Fermat's Library

Bill Bryson on atoms and ions

A sequence of six 9’s (known as the Feynman Point) begins immediately after the 761st decimal place of π. Nobel prize-winning physicist Richard Feynman expressed a wish to memorize the digits of π as far as that point so that when reciting them, he would be able to end with “. . . nine, nine, nine, nine, nine, nine, and so on.”

The mᵗʰ Fibonacci number

This week's paper is "Categorizing Variants of Goodhart’s Law" by David Manheim and Scott Garrabrant. Goodhart's law is often stated as "when a measure becomes a target, it ceases to be a good measure". The authors note that there are various ways in which systems can fail when optimized excessively based on specific metrics. This occurs when a metric, initially useful for system improvement, is pushed to a point where further optimization becomes either ineffective or counterproductive. The paper argues that these failure modes are often misunderstood, partly due to unclear terminology and partly because discussions using this ambiguous language fail to distinguish between different types of failures within this broader category. Goodhart's law can be observed in action across diverse domains, including public policy, corporate management, and artificial intelligence alignment. In their paper, Manheim and Garrabrant examine several distinct failure modes, providing mathematical explanations to elucidate each type. Paper: https://buff.ly/3y0Oi2n

This picture of Nikola Tesla, taken in 1894, is the first photograph ever taken by phosphorescent light. The time of exposure: eight minutes

Most of our greatest theories are able to explain several phenomena with a single core idea. John Wheeler summarized General Relativity: “Spacetime tells matter how to move; matter tells spacetime how to curve”

"In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual" - Galileo Galilei

The Hadwiger problem is the problem of finding the largest number of subcubes (not necessarily different) into which a cube cannot be divided by plane cuts. In 1977, this dissection was discovered, proving the answer was 47.

Euler on prime numbers

This is the only solution to a² b³ = c⁷ in positive integers