12 Matching Annotations
1. Apr 2024
2. papers.ssrn.com papers.ssrn.com

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3. phys.org phys.org

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4. Jan 2024
5. Local file Local file
1. Lenstra’s elliptic curve algorithm for fac-toring large integers. This is one of the recent applications of elliptic curvesto the “real world,” to wit, the attempt to break certain widely used public keyciphers.
2. the emphasis of this book is on number theoretic aspects ofelliptic curves, so we feel that an informal approach to the underlying geom-etry is permissible, since it allows us more rapid access to the number theory.

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

#### Annotators

6. Dec 2023
1. 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.

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8. Mar 2023
9. www.bbc.com www.bbc.com
1. But 150 alone doesn’t tell the whole story. Other numbers are nested within the social brain hypothesis too. According to the theory, the tightest circle has just five people – loved ones. That’s followed by successive layers of 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 the idea is that space has to be carved out for any new entrants.
• 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.

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10. Sep 2022
11. Local file Local file
1. A . Y A . K H I N C H I N

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

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

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12. Jan 2021
13. journals.plos.org journals.plos.org
1. 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

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14. Mar 2016
15. www.quantamagazine.org www.quantamagazine.org
1. 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

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16. Feb 2016
17. www.math-cs.gordon.edu www.math-cs.gordon.edu
1. Number Theory: In Context and Interactive, Karl-Dieter Crisman (online textbook)

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18. Nov 2013
19. www.economist.com www.economist.com
1. What makes number theory interesting is that problems that are simple to state (at least to mathematicians) are often fiendishly difficult to solve. The most famous such problem, Fermat's last theorem, was postulated in the 17th century. It took until 1993 to prove that it was true.