In 1978, Apéry [2] proved the irrationality of ζ(3) by constructing two explicit sequences of integers (un)n and (vn)n such that 0 6= unζ(3)− vn → 0 and un → +∞, both at geometric rates. He also deduced from this an upper bound for the irrationality exponent μ(ζ(3)) of ζ(3). In general, the irrationality exponent μ(ξ) of an irrational number ξ is defined as the infimum of all real numbers μ suc...

has infinitely many solutions in rational numbers p/q. It follows from the theory of continued fractions that μ(ξ) is always greater than or equal to 2, and an easy covering argument shows that μ(ξ) is equal to 2 for almost all real numbers ξ (with respect to the Lebesgue measure). Furthermore, Roth’s theorem asserts that the irrationality exponent of every algebraic irrational number is equal ...

This note collects the formal proof sketches that I have done. 1 Algebra: Irrationality of √ 2

In his recent book Bad Beliefs and Why They Happen to Good People, Neil Levy argues that conspiracy theories result from the same rational processes underlie epistemic success. While we think many of Levy’s points are valuable, like criticism myth individual cognition emphasis on importance one’s social environment, believe account overlooks some important aspects. We argue deference is an acti...

As soon as Maxwell published his first paper on the kinetic theory of gases in 1860, he noted that a major consequence of its principles, namely the equipartition theorem, conflicted with known empirical data about the specific heats of poly-atomic gases. According to the equipartition theorem, the total kinetic energy of an isolated system of particles at equilibrium is equally distributed amo...