"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: 3 + 4 = 5."
"But 3, 4, and 5 are special: they're the only consecutive whole numbers that obey the Pythagorean Theorem. In fact, the numbers 3, 4, and 5 are the only consecutive whole numbers that allow you to solve the equation a + b = c at all! However, 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 a + b + c = d + e:"
The Pythagorean Theorem states that for a right triangle the sum of the squares of the two legs equals the square of the hypotenuse. Integer Pythagorean triples exist in infinite families, with examples including (3,4,5), (5,12,13), (6,8,10), and (7,24,25). The triple 3, 4, 5 is unique among consecutive integers in satisfying the squared relation. Allowing more terms leads to the search for consecutive integers whose squared sums balance across more than two terms. A unique consecutive solution appears: the sum of the squares of 10, 11, and 12 equals the sum of the squares of 13 and 14. The phenomenon connects elementary geometry, number theory, and Diophantine reasoning under the label of Pythagorean runs.
Read at bigthink.com
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