Question
What is the Porian triple of 9 12 and 15
Answers
GPT-4o mini
To find the Porian triple of the numbers 9, 12, and 15, we first need to understand what a Porian triple is. A Porian triple refers to a set of three positive integers \( (a, b, c) \) such that \( a^2 + b^2 + c^2 = 3m^2 \) for some integer \( m \). This means that the sum of the squares of the three integers should be three times a perfect square.
Let's calculate the sum of the squares of 9, 12, and 15:
1. Calculate the squares:
- \( 9^2 = 81 \)
- \( 12^2 = 144 \)
- \( 15^2 = 225 \)
2. Sum these squares:
\[
81 + 144 + 225 = 450
\]
3. Now we check if 450 can be expressed as \( 3m^2 \):
\[
450 = 3m^2 \implies m^2 = \frac{450}{3} = 150
\]
4. Now we check if 150 is a perfect square. Since \( \sqrt{150} \approx 12.25 \), it's not an integer, and thus, 150 is not a perfect square.
Since the equation \( a^2 + b^2 + c^2 = 3m^2 \) does not hold for \( 9, 12, \) and \( 15 \), these numbers do not form a Porian triple.
In conclusion, the numbers 9, 12, and 15 do not constitute a Porian triple.
Let's calculate the sum of the squares of 9, 12, and 15:
1. Calculate the squares:
- \( 9^2 = 81 \)
- \( 12^2 = 144 \)
- \( 15^2 = 225 \)
2. Sum these squares:
\[
81 + 144 + 225 = 450
\]
3. Now we check if 450 can be expressed as \( 3m^2 \):
\[
450 = 3m^2 \implies m^2 = \frac{450}{3} = 150
\]
4. Now we check if 150 is a perfect square. Since \( \sqrt{150} \approx 12.25 \), it's not an integer, and thus, 150 is not a perfect square.
Since the equation \( a^2 + b^2 + c^2 = 3m^2 \) does not hold for \( 9, 12, \) and \( 15 \), these numbers do not form a Porian triple.
In conclusion, the numbers 9, 12, and 15 do not constitute a Porian triple.