Given three positive integers a, b and c such that a² + b² - c² = 1. Let the number of unique triangles formed with sides a, b and c with perimeter less or equal than 50 million represent the surface area of an ellipsoid of axes lengths n, 2n and 3n.

What is the square of the ceiled positive solution of n.

Where did you come across this problem? What is your maths background? This site is aimed towards helping you to solve the problem rather than just giving you an answer. But this is quite an intriguing question!

Initially I think you need to find how many positive integer triples a,b,c exist such that:-

a² + b² - c² = 1

a+b+c ≤ 50,000,000

You don't actually need to know the values of a,b,c that form the set of the solutions. Just work out how many solutions there are. However, I personally think that there's no shortcut, so you probably do need to work out what all the a,b,c triples are. Unless someone else on the forum has a clever method?

To start off, there are 24,999,999 solutions of the form b=1 and c=a.

And that's not all. Here are the additional solutions up to perimeter 50 with b≤a (since if a≠b then a and b can be trivially swapped to give another solution)

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do you mean add both of the equations together

because if you do the answer is 10x-5

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the answer is a i belaeve atleast thats what my calculator just gfave me

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