Proof that there are an infinite number of primes via Pi
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There's a simple logic mistake in this video:
x irrational does not imply x^2 irrational. Counterexample: x = sqrt(2) => x^2 = 2.
However, it is true that pi being transcendental does prove that pi^2/6 is irrational.
MyOverflow 2 months ago
@epsilon728 I would say that it follows from pi being transcendental. That is, because pi is transcendental, it does not solve any non-constant polynomial equations with rational coefficients. Therefore, since x^2 - a/b = 0 is a polynomial equation of rational coefficients, pi cannot be a solution, and, therefore, pi^2 cannot be rational. Also, dividing by any rational number does not change that (which is simple to prove). Therefore, pi^2/6 is irrational.
MyOverflow 2 months ago
how does it follow from pi being irrational that pi^2/6 is also irrational?
epsilon728 3 months ago
clever!~
I like it.
DivergentMind 3 months ago
very interesting observation. good insight
johndoe121213 8 months ago