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

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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
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lfdahl
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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.
 
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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
 
  • #3
Hi, kaliprasad!
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.

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