Integers Definition and 467 Threads

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?

Determine all possible integers $n$ for which $\dfrac{n^2+1}{\lfloor{\sqrt{n}}\rfloor^2+2}$ is an integer.

Prove that $\dfrac{378^3+392^3+1053^3}{2579}$ is an integer.

ok I don't don't know de jure on this so ... is it just plug and play?? find factors of -48 $-1(48)=-48$ $-2(24)=-48$ $-3(16)=-48$ $-4(12)=-48$ $-6(8)=-48$ check sums for positive number $-1+48=47$ $-2+24=22$ $-3+16=13$ $-4+12=8$ $-6+8=2$it looks like c. 5

Now, set of even integers is ## A = \{ \cdots, -4, -2, 0, 2, 4, \cdots \} ##. We need to prove that ## \mathbb{Z}^+ \thicksim A##. Which means that, we need to come up with a bijection from ##\mathbb{Z}^+## to ##A##. We know that ##\mathbb{Z}^+ = \{1,2,3,\cdots \} ##. I define the function ##f ...

I need to recursively generate a quadrature signal which fits exactly into a grid NxN, where N is a large power of two. After unsuccessful research, I decided to develop my own solution, starting from the waveguide-form oscillator taken from Pete Symons' book 'Digital wave generation, p. 100'...

Let (a, b, c) be some arbitrary positive integers such that: (q2^0 + q2^1+ . . . + q2^x), (q2^0 + q2^1+ . . . + q2^y), (q2^0 + q2^1 + . . . + q2^z), where: q = (1, 2), (x, y, z) = (1, 2, 3, . . ., n). In the case if and only if q = 2, we accept the following notation : [(q-1)2^0...

Determine if the number 7 is a natural number, an integer, a rational or irrational number. I know that integers include positive and negative numbers and 0. Let Z = the set of integers Z = {. . . -2, -1, 0, 1, 2, . . .} I also know that any integer Z can be written as Z/1 = Z. I will...

Are all complex integers that have the same norm associates of each other? I have seen definitions saying that an associate of a complex number is a multiple of that number with a unit. And I understand that the conjugate of a complex number is also an associate. But I am looking for a...

Homework Statement Do five positive integers exist such that the positive difference between any two is the greatest common divisor of those two numbers? Homework EquationsThe Attempt at a Solution I found four such numbers, ##\{6,8,9,12\}##. I did this in an ad hoc way though without any real...

Homework Statement Prove that the product of any three consecutive integers is divisible by 6. Homework EquationsThe Attempt at a Solution This doesn't seem true to me for any 3 consecutive ints. For example, let a_0 = 0 a_1 = 1 a_2 = 2 3 is not divisible by six. Assuming they meant a_x...

Give an example where a proposition with a quantifier is true if the quantifier ranges over the integers, but false if it ranges over rational numbers. I do not know where to go about when answering this, I know that an integer can be a rational number, for example 5 is an integer but can also...

I am exploring the behaviors of complex integers (Gaussian and Eisenstein integers). My understanding is that when a complex integer z with norm >1 is multiplied by itself repeatedly, it creates a series of perfect powers. For instance, the Gaussian integer 1+i generates the series 2i, -2+2i...

Homework Statement Let ##a, b, c \in \mathbb{N \setminus \{0 \}}##. Show that for all ##n \in \mathbb{Z}## we have $$n^{11a + 21b + 31c} \equiv n^{a + b + c} \quad (mod \text{ } 11).$$ Homework EquationsThe Attempt at a Solution We have to show that ##11 | (n^{11a + 21b + 31c} - n^{a + b +...

How do computers evaluate the gcd of two integers?

Do they use the Euclidean Algorithm?

Alright, so this might be a stupid question, but nevertheless, I ask. I am to consider whether the quadratic form ## P(x,y) = a x + b y + d xy ## can map the integers onto the integers. So through a change of basis, I re-express this as ## P'(u,v) = Au^2 + Bv^2 ## for rational A and B...

for the equation $8\cdot8\cdot 8=4$. Find all integers $n$ for which this statement is true, modulo $n$. ok so $$8^3-(4)=508$$ 508/4=127 508/127=4 then 2^2\cdot 127 = 508 ok I'm sure this is not the proper process

Suppose a and b are integers that divide the integer c If a and b are relatively prime, show that $ab / c$ Show by example that if a and b are not relatively prime, then ab need not divide c let $$a=3 \quad b=5 \quad c=15$$ then $$\frac{15}{3\cdot 5}=1$$ let $$a=4 \quad b=6 \quad c=15$$ then...

This is a simple computational question. Let ##n \in [0, 36)##. What's the fastest way to list all ##n## s.t ##36## divides ##48n##?

If you have 2 integers n and n+1, it is easy to show that they have no shared prime factors. For example: the prime factors of 9 are (3,3), and the prime factors of 10 are (2,5). Now if we consider 9 and 10 as a pair, we can collect all their prime factors (2,3,3,5) and find the maximum, which...

Homework Statement Let ##\phi : G \to H## be a homomorphism. Prove that ##\phi (x^n) = \phi (x)^n## for all ##n \in \mathbb{Z}## Homework EquationsThe Attempt at a Solution First, we note that ##\phi (x^0) = \phi(x)^0##. This is because ##1_G \cdot 1_G = 1_G \implies \phi (1_G 1_G) = \phi...

I am exploring Gaussian integers in terms of roots, powers, primes, and composites. I understand that multiplying two integers with norm 5 result in an integer with norm 25. I get the impression that there are twelve unique integers with norm 25, and they come in two flavors: (1) Four of them...

Question: There are three non-negative integers with the following property: If you multiply any two of the numbers and subtract the third number, the result is a perfect power of 2. Find these three numbers that satisfy this property. My attempt: I worked out that the three numbers must be...

Homework Statement Can you partition the positive integers in such a way that if x, y are member of A, then x+y is not a member of A. x and y have to be distinct. That is, {1, 2, 3} cannot be in the same set, since 1+2 = 3, but 1 and 2 can be, since 1+1=2, but 1 and 1 are not distinct...

Determine the number of integers $n \geq 2$ for which the congruence $x^{25} \equiv x$ $(mod \;\; n)$ is true for all integers $x$.

$f(x)$ is a degree 10 polynomial such that $f(p)=q$, $f(q)=r$, $f(r)=p$, where $p$, $q$, $r$ are integers with $p<q<r$. Show that not all the coefficients of $f(x)$ are integers.

Three times the smaller of two consecutive EVEN integers IS four less than twice the larger. What are the two integers? My set up: x and x + 2 are the two consecutive even integers. True? Here, x is the smaller integer and (x + 2) the bigger integer. True? The equation is 3x = 2(x + 2) - 4. Yes?

Dear Everyone, Directions: Decide whether the statement is a theorem. If it is a theorem, prove it. if not, give a counterexample. There exists a unique integer n such that $$n^2+2=3$$. Proof: Let n be the integer. $$n^2+2=3$$ $$n^2=1$$ $$n=\pm1$$ How show this is unique or not? Please...

Find three consecutive odd integers such that the square of the first plus the square of the third is 170. See picture for the set up. Is the set up correct?

Homework Statement Let ##\alpha \in \mathbb{R}## and ##n \in \mathbb{N}##. Show that exists a number ##m \in \mathbb{Z}## such that ##\alpha - \frac {m}{n} \leq \frac{1}{2n}## (1).The Attempt at a Solution If I take ##\alpha= [\alpha] +(\alpha)## with ##[\alpha]=m## (=the integer part) and...

Homework Statement The set ##\Bbb{A}## of all the algebraic integers is a subring of ##\Bbb{C}## Homework EquationsThe Attempt at a Solution Here is an excerpt from my book: "Suppose ##\alpha## an ##\beta## are algebraic integers; let ##\alpha## be the root of a monic ##f(x) \in...

<Moderator's note: Moved from a technical forum and therefore no template.> Hi everybody I've been trying to solve this problem all the afternoon but I haven't been able to do it, I've written what I think the answers are even though I don't know if they're correct, so I've come here in order...

$(1)$ $\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{7}{8}(x,y\in N)$ $find$: $x,y$ $(2)$ $\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{6}{12}(x,y\in N)$ $find$: $x,y$ $(3)$ $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac {1}{z}=\dfrac{6}{24}(x,y,z\in N)$ $find$: $x,y,z$

Hello! (Wave)Let $b_1< b_2< \dots< b_{\phi(m)}$ be the integers between $1$ and $m$ that are relatively prime to $m$ (including 1), and let $B=b_1 b_2 b_3 \cdots b_{\phi(m)}$ be their product. I want to show that either $B \equiv 1 \pmod{m}$ or $B \equiv -1 \pmod{m}$ . Also I want to find a...

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Section 1.4: Entry 2: Axioms for the Integers ... In this section Bloch defines the integers as an ordered integral domain that satisfies the Well Ordering Principle ... rather than defining the...

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Section 1.4: Entry 2: Axioms for the Integers ... In this section Bloch defines the integers as an ordered integral domain that satisfies the Well Ordering Principle ... rather than defining the...

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Chapter 1: Construction of the Real Numbers ... I need help/clarification with an aspect of Theorem 1.3.7 ... Theorem 1.3.7 and the start of the proof reads as follows: n the above proof we...

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Chapter 1: Construction of the Real Numbers ... I need help/clarification with an aspect of Theorem 1.3.7 ... Theorem 1.3.7 and the start of the proof reads as...

Is there a set of $2015$ consecutive positive integers containing exactly $15$ prime numbers?

At present I am trying to understand the construction of the number systems ... natural, integers, rationals and reals ... What do members of PFs think is the clearest, most detailed, most rigorous and best treatment of number systems in a textbook or in online notes ... ? NOTE: I am currently...

At present I am trying to understand the construction of the number systems ... natural, integers, rationals and reals ... What do members of MHBs think is the clearest, most detailed, most rigorous and best treatment of number systems in a textbook or in online notes ... ? NOTE: I am...

Homework Statement determine the number of pairs of integers (a,b) 1≤b<a<200 such that the sum ## (a+b) + ( a-b) + ab + \frac a b\ ## is the square of an integer i have the solution to the problem this was the given solution the given equation is equivalent to ## \frac {a*(b+1)^2} b\\ ##...

how many integers are between \sqrt{19} and \sqrt{90}

In the textbook, the author showed that 8 + (a - 2)^3 factors out to be a(a^2 - 6a + 12). The author goes on to say "...the expression a^2 - 6a + 12 is irreducible over the integers." What does the author means by the statement?

How many integers are there between √19 and √90