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| The equation 1⁰² + 1¹² + 1²² = 1³² + 1⁴², whose answer is that both sides equal 365, was immortalized in a different form in this 1895 painting: “Mental Arithmetic. In the Public School of S. Rachinsky.” Photo: NIKOLAY BOGDANOV-BELSKY |
One
of the first theorems anyone learns in mathematics is the Pythagorean
Theorem: if you have a right triangle, then the square of the longest
side (the hypotenuse) will always equal the sums of the squares of the
other two sides. The first integer combination that this works for is a
triangle with sides 3, 4, and 5: ³² + ⁴² = ⁵². There are other
combinations of numbers that this works for, too, including:
- 5, 12, and 13,
- 6, 8, and 10,
- 7, 24 and 25,
and
infinitely more. But 3, 4, and 5 are special: they’re the only
consecutive whole numbers that obey the Pythagorean Theorem. In fact,
they’re the only consecutive whole numbers that allow you to solve the
equation a² + b² = c² at all.
But if you allowed yourself the freedom to include more numbers, you
could imagine that there might be consecutive whole numbers that worked
for a more complex equation, like a² + b² + c² = d² + e². Remarkably, there’s one and only one solution: 10² + 11² + 12² = 13² + 14². Here’s why.
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| If you take the sum of the squares of any two legs of any right triangle, it will always equal the square of the hypotenuse. But there is much more to this relation than a simple equation. Photo: HISTORYOFPYTHAGOREANTHEOREM.WEEBLY.COM |
Pythagoras just started with a² + b² = c²,
which has 3, 4, and 5 as the only set of consecutive numbers that solve
it. We can extend this as long as we like, however, and for each
equation with an odd number of terms we can write down, there’s only one
unique solution of consecutive whole numbers. These Pythagorean Runs
have a clever mathematical structure governing them, and by
understanding how squares work, we can see why they couldn’t possibly
behave in any other way.
Source: Medium
