unclean coincidences

At the end of the 19th century, a German otolaryngologist named Wilhelm Fliess, a close friend of Sigmund Freud, published one of the greatest numerological nonsense of all time. According to Fliess, human life is governed by cycles of 23 and 28 days. The 23-day cycle is “masculine” and prevails in men, and the 28-day cycle is “feminine” and prevails in women, although both cycles affect one another. But, not content with this, Fliess’s numerological rage led him to see the numbers 23 and 28, together with their multiples and combinations, everywhere.

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For example, the sum of both, 51, expressed, in years, a decisive turning point in a man’s life (and influenced by this trickery, Freud became obsessed with the idea that he would die at 51). Ultimately, in the simple Diophantine equation 23x + 28y = n, where x and y are integers, practically everything was enclosed, and proof of this, according to Fliess, was that any number could be obtained from the formula simply by giving ax and the suitable values. For example, with x = 11 and y = -9 we get 1:

23 x 11 – 28 x -9 = 253 – 252 = 1

Can you obtain, from Fliess’s formula, the following natural numbers: 2, 3, 4…? Is it just a coincidence that 23 and 28 allow you to get any number using this formula, or is there some hidden reason?

As the German mathematician Roland Sprague pointed out in the middle of the last century, all natural numbers (integers and positives), starting from a certain value, can be expressed using the Fliess formula with x and y also positive. Obviously, the smallest number that can be expressed with positive x and y is when both are 1, that is, 51; but what is the largest number that cannot be expressed with x and y both being positive?

Numerical coincidences, pure or impure, have always attracted attention, and not just that of mathematicians. Let’s see some:

The number e is usually expressed as 2.71828…, since the sixth decimal is 1 and neglecting it supposes an error of just one millionth; but if we increase the number of decimals we find a curious coincidence:

e = 2,718281828…

In the first nine decimal places the group 1828 is repeated. Is it a “pure” coincidence or does the repetition have to do with some obscure property of e?

Moving from 1828 to 1928, that year Scott Fitzgerald wrote to his colleague Shane Leslie: “Bernard Shaw is 61 years old; HG Wells, 51; GK Chesterton, 41; you are 31 and I am 21: the great writers of the world in arithmetic progression”. Aside from revealing Fitzgerald’s remarkable self-esteem, can any conclusions be drawn from this coincidence?

The square root of 0.999 is 0.9994…, and the square root of 0.9999999 is 0.99999994… Is it pure coincidence, or will the same thing happen with another number of nines instead of 3 or 7?

branching jokes

Regarding the unfinished jokes of last week, here are the proposals of José Luis Cruz:

What does the dot tell the asterisk?

Cut your hair, pimp!

What is the height of a bald man?

Let them nail it calmly.

Why don’t elephants play tennis?

Because they would spend the game on the tile break.

How are a lifeguard and a waiter alike?

In which the two take away what you have drunk.

And these are the “official” versions of the jokes:

Where are you going with those hairs? Having crazy ideas. Because there are no round shoes. In which the two work where others enjoy.

Personally, Cruz’s fourth option seems as good or better than the official one. What do my astute readers think?

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