An integer (from the Latin integer meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and √2 are not.
The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, ...), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., −1, −2, −3, ...). The set of integers is often denoted by the boldface (Z) or blackboard bold
(
Z
)
{\displaystyle (\mathbb {Z} )}
letter "Z"—standing originally for the German word Zahlen ("numbers").ℤ is a subset of the set of all rational numbers ℚ, which in turn is a subset of the real numbers ℝ. Like the natural numbers, ℤ is countably infinite.
The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.
My interest is on the highlighted part ...
Now to my question,
in what cases do we have ##mn<(m,n)[m,n]?##
I was able to use my example say,
Let ##m=24## and ##n=30## for example, then
##[m,n]=120## and ##(m,n)=6## in this case we can verify that,
##720=6⋅120## implying that, ##mn≤...
Hello guys,
I am able to follow the working...but i needed some clarification. By rounding to the nearest integer...did they mean?
##z=1.2-1.4i## is rounded down to ##z=1-i##?
I can see from here they came up with simultaneous equation i.e
##(1-i) + (x+iy) = \dfrac{6}{5} - \dfrac{7i}{5}## to...
Given the definition of whole numbers as integers, https://www.google.com/search?q=what+is+a+whole+number&rlz=1C1VDKB_en-GB&oq=what+is+a+whole+number&aqs=chrome..69i57j0i512l9.11619j0j15&sourceid=chrome&ie=UTF-8
Is it known why atom vibrations are only at whole numbers ( ref plank’s constant)...
So, a friend of mine has attempted a solution. Unfortunately, he's having numbers spawn out of nowhere and a lot of stuff is going on there which I can't make sense of. I'm going to write down the entire attempt.
$$
0 \in X \; \text{otherwise no subgroup since neutral element isn't included}...
Is there a name for the union of {prime numbers} and {integers that are not powers of integers}?
For example, we would include 2, 3, 5, 7, 11... And also 6, 10, 12...
But we exclude 2^n, 3^n, ... and 6^n , 10^n , etc.
What are some interesting contexts where this set crops up?
Problem Statement : I copy and paste the problem as it appears in the text (Lang, Basic Mathematics, 1971).
Attempt : There are several questions in both a) and b) above. I type out the question and my answer each time.
a) (i) Show that addition for ##E## and ##I## is associative and...
Observe that ## \phi(1)=\phi(2)=1 ##.
This implies ## \phi(1)\mid 1 ## and ## \phi(2)\mid 2 ##.
Thus ## n=1 ##.
Let ## n=p_{r}^{k_{1}}\dotsb p_{s}^{k_{s}} ## be the prime factorization of ## n ##.
Then ## \phi(n)=n\prod_{p\mid n} (1-\frac{1}{p}) ##.
Suppose ## \phi(n)\mid n ##.
Then ##...
Let ## a, b, c ## and ## d ## be integers such that ## 225a+360b+432c+480d=3 ##.
Then ## 75a+120b+144c+160d=1 ##.
Applying the Euclidean algorithm produces:
## gcd(75, 120)=15, gcd(120, 144)=24 ## and ## gcd(144, 160)=16 ##.
Now we see that ## 15x+24y+16z=1 ##.
By Euclidean algorithm, we have...
Suppose that ## n=p_{1}^{k_1}p_{2}^{k_2}\dotsb p_{r}^{k_r} ## satisfies ## \phi(n)=k ##.
Then ## n=\frac{k}{\prod(p_{i}-1)}\prod p_{i} ##.
Note that the integers ## d_{i}=p_{i}-1 ## can be determined by the conditions
## (1) d_{i}\mid k, (2) d_{i}+1 ## is prime, and ## (3) \frac{k}{\prod d_{i}}...
im thinking i should just integrate (binominal distribution 1-2000 * prime probability function) and divide by integral of bin. distr. 1-2000.
note that I am looking for a novel proof, not just some brute force calculation.
(this isn't homework, I am just curious.)
Hey! 😊
I want to write a one-line Python generator or iterator expression that returns the sequence of integers generated by repeatedly adding the ascii values of each letter in the word “Close” to itself. The first 10 integers in this sequence are: 67, 175, 286, 401, 502, 569, 677, 788, 903...
Proof:
Let ## a ## be any integer.
Then ## a\equiv 0, 1, 2, 3, 4, 5, 6, 7, 8 ##, or ## 9\pmod {10} ##.
Thus ## a^{3}\equiv 0, 1, 8, 27, 64, 125, 216, 343, 512 ##, or ## 729\pmod {10} ##.
Therefore, anyone of the integers ## 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ## can occur as the units digit of ##...
Proof:
Suppose for the sake of contradiction that ## r, s\in {0, ..., n-1} ## for ## r<s ## where ## c+ra\equiv c+sa\pmod {n} ##.
Then ## c+ra\equiv c+sa\pmod {n}\implies ra\equiv sa\pmod {n}\implies r\equiv s\pmod {n} ##.
Thus ## n\mid (r-s)\implies n<r-s ##, which is a contradiction because...
From my research I have found that since Fermat proved his last theorem for the n=4 case, one only needs to prove his theorem for the case where n=odd prime where c^n = a^n + b^n. But I am not clear on some points relating to this. For example, what if we have the term (c^x)^p, where c is an...
Proof:
Let ## n>3 ## be an integer.
Applying the Division Algorithm produces:
## n=3q+r ## for ## 0\leq r< 3 ##,
where there exist unique integers ## q ## and ## r ##.
Suppose ## n ## is prime.
Then ## n=3q+1 ## or ## n=3q+2 ##, because ## n\neq 3q ##.
Now we consider two cases.
Case #1...
Proof:
Consider all primes ## p\leq \sqrt{1949} \leq 43 ## and ## q\leq \sqrt{1951} \leq 43 ##.
Then we have ## p\nmid 1949 ## and ## q\nmid 1951 ## for all ## p\leq 43 ##.
Thus, ## 1949 ## and ## 1951 ## are both primes.
By definition, twin primes are two prime numbers whose difference is ## 2...
## 1234=2\cdot 617 ##
## 10140=2\cdot 2\cdot 3\cdot 5\cdot 13\cdot 13 ##
## 36000=2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 3\cdot 3\cdot 5\cdot 5\cdot 5\cdot ##
Are the answers above correct? Or do I need to put them in canonical form as below?
## 1234=2\cdot 617 ##
## 10140=2^{2}\cdot 3\cdot 5\cdot...
Sometimes I have seen a process to build integers and rationals via a sort of Grothendieck product, Z being classes of equivalence in N x N, and Q being classes of equivalence in Z x Z.
Now, I was wondering if it makes sense to consider the integers as the extension of ##\mathbb{N}## by ##\sqrt...
The following Python 3 code is provided as the solution to this problem (https://leetcode.com/problems/subsets/solution/) that asks to find all subsets of a list of integers. For example, for the list below the output is [[], [1], [1, 2], [1, 2, 3], [1, 3], [2], [2, 3], [3]].
I am not familiar...
Hello, I've been trying to solve this problem for a while, and I found a technical solution which is too computationally intensive for large numbers, I am trying to solve the problem using Combinatorics instead.
Given a set of integers 1, 2, 3, ..., 50 for example, where R=50 is the maximum...
a)
Yes.
One surjection from ##\mathbb{Z}## to ##\mathbb{N}## is the double cover of ##\mathbb{N}## induced by ##f:\mathbb{Z}\longmapsto\mathbb{N}## with
$$f(z)=\begin{cases}
-z & ,\forall z<0\\
z+1 & ,\forall 0\leq z
\end{cases}$$
b)
Yes.
One surjection from ##\mathbb{R}## to ##\mathbb{N}## is...
The probability that two randomly chosen integers to be coprimes is known to be equal to ## \prod_{2}^{\infty}(1-\frac{1}{p^2})=6/\pi^2##
I tried to conceptualize the problem in the following way but got different results.
Suppose that we pick up an integer at random which could be either prime...
First, read in an input value for variable inputCount. Then, read inputCount integers from input and output each integer on a newline after the string "value - ".
Ex: If the input is 2 25 55, the output is:
value - 25
value - 55
Code:
#include <iostream>
using namespace std;
int main() {...
Hey! :giggle:
Consider a representation of signed integers of base $B$, in which the digits are listed in descending order of importance, with the least significant digit corresponding to a positive, and the next digits to an alternate negative and positive value. Thus, a number of this...
For fun, I have decided to implement a simple XOR encryption algorithm. The first step is to convert messages into bytes to perform XOR operation on each bit. The problem has started here. For instance, I want to encrypt this message.
I hiked 24 miles.
Now I need to turn this text into binary...
Say for example I have a dataset (X, Y) which I need to fit to the function y = Ax^2 + By + Cxy.
How do I retrieve values of A, B, and C such that they can only be integers? As of now I'm doing grid search which is so taxing.
Hi,
The original problem was : for a given number k = d + n/d, where d is a divisor of another number n, how many k <= N are prime?
When I looked at this problem, for k to be prime > 2, it has to be odd.
This implies d and n/d can't both be even or odd. If d = 2, then d is even and n/d has...
Personal Question:
Internet says the standardized math symbol for integers is ## \mathbb {Z}##. However, my Alberta MathPower 10 (Western Edition) textbook from 1998 says the symbol is I. I'm guessing that textbook is wrong? Or are both answers correct?
I've written a program to factorise large numbers (although not that large). The following arithmetic operation goes wrong:
x= int(912_321_155_211_368_155/(5))
The result is
182_464_231_042_273_632
Which is clearly not right (should end in 631). The maximum integer on the 64-bit version is...
I've attempted and I can retrieve a 0 when the first input is greater than the second using BRP. I can't seem to get the output of 1 if the first input is equal to the second input. I also don't get a 2 output with the first input being less than the second. I also have no idea how to loop a...
Given : The quadratic equation ##x^2+px+q = 0## with coefficients ##p,q \in \mathbb{Z}##, that is positive or negative integers. Also the roots of the equation ##\alpha, \beta \in \mathbb{Q}##, that is they are rational numbers. To prove that ##\boxed{\alpha,\beta \in \mathbb{Z}}##, i.e. the...
A proper number is expressed in \pi in a similar way as a decimal integer is expressed in base 2. For example, 4375_{\pi} = 4{\pi^3}+3{\pi^2}+7{\pi}+5. The only exception I make is that the 10 digits are included when expressing a number with \pi. To clarify, the first positive such numbers are...
Let $a,\,b,\,c,\,d,\,e,\,f$ be positive integers and $S=a+b+c+d+e+f$. Suppose that the number $S$ divides $abc+def$ and $ab+bc+ca-de-ef-df$, prove that $S$ is composite.
I am trying to get one step further with my search for \sum_{n=1}^{\infty}\frac{1}{n^{2s+1}} . Part of the way is to calculate some algebraic expressions containing fractions with really huge numbers (as in (\frac{1}{5^{9}}+\frac{1}{7^{9}}-\frac{1}{17^{9}}-\frac{1}{19^{9}})\div...
Find the number of integers $k$ in the set ${0,\,1,\,\cdots,\,2012}$ such that the combination number $\displaystyle {2012\choose k}=\dfrac{2012!}{k!(2012-k)!}$ is a multiple of 2012.
Hey.
Im pretty new to Python (and programming in general in fact). I have received two different files, each containing 10 000 integers mixed up with some commas, colons, and \n, and in one of the files there are also negative numbers. I have tried all day retrieving those numbers using all...
Summary:: Interested in the history of the proof.
Consecutive integer numbers are always relatively prime to each other. Does anyone know when this was proved? Was this known since Euclid's time or was this proved in modern times?
Hello all,
This is a problem of a different flavour from my usual shenanigans. I'm looking at a function
$$f(m,n)=\frac{m^2n^2}{(m+n)(m-n)}$$
and am trying to determine if there are any two pairs of values ##(m_1,n_1)## and ##(m_2,n_2)## which evaluate to the same result. Assume that...
It is pretty obvious that all right-angled triangles whose sides are integers will have areas which are also integers. Since either the base or height will be an even number, half base x height will always come out exactly.
However, I have only found one non-right-angled triangle where this is...
How many of the first 1000 positive integers can be expressed in the form $\lfloor 2x \rfloor+\lfloor 4x \rfloor+\lfloor 6x \rfloor+\lfloor 8x \rfloor$, where $x$ is a real number?