Given that this can be quite a tricky problem for most people, perhaps we could motivate them a bit with a story?
You are an accomplished problem-solver, known for your ability to tackle the most challenging mathematical puzzles. One evening, you receive an urgent letter from an inventor called the Triangle Tinkerer, who is working on a groundbreaking new project. The letter reads:
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Dear Mathematician,
I need your help. I am designing a device that relies on the unique properties of Pythagorean triplets — sets of three integers that satisfy the equation a² + b² = c². This device will revolutionize navigation, but for it to work, I must program it with every possible triplet where the sum of a, b, and c equals a specific number, N. Calculating these triplets by hand would take me years, but I hear you are more than up to the task.
Time is of the essence. The future of my invention — and perhaps even the future of mathematical innovation — rests on your ability to solve this problem.
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Motivated by the importance of the task, you set out to find all Pythagorean triplets that satisfy the condition. Your work could have far-reaching implications, unlocking new possibilities in science and engineering. Can you rise to the challenge and make history?
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