logarithm

A newly shared preprint claims to prove that logarithms of rational numbers have irrationality exponent 2, a measure of how well such numbers can be approximated by rationals. The result would sharpen known bounds for numbers like log 2. The paper is circulating among mathematicians, with early discussion focused on verifying the argument and assessing whether the proof holds up.

A new paper by GitHub user jdb1729 proves that the logarithms of rational numbers have irrationality exponent exactly 2, meaning they cannot be approximated by rationals much better than generic irrational numbers. The result, shared as a self-published PDF, is drawing attention from mathematically inclined readers who are discussing the proof techniques and how the bound compares to known results for related constants.

The Bitcoin Rainbow Chart, a long-standing logarithmic price model that bands Bitcoin's value from 'maximum bubble territory' to 'fire sale', has been used to project where BTC could trade on October 31, 2026. The model extrapolates historical price behaviour into colored zones, and traders regularly cite its bands when debating whether Bitcoin is overbought or undervalued. Crypto watchers are sharing the projection as Bitcoin's long-term trajectory remains a topic of debate.