# Prove this relation between two numbers (Number theory)

• LCSphysicist
In summary, Bézout's identity states that if a and b are coprimes, then there exist integers x and y such that ax + by = d.
LCSphysicist
Homework Statement
If (u,v) = 1, prove that (u+v,u-v) is either 1 or 2.
u and v are integers.
Relevant Equations
.
If (u,v) = 1, prove that (u+v,u-v) is either 1 or 2.
Where (,) means:

$$ux_1 + vx_2 = 1$$
$$u + v(x_2/x_1) = 1/x_1, u(x_1/x_2) + v = 1/x_2$$
$$u + v = 1/x_1 + 1/x_2 - v x_2/x_1 - u x_1/x_2$$
$$u - v = 1/x_1 - 1/x_2 + u x_1/x_2 - v x_2/x_1$$

Now we can express (u+v,u-v). But i am not sure if it will give us an answer.
I think the key is to know what x1 and x2 are, knowing that u and v are coprimes. But i don't know how to find it.

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I am missing the
Homework Statement:
and
Relevant Equations:

For part of the first I can substitute " If (u,v) = 1, prove that (u+v,u-v) is either 1 or 2."
But one really needs a definition of (u,v).
And one needs to know what u and v are. Uruguaians, Venezolans ?

LCSphysicist
BvU said:
I am missing the
Homework Statement:
and
Relevant Equations:

For part of the first I can substitute " If (u,v) = 1, prove that (u+v,u-v) is either 1 or 2."
But one really needs a definition of (u,v).
And one needs to know what u and v are. Uruguaians, Venezolans ?
u and v are integer numbers hahaha
You are right, i forgot to write the meaning of (,).
I will edit

I don't understand how that definition of (u,v) that is given in post #1 is equivalent to that u,v are co primes.

Delta2 said:
I don't understand how that definition of (u,v) that is given in post #1 is equivalent to that u,v are co primes.
(u,v) = 1
ux + vy = 1 (1)​

" Bézout's identity — Let a and b be integers with greatest common divisor d. Then there exist integers x and y such that ax + by = d. More generally, the integers of the form ax + by are exactly the multiples of d. "

if a and b are co primes, gcd = 1
ax + by = 1

As in our case (1)

Sorry there is something i don't understand, let's take 5 and 2 that are coprimes, ##5\times10+2\times25=100\neq 1##

Something that you don't explain well, you saying for every ##x_1,x_2## or do you mean there exist ##x_1,x_2## such that

u and v are integers. Interesting.

Then what does it mean if you write "(u,v)=1" ?
That the set of all integers of the form ux1 + vx2 consists of one and only one element, namely 1 ?

Delta2
1 here represents the ideal generated by 1, or to put it a bit simply the set of all multiples of 1. This is equivalent to saying 1 is the smallest natural number that can be represented this was.

I think the solution to this is once you have a way of showing (u,v)=1, it's easy to give a representation that shows (u+v,u-v) contains 2, and hence is either 2 or 1.

LCSphysicist

## 1. What is number theory?

Number theory is a branch of mathematics that deals with the properties and relationships of numbers, particularly integers.

## 2. What is a relation between two numbers?

A relation between two numbers is a statement that describes how the two numbers are related or connected to each other.

## 3. How do you prove a relation between two numbers?

To prove a relation between two numbers, you must provide evidence or logical steps that show how the two numbers are related. This can be done through mathematical equations, proofs, or examples.

## 4. What are some common relations between two numbers in number theory?

Some common relations between two numbers in number theory include divisibility, prime numbers, factors, and multiples.

## 5. Why is proving relations between two numbers important in number theory?

Proving relations between two numbers is important in number theory because it helps us understand the fundamental properties and patterns of numbers. It also allows us to solve more complex mathematical problems and make connections between different areas of mathematics.

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