The article in question makes the point that it is not enough to make easy things easy and hard things possible. It is also necessary to make trivial things trivial. This is a very nice observation, and one that, in his own way, Arnold Bennett expressed in his book ‘How to live on 24 hours a day’, and has an obvious application to educational curricula as well.

We can model this situation by f(x) = exp(ax) – b, where a is a positive constant, and b is a constant in [0, 1]. Making easy things easy and hard things possible means having a close to 0, whereas making trivial things trivial means having b close to 1.

Here’s the link to the article.

keywords:

[taxicab numbers]

[Hardy’s taxicab number]

[1729]

# Day: July 16, 2020

# Boolean Pythagorean Triples Problem

Here is the link to the Wikipedia article on this topic.

The construction is possible up to, and including, 7824. That is, the first breaking point, or failure point, is 7825.

keywords:

[Pythagoras]

[Mathematics]

[partition]

[sum of squares

[hypotenuse]

[legs of a triangle]

[coloring, in the mathematical sense]

# the lunar eclipse of a number

(suggested new terminology)

Definition. The lunar eclipse of a 4-digit number (padded with leading zeros, if necessary) is the result of creating the largest number possible from the digits of the number and creating the smallest number possible from the digits of the number and then subtracting the smaller number from the larger number.

For example, the lunar eclipse of 1024 is 4086, because the largest number possible using its digits is 4210, and the smallest number possible using its digits is 0124, and 4210 – 0124 = 4086.

Another example (involving padding with leading zeros): The lunar eclipse of 32 is 3177, because the lunar eclipse of 32 is the lunar eclipse of 0032, and the largest number possible from the digits of 0032 is 3200, and the smallest number possible from the digits of 0032 is 0023, and 3200 – 0023 = 3177.

keywords:

[Recreational Mathematics]

[daily exercise in mathematics]

[Number Theory]

[Kaprekar process]

[subtraction]

# Folkman’s theorem

Definition. (suggested new terminology) A set of natural numbers is said to be of critical mass fif there exist arbitrarily-large sets of natural numbers, all of whose sums belong to it.

Theorem. (Folkman’s theorem) Every finite partition of the set of natural numbers contains a member of critical mass.

Here is the link to the Wikipedia article on Folkman’s theorem.

keywords:

[Mathematics]

[Number Theory]

[Ramsey Theory]

[partitions]

[strength]

[size]

[summation]

# medical condition: deep vein thrombosis (DVT)

Here’s the link to the May Clinic article on this topic.

keywords:

[blood clot]

[pulmonary embolism]

[human physiology]

[pathology]

[lack of exercise]

# list: first 1000 numbers whose cototient is a square

Here’s the link.

keywords:

[Mathematics]

[Number Theory]

[Euler’s totient function]

[Euler’s φ function]

# cototients

Here’s the link to the Wikipedia article on this topic.

keywords:

[Mathematics]

[Number Theory]

[Euler’s totient function]

[Euler’s φ function]