01 Oct 2025 Daily Log

Fermat Saga Continues & Domino Effect

2 min read

Maths

z2=x2+y2z^2 = x^2 + y^2

Fermat Saga Continues, looking at the equation, one can see the similarities with Pythagoras Theorem from high school geometry classes. The question is whether we can find whole number of units in length for all sides of a right angled triangle?

The book shows a common solution i recollect from my Mathematics Practice in high school i.e

52=32+425^2 = 3^2 + 4^2

25=9+1625 = 9 + 16

Knowing this we could deduce infinitely number of whole number solutions by simply multiplying the three whole numbers by any natural number, for example we can multiple by 3 and get

152=92+12215^2 = 9^2 + 12^2

225=81+144225 = 81 + 144

In addition, a complete resolution to this problem was proposed by Euclid in his book Elements x=2stx = 2st , y=s2+t2y = s^2 + t^2 , z=s2−t2z = s^2 - t^2

This formula can generate all primitive solutions to the original equation, as s and t vary over all natural numbers such that:

  1. s>ts > t
  2. ss and tt have no common factor
  3. one of ss, tt is even, the other odd

Notes from The Language of Mathematics: Keith Devlin

Prayer

Bible verse of the day was Isaiah 40:11 KJV.

He shall feed his flock like a shepherd: he shall gather the lambs with his arm, and carry them in his bosom, and shall gently lead those that are with young.

Language

Day 61 on Duolingo, Duo is happy