all over the domain of all integers. (∀a,∀b)[a>b] (∀a,∃b)[a>b] (∃a,∀b)[a>b] (∃a,∃b)[a>b] I feel like they're all asking practically similar things, ...
1
vote
1answer
52 views

Prove that $a^2 \equiv 4 \mod 3^m$ if and only if $a \equiv 2 \mod 3^m$ or $a \equiv −2 \mod 3^m$ [duplicate]

Let $m ≥ 1$, and let $a$ be an integer. Prove that $a^ 2 \equiv 4\mod 3^m$ if and only if $a \equiv 2\mod 3^m$ or $a \equiv −2\mod 3^m$. I know that i'm supposed to find $m$ factors $3$ in $a^ 2 − 4 =...
-2
votes
1answer
36 views

What integer solutions does the equation $abc + abd + acd + bcd = efg + efh + egh + fgh$ have? [closed]

For which sets {$a,b,c,d$}, {$e,f,g,h$} do we have $abc + abd + acd + bcd = efg + efh + egh + fgh$?
0
votes
2answers
60 views

To prove identity $P(n,3)= \lfloor n^2/12 \rfloor$

Suppose $P(n,k)$ is number of partitions of positive integer n by k positive integers with no duplicative tuples. And $\lfloor r\rfloor$ is largest of integers equal or less than real number $r$ If $...
-1
votes
0answers
30 views

What is wrong with the following proof of the well-ordering property?

Let $A = \{m\}$. Then $m\leq m$, so the base case holds. Now let $A = \{a_1, a_2, ..., a_n\}$ where $n$ is a positive integer. Pick $a_1 = m$. If $a_2 \leq m$, then $a_2 = m$. This continues until $...
0
votes
2answers
76 views

How many numbers from 0 to 99999 contain the digits 2, 5 and 8?

I have this problem: How many integers between 0 and 99999 contain the digits 2, 5 and 8? I've tried a lot, but I don't know how resolve it. P.S. The solution should be 4350.
0
votes
0answers
18 views

Detect intersection in constant time?

I'm working in an algorithm and one of the steps requires me to check whether or not 2 sequences of integers are what I call 'compatible' this is, their intersection is empty. I managed to make this ...
0
votes
2answers
37 views

Can someone give me a hint on how to prove this?

I'm supposed to prove that, for every integer $n > 0,$ it is true that $(1 + 2 + ... + n)$ divides $3(1^2 + 2^2 + ... + n^2)$. Should I use induction? This was given as an exercise in a chapter ...
3
votes
1answer
61 views

Proof by induction: $3^n|a_n\ $

$$Problem$$ Given $ (a_n)_{n\in N}$ the sequence defined by: $$a_1=15,\ a_2=18,\ a_{n+2}=6a_{n+1}-7a_n^4,\ \forall n\in \mathbb{N}$$ Prove by induction that $\forall n \in N$ a) $3^n|a_n$ b) $3^{...
1
vote
0answers
15 views

Non-reversible primitive operations on integers [closed]

Along the lines of How to map 256 unique strings to 256 unique but effectively arbitrary integers, I am wondering how to generate basically a hashing function. For this question I am wondering if ...
1
vote
2answers
47 views

Find the smallest value of positive integer !!

Find the smallest positive integer with exactly 30 positive factor First, I use function $\tau$ to find the exponential that gives $2×3×5$ and I want to find the smallest value. How can I find it use ...
2
votes
3answers
60 views

Is this really an integer solution to $n^{1.01} = n*\ln(n)$

Wolfram alpha says a very large number ($n > 10^{100}$) is an integer solution to $ n^{1.01} = n*\ln(n) $. I'm skeptical and have no way computationally viable way to verify this. Is it true?
3
votes
2answers
71 views

Prove there are infinitely many (x, y, z) positive integers satisfying $x^5 + y^7 = z^9$

Prove there are infinitely many (x, y, z) positive integers satisfying $x^5 + y^7 = z^9$ I have reduced the problem to finding only one solution $(x_0,y_0,z_0)$ and then using the fact that there are ...
0
votes
1answer
14 views

System of divisibility constraints

Let us consider integers $a,b$ and $c$ such that all of them are greater than 1. I am trying to figure out whether the following three divisibility conditions can be satisfied together $a|b^{a-1}$ $...
-2
votes
1answer
33 views

Could someone explain how to solve this problem?

Let $$f(x)=x^2-3$$ For how many integer values of x is $f(f(f(x)))$ divisible by x?
2
votes
0answers
34 views

Quotient cancellation for invertible ideals of orders in quadratic fields

Let $K/\mathbb{Q}$ be an imaginary quadratic field, $m\ge 1$ be a positive integer and let $\mathcal{O}=\mathbb{Z}+m\mathcal{O}_K\subset \mathcal{O}_K$ be the unique order of index (equivalently, ...
1
vote
1answer
88 views

How to find the integer solution for $6x + 25 = 7y$?

When I type this equation ($6x + 25 = 7y$) into WolframAlpha, it is able to tell me that the integer solution for this equation is: $x = 7n + 4$, $y = 6n + 7$, where n in the set of all integers ...
0
votes
0answers
30 views

Is there a natural number satisfying the given condition? [duplicate]

Is there a positive integer $n$ such that $\sum_{k=0}^{n}\sqrt{n+k}$ is also an integer?
2
votes
2answers
38 views

Find the number(s) $k$ such that $k\mathbb{Z} = \mathbb{Z}$

I am just starting out with Discrete Math and there is a question in my book that is Find the number(s) $k$ such that $k \mathbb{Z} = \mathbb{Z}$. The answer is -1 and 1. I understand why it is 1 ...
3
votes
4answers
96 views

Math formula to check two integers [closed]

I was just wondering if there is a way to check that two unknowns are integers as follows:- if x and y are two values, and I want to know if these two values are integers by using a formula, I tried ...
0
votes
2answers
55 views

A well-defined map from rational numbers to integers

I am trying to come up with a well-defined map from $\mathbb{Q}$$\to$ $\mathbb{Z}$ i.e. find a map $f$ such that it maps $\frac{a}{b}$ $\epsilon$ $\mathbb{Q}$ to an integer in $\mathbb{Z}$. I tried a ...
4
votes
0answers
69 views

For which even integers $k$ has $\varphi(n+1)-\varphi(n)=k$ a solution?

For which even integers $k$ does the equation $$\varphi(n+1)-\varphi(n)=k$$ have a solution ? $\varphi(n)$ denotes the totient function and $n$ is a positive integer. For the following $|k|\le 1\ ...
2
votes
6answers
67 views

Integer $a$ that divides $bc$ but does not divide $b$ or $c$

I know this to be true but I do not know why I can't prove it. Here is my proof so far: $a$ doesn't divide $b$ means $b = a k_1 + r_1$ for some $0 < r_1 < a$. $a$ doesn't divide $c$ means $c =...
1
vote
1answer
56 views

Find all $n$ such that the following is prime [closed]

Find all positive integers $n$ for which $(1+n+n^2+...+n^n)^2-n^n$ is prime.
1
vote
2answers
50 views

The number of prime pairs of $x^2-2y^2=1$

How to find the number of pairs of positive integers $(x,y)$ where $x$ and $y$ are prime numbers and $x^2-2y^2=1$? I am not getting any clue here.
1
vote
0answers
28 views

Determine the quotient and the remainder of the division:

Determine the quotient and the remainder of the division: ($1$).of $f\in \mathbb K[x]$ by $x^2-a$ in $\mathbb K[x],$Where $\mathbb K$ is a field. ($2$).of $x^m-1$ by $x^n-1$ in $\mathbb Z[...
-1
votes
2answers
74 views

When extending the natural numbers to the integers when is it legal to set a natural number equal to an integer.

My source BBFSK I need to add that natural numbers in this context are defined as starting with 1. I didn't think that would impact the answer, but apparently it does. $n-0$ provides a "bridge" ...