To find a Pythagorean triplet $(a, b, c)$ where $a < b < c$, $b + c = a^2$, and $a > 7$, let's go through the conditions step-by-step.

We have:

1. $a$, $b$, and $c$ must be positive integers.
2. They should satisfy the Pythagorean theorem: $a^2 = b^2 + c^2$.
3. Additionally, $b + c = a^2$.
4. We want $a > 7$.

Let's explore possible values for $a$, starting from $a = 8$, and check if we can find suitable $b$ and $c$ values.

Step-by-Step Solution

1. For $a = 8$:

   • $a^2 = 8^2 = 64$
   • We need to find $b$ and $c$ such that: $b + c = 64 \quad \text{and} \quad b^2 + c^2 = 64$
   • Solving this does not yield integers for both $b$ and $c$, so $a = 8$ does not work.

2. For $a = 9$:

   • $a^2 = 9^2 = 81$
   • We need $b + c = 81$ and $b^2 + c^2 = 81$.
   • After checking various possibilities, we don’t get integer values for both $b$ and $c$.

3. For $a = 10$:

   • $a^2 = 10^2 = 100$
   • We need $b + c = 100$ and $b^2 + c^2 = 100$.
   • By testing possible values for $b$ and $c$, we find that there is no integer solution here as well.

After testing more values for $a$ beyond this, we do not find an integer solution that satisfies all conditions, which suggests that there may not be a Pythagorean triplet meeting these exact criteria.

If you’d like to explore further with a specific approach, or if there's an additional condition to adjust, feel free to let me know!

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Would you like details on each calculation, or have questions?

Related Questions

1. How are Pythagorean triplets generally derived?
2. Can you find other conditions where $a < b < c$ is satisfied with a Pythagorean triplet?
3. What are common methods to generate Pythagorean triplets?
4. Is there a geometric interpretation of Pythagorean triplets?
5. Can we have conditions where only two out of three conditions apply?

Tip: Pythagorean triplets are often found by formulas, such as $a = m^2 - n^2$, $b = 2mn$, and $c = m^2 + n^2$, for integers $m > n > 0$.