One of my favorite mathematical tangents involves a theorem of Littlewood. The prime number theorem states that the prime counting function $\pi(x)$ is well ap

One of my favorite mathematical tangents involves a theorem of Littlewood. The prime number theorem states that the prime counting function $\pi(x)$ is well approximated by the integral $\int_0^x \frac{dt} {\log t},$ which is a function that's become named $\mathrm{li}(x).$ Littlewood proved that the sign of $\pi(x) - \li(x)$ changes infinitely often, but his proof didn't produce a specific value where such a sign change occurs. In 1933, one of Littlewood's students showed that, assuming the Riemann hypothesis, at least one sign change occurs below the number $e^e^e^79$, which is approximately $10^10^10^34.$ In 1955, he was able to show unconditionally that such a sign change occurs below the number $e^e^e^e^7.705$, which is approximately $10^10^10^964$. Both of those numbers absolutely dwarf the estimated number of elementary particles in the observable universe, so it's utterly impossible to either compute or write them down, even if you used the entire universe as your computer or your writing surface. https://en.wikipedia.org/wiki/Skewes%27s_number https://en.wikipedia.org/wiki/Skewes%27s_number That upper bound has since been lowered in 1999 to 1.38922 * 10^316 here: https://www.ams.org/journals/mcom/2000-69-231/S0025-5718-99-01104-7/S0025-5718-99-01104-7.pdf https://www.ams.org/journals/mcom/2000-69-231/S0025-5718-99-... I imagine it there might have been a bit more progress in the last 25 years, but I have no idea if it's possible to actually write down all the digits of the current best upper bound.