ᔥ[[Michael Gibb]] in Breakthrough in prime number theory demonstrates primes can be predicted

Breakthrough in prime number theory demonstrates primes can be predicted by [[Michael Gibb]]

The approach of the book is underpinned more by number theory rather than geometry, though the geometry is used for some basic motivations.

How to fold and cut a Christmas star<br /> Christian Lawson-Perfect https://www.youtube.com/watch?v=S90WPkgxvas

What a great simple example with some interesting complexity.

For teachers trying this with students, when one is done making some five pointed stars, the next questions a curious mathematician might ask are: how might I generalize this new knowledge to make a 6 pointed star? A 7 pointed star? a 1,729 pointed star? Is there a maximum number of points possible? Is there a minimum? Can any star be made without a cut? What happens if we make more than one cut? Are there certain numbers for which a star can't be made? Is there a relationship between the number of folds made and the number of points? What does all this have to do with our basic definition of what a paper star might look like? What other questions might we ask to extend this little idea of cutting paper stars?

Recalling some results from my third grade origami days, based on the thickness of most standard office paper, a typical sheet of paper can only be folded in half at most 7 times. This number can go up a bit if the thickness of the paper is reduced, but having a maximum number of potential folds suggests there is an upper bound for how many points a star might have using this method of construction.

• Paraphrase
• 150 alone doesn’t tell the whole story.
• Other range numbers are nested within the social brain hypothesis.

• curiously, Dunbar recognized they were all multiples of 5.

  • the tightest circle has just 5 people (loved ones).
  • 15 (good friends),
  • 50 (friends),
  • 150 (meaningful contacts),
  • 500 (acquaintances) and
  • 1500 (people you can recognise).

• People migrate in and out of these layers,

• but that space has to be carved out for any new entrants.

Aleksandr Yakovlevich Khinchin https://en.wikipedia.org/wiki/Aleksandr_Khinchin

Khinchin, Aleksandr Yakovlevich. Continued Fractions. 3rd ed. Chicago: University of Chicago Press, 1964.

Parag. K. V., Donnelly. C. A., (2020) Using information theory to optimise epidemic models for real-time prediction and estimation. PLOS. Retrieved from https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1007990

New property of prime numbers discovered. Primes greater than 5 can end with 1, 3, 7, or 9. The next prime is less likely to end with the same digit, and biased toward one of the remaining three. For instance, a prime ending in 3 is most likely to be followed by a prime ending in 9. The bias evens out as the primes get larger, but only very slowly.

http://arxiv.org/abs/1603.03720

Number Theory: In Context and Interactive, Karl-Dieter Crisman (online textbook)