I always thought that the famous equation
102 + 112 + 122 = 132 + 142
is sort of a miracle, a random fluke. I enjoyed this cute equation, but never really thought about it seriously. Recently, when my son Sergei came home from MOP, he told me that this equation is not a fluke; and I started thinking.
Suppose we want to find five consecutive integers such that the sum of the squares of the first three is equal to the sum of the squares of the last two. Let us denote the middle number by n, which gives us the equation:
(n–2)2 + (n–1)2 + n2 = (n+1)2 + (n+2)2.
After simplification we get a quadratic equation: n2 – 12n = 0, which has two roots, 0 and 12. Plugging n = 0 into the equation above gives us (–2)2 + (–1)2 + 02 = 12 + 22, which doesn’t look like a miracle at all, but rather like a trivial identity. If we replace n with 12, we get the original miracle equation.
If you looked at how the simplifications were done, you might realize that this would work not only with five integers, but with any odd number of consecutive integers. Suppose we want to find 2k+1 consecutive integers, such that the sum of the squares of the first k+1 is equal to the sum of the squares of the last k. Let us denote the middle number by n. Then finding those integers is equivalent to solving the equation: n2 = 2k(k+1)n. This provides us with two solutions: the trivial solution 0, and the non-trivial solution n = 2k(k+1).
So our miracle equation becomes a part of the series. The preceding equation is the well-known Pythagorean triple: 32 + 42 = 52. The next equation is 212 + 222 + 232 +242 = 252 + 262 + 272. The middle numbers in the series are triangular numbers multiplied by four.
Actually, do you know that 102 + 112 + 122 = 132 + 142 = 365, the number of days in a year? Perhaps there are miracles or random flukes after all.Share: