Perfect Planet

  On our journey through the Pythagorean Table ( see Correio do Minho, 12/12/2024 ), we observed that the table displayed regular patterns when analyzed using the elimination of the nine methods ( modulo 9 ). Surely, you must have heard of the mathematical trick “ Casting out Nines .” This method works because, in the decimal system (base 10) , there is a fundamental property: 10 ≡ 1 (mod 9). This rule allows us to  replace a number with the sum of its digits to determine the remainder when divided by 9. Modular arithmetic exposes intrinsic patterns within numbers, which are essential both in the elimination of nine methods and in the structure of the Pythagorean Table . For example, as analyzed in “A Journey through the Pythagorean Table,”  we identify predictable cycles in the values obtained by recursively reducing the results of multiplications. This reflects the periodic nature of modular arithmetic , which fascinated ancient mathematicians, including the ...

  The sun was setting on the horizon. It was a late afternoon on one of those days still under the old time, meaning an hour ahead of where we are now. Despite the time being ahead, I will not talk about the future. On the contrary, I will speak of two recent and distant pasts. In the recent past, while traveling on a highway, I was suddenly struck by a beam of light (fortunately, not a lightning bolt) that made me reflect on the multiplication table and transported me to a distant past. Suddenly, as if in a leap through time, I found myself in the 6th century BC, on an island in ancient Greece, face to face with Pythagoras. Who does not know the well-known rhyme that describes his famous theorem? "On the way to Syracuse, Pythagoras told his grandchildren: The square of the hypotenuse is equal to the sum of the squares of the other two sides." Although the theorem is fascinating, it is irrelevant to today’s topic. Beyond the theorem and many other significant contributions...