# Given a,b∈T, a^2−ab+b^2 divides a^2b^2, Prove that T is finite

• MHB
• lfdahl
In summary, T refers to a finite set of numbers that satisfy the given conditions, which involve a and b belonging to the set and the expression a^2−ab+b^2 dividing a^2b^2. To prove that T is finite, mathematical proofs such as contradiction or induction can be used to show that there are only a finite number of values for a and b that satisfy the conditions. An example of such a set is the set of all positive integers, where taking a=2 and b=3 does not satisfy the conditions. The fact that the expression a^2−ab+b^2 divides a^2b^2 limits the possible values of a and b in T, making it a finite set. This
lfdahl
Gold Member
MHB
Let $T$ be a set of natural numbers such that for any $a, b \in T$, $a^2 − ab + b^2$ divides $a^2b^2$.

Prove, that $T$ is finite.

lfdahl said:
Let $T$ be a set of natural numbers such that for any $a, b \in T$, $a^2 − ab + b^2$ divides $a^2b^2$.

Prove, that $T$ is finite.

For the above to be valid a and b should be co-prime or one of them 1 else if (a,b) is a solution then (na,nb) is also a sloution for integer N

In order to answer your comment, I´ll suppose, that $a,b \in T$ and $a$ and $b$ are not coprimes.

Let $d = gcd(a,b)$. Then we have:

$a = da_1$ and $b = db_1$, where $a_1$ and $b_1$ are coprimes.

Then: $a_1^2-a_1b_1+b_1^2$ divides $d^2a_1^2b_1^2$, but $gcd(a_1^2-a_1b_1+b_1^2,a_1b_1) = 1$. Hence, $a_1^2-a_1b_1+b_1^2$ divides $d^2$, i.e. $a^2-ab+b^2$ divides $d^4$.

Since $d \leq a$, we have $a^2-ab+b^2 \leq a^4$. If you fix any $a \in T$, $b$ can only take on a finite number of distinct values.

## 1. What is the meaning of T in the given statement?

T in this statement refers to a set of numbers that satisfy the given conditions, namely a and b belong to this set and the expression a^2−ab+b^2 divides a^2b^2.

## 2. How do we prove that T is finite?

To prove that T is finite, we need to show that there are only a finite number of values for a and b that satisfy the given conditions. This can be done by using mathematical proofs such as contradiction or induction.

## 3. Can you provide an example to illustrate this statement?

Yes, for example, let T be the set of all positive integers. If we take a=2 and b=3, then a^2−ab+b^2=4−6+9=-2 does not divide a^2b^2=4*9=36. Therefore, (2,3) does not belong to T. This shows that T is a finite set.

## 4. How does the fact that a^2−ab+b^2 divides a^2b^2 relate to T being finite?

The fact that a^2−ab+b^2 divides a^2b^2 means that for every pair of numbers a and b in T, the result of this expression will always be a factor of a^2b^2. This limits the possible values of a and b that can belong to T, making it a finite set.

## 5. Can we apply the same proof for any other set instead of T?

No, the given statement specifically refers to a set T that satisfies the given conditions. The proof may vary for different sets, depending on the conditions given. It is not applicable to all sets in general.

• General Math
Replies
1
Views
771
• General Math
Replies
1
Views
728
• General Math
Replies
1
Views
742
• General Math
Replies
19
Views
2K
• General Math
Replies
1
Views
973
• General Math
Replies
1
Views
899
• General Math
Replies
1
Views
622
• General Math
Replies
3
Views
890
• General Math
Replies
1
Views
936
• General Math
Replies
1
Views
874