Stack Exchange Network

Stack Exchange network consists of 174 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 $\mathbb{Z}$.

0
votes
0answers
12 views

Can we re-read natural numbers as a “binary operation” of Integers Numbers? [on hold]

I start from Binary Operator definition I see an affinity between this definition and the 'difference' between Integers and Natural numbers. I ask if I can value in this way +1 -1 as $x, y$ |1| as $...
0
votes
2answers
41 views

Find the cardinality of set $\{(x,y,z):x,y,z \in \mathbb{Z}^+ \wedge xyz=108\}$

Find the cardinality of set $\{(x,y,z): x,y,z\text{ are positive integers and }xyz=108\}$ my attempt: I write all factors of $108$. $$1,2,3,4,6,9,12,18,27,36,54,108.$$ First observe that $xyz $ has ...
6
votes
4answers
63 views

Need help with the proof {needed for my algorithm}

I am not from pure math background. I am working on an algorithm which works good for all the practical reasons based on the following assumption. that, if ab = cd and a+b = c+d then, either a = c ...
1
vote
0answers
38 views

Identifying a group involving transformation of 4 integers

I'm looking at some messy data, and I found by accident that some properties seem preserved under some transformations of a subset of the variables. After some work I think I found a base set of ...
0
votes
3answers
59 views

Find the pairs $(m,k)$ solving the diophantine equation $m^2=7k+9$

Solve $$m^2=7k+9$$ over the integers First i rearrange got $m^2-9=7k$ And $(m^2-9)/7=k$ So first $m^2-9$ must be divisible by $7$ So suppose $m=7n , 7n+1 ,...,7n+5$ But it doesn't work ....
1
vote
1answer
70 views

what's the best programming language (and development community) for integer mathematics? [closed]

I was wondering what's the best programming language (and development community) for integer mathematics? I'm an R supporter. However, I'm interested in primality testing and integer factorization, ...
3
votes
2answers
54 views

Why do Fibonacci Digits always return to either 11 or 14?

I recently posed a question on Code Golf on what I'm terming "Fibonacci Digits", where each number is the sum of the previous two digits, not necessarily the previous two numbers. So, for instance, ...
0
votes
0answers
19 views

On largest box expected size not containing integer vector solutions

I am trying to understand the largest cube in $\Bbb Z^8$ around origin not containing integer vector $(a,b,c,d,a',b',c',d')$ solutions to $$aBDE+bBCE+cADE+dACE+a'BDF+b'BCF+c'ADF+d'ACF=0$$ where $A,B,C,...
0
votes
0answers
31 views

How to calculate the probability in the following problem?

Let us consider $f(x)=202+37x+243x^2+a_3x^3$$\pmod{257}$, where $a_3$ is randomly chosen from $\Bbb{Z}_{257}$. I want to calculate all such $S=\{f(1),f(2),\dots, f(n)\}, n<255$ such that $f(t)\...
1
vote
2answers
54 views

Determinant of a Hankel matrix with sequence (1, 2, 3,…, n, 1, 2,…, n-1)

I'm looking for a closed form of the determinant of matrices like $\begin{bmatrix}1 & 2 & 3\\2 & 3 & 1\\ 3& 1 &2\end{bmatrix}$ or $\begin{bmatrix}1 & 2 & 3 &4\\...
1
vote
3answers
25 views

Proof in mirror numbers / multiples of 3

Happy 2019, What's the proof for: any integer number subtracted by its mirror number is always a multiple of 3? This is, abfc – cfba will always be a multiple of 3 (abfc, integer number). Thanks
-1
votes
1answer
44 views

$\omega^\omega$ correspondence with $\mathbb R$-irrationality

Here in the second comment I do not understand why $\omega^\omega$ corresponds to irrational numbers? : In my experience one typically identifies $ω^ω$ with the irrational elements of R; and then we ...
0
votes
0answers
21 views

Identity involving floor, ceiling and nearest integer functions

For $n\geqslant0$, $m>0$, $s>t\geqslant0$, $n,m,s,t$ - integers we have $$\sum\limits_{k=0}^{m-1}\left\lfloor{n+ks+t\over ms}\right\rfloor=\left\lfloor{n+t\over s}\right\rfloor$$ $$\sum\limits_{...
2
votes
1answer
86 views

Goldbach conjecture: Every integer $n>3$ is halfway between $2$ primes.

Prove that the following conjecture is equivalent to the strong Goldbach conjecture: Every integer $n>3$ is halfway between $2$ primes. I'm able to prove it, but i don't have much experience in ...
3
votes
0answers
93 views

Methods for proving a function outputs an infinite number of integers

I have a function involving polynomials and the centre of the Binomial Triangle and I'd like to prove that the function produces a positive integer infinitely many times. I don't have any interest in ...
0
votes
2answers
54 views

Find common factors of three rounded integers

I'm trying to find the common factor(s) of three integers which have each been rounded up to the next multiple of 8: i = round(n * a) j = round(n * b) k = round(n * c) I'm given i, j, and k and I'm ...
-2
votes
0answers
151 views

All integers from 1 to 73 are recorded in a sequence such that each number

All integers from 1 to 73 are recorded in a sequence such that each number (from the second onwards) divides the sum of all previous numbers. What numbers can be in the third place and why? The ...
5
votes
5answers
69 views

Show that $\gcd(a, 0)$ exists and equals $|a|$ for all $a$ in $\mathbb Z$

I came up with the proof in the paragraph below. My question is about how I expressed the proof, and about the first part of the question above. For one, my proof seems to me very wordy compared to ...
1
vote
1answer
53 views

$\omega^\omega$ correspondence with $\mathbb R$

How does the natural continuous bijection between $\omega^\omega$ and $\mathbb R$ look like? I.e. why elements of $\omega^\omega$ are called reals?
3
votes
1answer
57 views

Representation of an integer / Euclidean algorithm

Let $r \in \mathbb{N}$ be a natural number. Let $$L \geq 2(r-1)²$$ A paper (on quantum information theory, I'm not an expert in number theory or so...) I'm recently reading now says "One can easily ...
0
votes
0answers
23 views

Trying to extend distributive property of modulo operation to real numbers

Here Wikipedia states that modulo operation is distributive: $$a \cdot b\ mod\ n = (a\ mod\ n)\cdot (b\ mod\ n)\ mod\ n$$ Which is true for every natural number. Unfortunately it is not for rational ...
2
votes
3answers
33 views

Division Algorithm and Negative Divisor

I'm learning some basic number theory from Strayer's 'Elementary Number Theory.' I've arrived at what seems to be a very basic problem, albeit complete with a nasty twist: Let $a,b,c \in \mathbb Z$ ...
0
votes
0answers
21 views

Positive solution of a linear system with integer coefficients

Suppose that $b_1,\dots,b_m \in \mathbb Z^n$ are not linearly independent over $\mathbb Z$ (otherwise the problem is trivial). Given another element $w\in \mathbb Z^n$ is there a way to determine (...
12
votes
2answers
106 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 ...
1
vote
0answers
23 views

Sequence of Number System Construction

After constructing the naturals, why construct integers before rationals? Is there a historical explanation? Couldn't ordered pairs of fractions constructed from the naturals be used to represent ...
0
votes
0answers
33 views

Roots of polynomials are Gaussian integers

I have got a question. I want to show the following: Let P be a normalized polynomial with integer coefficients and let w be a root of this polynomial (in $\mathbb{Q}[i]$), then w is a Gaussian ...
19
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 ...
3
votes
2answers
44 views

Factorization in prime elements of $\mathbb{Z}\left[\sqrt{p}\right]$ for a prime number $p$

I'm having troubles with the following problem: Let $p$ be a prime number in $\mathbb{Z}$, and $\alpha\in\mathbb{Z}\left[\sqrt{p}\right]$ which is not a unit. Prove that $\alpha$ have a factorization ...
0
votes
1answer
26 views

How to prove #R + #P = #R

I have already started this. I redefined the Reals as the Reals minus the Positive Integers (to make the two sets disjoint) so that I could prove that #(R - P) + #P = #R. I know that to prove this I ...
9
votes
1answer
136 views

Is it possible to mathematically skip numbers containing 666?

I once had a project which involved taking actual real social security numbers and anonymizing them into unique real-looking fake SSNs. One of the rules for SSNs is that it cannot contain a run of 666 ...
0
votes
0answers
16 views

How many balls will be left at the end of this process?

Consider having $N$ colored balls. Each color has at least $N/2k$ and at most $N/k$ balls in the beginning, for some parameter $k\ll N$. At each iteration, we remove $k$ balls with different colors, ...
1
vote
1answer
58 views

Natural numbers as a subset of integer numbers: $\mathbb{N}\subset\mathbb{Z}$.

Within set theory, having the natural numbers $\mathbb{N}$ built as the minimal inductive set with the corresponding additive and multiplicative operations defined, integers $\mathbb{Z}$ can be set as ...
2
votes
0answers
57 views

When do integers satisfy simple quadratic relation?

$$((w+x)(y+z)-(w-x)(y-z))^2=4(wy+xz)^2-4(w^2-x^2)(y^2-z^2)$$ is true and so if $(a-b)^2=4(e+f)^2-4ab$ then under what additional minimal conditions can we say $e=wy$ and $f=xz$ while $a=(w+x)(y+z)$ ...
1
vote
2answers
25 views

Smallest integer power for an inequality to hold

so I have this inequality: Given integers $m, k\geq1$. $$2^{m/k} > \frac{3}{2}$$ I'm interested in finding the smallest integer power $m$, as a function of $k$, that will make this inequality ...
1
vote
1answer
19 views

if $n = a_1a_2 \cdots a_r + 2$, then $a_i \nmid n$ for each integer $i (1 \leq i \leq r)$.

Let $a_1, a_2, \cdots , a_r$ be odd integers where $a_i > 1$ for $i = 1, 2, \cdots , r$. Prove that if $n = a_1a_2 \cdots a_r + 2$, then $a_i \nmid n$ for each integer $i (1 \leq i \leq r)$. Let $...
6
votes
2answers
71 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, ...
0
votes
1answer
30 views

Two sequences share a common (yet unknown) term. What is it?

Say you have two sequences which are given by a polynomial of some degree. Both of these sequences share a term. Is there a way to find the missing term, and what is the minimum number of values (or ...
2
votes
3answers
48 views

Quickest way to find a number between 0 and 100 if you can verify if it's bigger (or smaller) than another number

If there is a number somewhere between 0 and 100 and you have to find it with the least attempts possible. Every attempt consists of you checking if the number is smaller (or bigger) than a number in ...
0
votes
0answers
22 views

Poster Block Heading

In my poster, I review some binary operations from addition, subtraction, and multiplication. The title of the Example's block is the following: Operations in the Integers. The following examples: ...
1
vote
2answers
70 views

Show that these non-linear recursions produce integers only

The recurrence is of third order: Start with \begin{align*}a_0(x)&=1\\ a_1(x)&=1\\ a_2(x)&=x \end{align*} and then \begin{align*}a_{n+3}(x)&=\frac{a_{n+2}^2(x)-a_{n+1}^2(x)}{a_{...
0
votes
1answer
36 views

Existence of $T$ such that $[(T\circ T)(f)](n)=f(n+1)$

Defined $A$ as the set of all the functions $f:\mathbb Z\rightarrow \mathbb R$ exists a function $T:A\rightarrow A$ such that $$ (T^2f)(n)=f(n+1)\text{ for all }n\in\mathbb Z\text{ and }f\in A $$ or ...
1
vote
1answer
24 views

Uniqueness of set where subset exactly sums to 1

Let $A = \{a_1, a_2, \ldots, a_I\}$ be a set of real numbers, where for all $i \in \{1,2,\ldots,I\}$, $0<a_i\le1$ and $\sum_{j \ne i} a_j \ge 1$. I am interested in the set (or sets) $A$ ...
-1
votes
1answer
33 views

$f(x,y)=(x/2^y) \mod 16$ a Bivariate function?

I have a two-input function on the integers or naturals. Is a bivariate function a function that takes two inputs or is there anything more to it? For example take the function: $f(x,y)=(x/2^y) \mod ...
1
vote
2answers
56 views

When is $\prod_{n=2}^{k} n-\frac{1}{n}$ not an integer?

For which values of $k$ does the following product not evaluate to an integer: $$\prod_{n=2}^{k} n-\frac{1}{n}$$ I find it somewhat surprising that it not only can evaluate to an integer, but also ...
0
votes
2answers
20 views

Equivalence Relation with dividing integers

Define $x\sim y$ if $4$ divides $(x+3y)$ for $x$ and $y$ integers. Show that is an equivalence relation. Equivalence relation means it satisfies reflexity, symmetry, and transitivity. Can anyone ...
0
votes
0answers
21 views

“Intersection” of quotient groups, modulo $\mathbb{Z}^n$

Let's say that $M,N$ are two invertible $n\times n$ integer matrices, such that all their eigenvalues are greater than $1$ in magnitudes. Then we define $$K_M:=(M^{-1}\mathbb{Z}^d)\cap[0,1)^n,\quad ...
1
vote
7answers
62 views

How can you prove that $b=2, m=3$ are the only positive integer solutions to $4b+3m=17$?

How can you prove that $b=2, m=3$ are the only positive integer solutions to $4b+3m=17$ without a proof by exhaustion?
2
votes
1answer
32 views

Sum of squares related with integers

Let $x_1,x_2,x_3,...,x_{19}$ be positive integers satisfying $\sum\limits_{i=1}^{19}x_i=2020$ and $x_i \geq 2$ for $i=1,2,...,19$. Find the smallest value of $$P=\sum\limits_{i=1}^{19} x_i^2.$$ I've ...
0
votes
2answers
24 views

Is it possible to find all arithmetic progressions that exist in a set of three integers?

Is it possible to find all arithmetic progressions that exist in a set of say 3 integers. I know that the simplest arithmetic progression would be $$ a_n = a_1 + (n-1)d\:\text{ with }\:d = 1. $$ ...
0
votes
1answer
33 views

Difference between negative and positive temperatures written as absolute value or the negative integer?

**In a chemistry class, students were learning about boiling point of various elements.Reuben has compiled a table that gives boiling points of each of various elements. The lowest boiling point is ...