Infinite descent principle

Description

• There is no infinite sequence of strictly decreasing non-negative integerstheorem
• Used to show that a statement cannot possibly hold for any number
  • by showing that if the statement were to hold for a number
  • then the same would be true for a smaller number
  • leading to an infinite descent and ultimately a contradiction.
• The method relies on Well ordering principle, so only a finite number of non-negative integers are smaller than any given one
• A method of Proof by contradiction
• Used to solve Fermat’s Last Theorem for n = 3, 4

Proof by contradiction

• Let P is a property for nonnegative integers
• Assume P(0) is true
• for all $k\in \mathbb{N}$, if $P(k)$ is false, there exist an $m<k$ that $P(m)$ is also false → contradicting to the WOP
• Then, $P(n)$ is true for all $n\in \mathbb{N}$, including $k$
• Example:
  • P(n): “n is a positive integer that is not divisible by any square number other than 1”
  • assume there exists a k satisfies P(n)
  • k is not divisible by any square number other than 1.
  • but every positive number is divisible by 1^2 (which is 1) → there exists a smaller number m where m = k/ 1^2 = k satisfies P(n).
  • but this is still k and we are not actually finding a new number, contradicts that we could keep finding a smaller numbers satisfies P, leading to infinite descent
• Core situation:
  • we can keep finding smaller and smaller natural numbers violating a property >< in a set of natural numbers, there is a smallest element (WOP)
  • if $P(k)$ is true, there will be $P(m)$ and infinite descent numbers of $m$, contradicting with the WOP