0:00 Generating primes 1:26 Shortcut 4:42 What if 2 n - 1 is prime? 9:31 What if 2 n - 1 isn't prime? 14:46 Trial division
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Animated with Manim. https://www.manim.community Music by Callistio. Audio recorded at the Lawrence Herbert School of Communication at Hofstra University. https://www.hofstra.edu/communication/
Why Does This Recurrence Generate Primes?Eric Rowland2024-02-06 | The recurrence R(n) = R(n - 1) + gcd(n, R(n - 1)) generates primes. But why? It turns out it's essentially implementing trial division in disguise.
0:00 Generating primes 1:26 Shortcut 4:42 What if 2 n - 1 is prime? 9:31 What if 2 n - 1 isn't prime? 14:46 Trial division
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Animated with Manim. https://www.manim.community Music by Callistio. Audio recorded at the Lawrence Herbert School of Communication at Hofstra University. https://www.hofstra.edu/communication/
Web site: ericrowland.github.io Twitter: twitter.com/ericrowlandNew Breakthrough on a 90-year-old Telephone QuestionEric Rowland2024-09-05 | What numbers do you get when you iteratively scale a table? Approximations of these numbers have been used since the 1930s to predict telephone traffic and in other applications. But mathematically, the exact values are extremely complicated!
Our paper identifying the structure behind these numbers: Eric Rowland and Jason Wu, The entries of the Sinkhorn limit of an m × n matrix (25 pages). arxiv.org/abs/2409.02789
J. Kruithof, Telefoonverkeersrekening, De Ingenieur 52 (1937) E15-E25. English translation by Pieter-Tjerk de Boer: https://wwwhome.ewi.utwente.nl/~ptdeboer/misc/kruithof-1937-translation.html See Appendix 3d.
Melvyn B. Nathanson, Alternate minimization and doubly stochastic matrices, Integers 20A (2020) Article A10 (17 pages). https://math.colgate.edu/~integers/uproc10/uproc10.pdf
Richard Sinkhorn, A relationship between arbitrary positive matrices and doubly stochastic matrices, The Annals of Mathematical Statistics 35 (1964) 876-879. doi.org/10.1214/aoms/1177703591
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0:00 Predicting telephone traffic 1:16 Kruithof's example 6:02 2x2 tables 8:35 3x3 tables 15:56 Rewriting the equation for 3x3 tables 22:20 Compact equation for 3x3 tables 24:47 Larger tables 27:40 Answer to Kruithof's example
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Animated with Manim. https://www.manim.community Music by Callistio. Audio recorded at the Lawrence Herbert School of Communication at Hofstra University. https://www.hofstra.edu/communication/
Web site: ericrowland.github.io Twitter: twitter.com/ericrowlandIn 2003 We Discovered a New Way to Generate PrimesEric Rowland2023-01-20 | There is a Fibonacci-like recurrence that seems to generate primes! It was discovered in 2003, but at the time no one understood why it worked. A few years later, I plotted the primes in a way that reveals some hidden structure. This is a tale of logarithmic scale.
Fernando Chamizo, Dulcinea Raboso, and Serafín Ruiz-Cabello, On Rowland's Sequence, The Electronic Journal of Combinatorics 18(2) (2011) P10 (10 pages). doi.org/10.37236/2006
Benoit Cloitre, 10 conjectures in additive number theory (2011) (46 pages). arxiv.org/abs/1101.4274
Serafín Ruiz-Cabello, On the use of the least common multiple to build a prime-generating recurrence, International Journal of Number Theory 13 (2017) 819–833. doi.org/10.1142/S1793042117500439 Open access: arxiv.org/abs/1504.05041
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0:00 Recurrence 2:59 Doubling relations 4:03 Plotting locations of primes 6:24 Clusters of primes 9:49 Predicting primes in each cluster 15:22 Answers to burning questions 18:19 Changing the initial term 20:08 Cloitre's lcm recurrence
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Animated with Manim. https://www.manim.community Thanks to Ken Emmer for supplying the microphone.
Web site: ericrowland.github.io Twitter: twitter.com/ericrowlandAn Exact Formula for the Primes: Willans FormulaEric Rowland2022-09-22 | Formulas for the nth prime number actually exist! One was cleverly engineered in 1964 by C. P. Willans. But is it useful?
0:00 A formula for primes? 1:24 Engineering a prime detector 4:00 Improving the prime detector 5:46 Counting primes 6:29 Determining the nth prime 9:42 The final step 11:36 What counts as a formula? 12:56 What's the point? 13:51 Who was Willans?
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Animated with Manim. https://www.manim.community Thanks to Ken Emmer for supplying the microphone.
Web site: ericrowland.github.io Twitter: twitter.com/ericrowland1 Billion is Tiny in an Alternate Universe: Introduction to p-adic NumbersEric Rowland2022-08-05 | The p-adic numbers are bizarre alternative number systems that are extremely useful in number theory. They arise by changing our notion of what it means for a number to be large. As a real number, 1 billion is huge. But as a 10-adic number, it is tiny! #SoME2
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Notes and references:
The last 30 digits of 2^1000000 and other large powers can be computed using modular arithmetic, by working modulo 10^30. In Mathematica, use the function PowerMod. In Python, use the third argument of pow. These functions implement the method of repeated squaring or one of its variants: en.wikipedia.org/wiki/Exponentiation_by_squaring en.wikipedia.org/wiki/Modular_exponentiation
The 2-adic limits arising from the (2^n)th Fibonacci numbers were established on page 216 of this paper: Eric Rowland and Reem Yassawi, p-adic asymptotic properties of constant-recursive sequences, Indagationes Mathematicae 28 (2017) 205–220. doi.org/10.1016/j.indag.2016.11.019
0:00 Introduction 2:16 Properties of the real numbers 3:19 10-adic integers 6:55 Properties of the 10-adic integers 10:06 Division? 12:47 Limit points 13:50 5-adic limit 15:36 Fibonacci numbers 16:31 Square roots of -1 18:25 What are p-adics good for?
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Animated with Manim. https://www.manim.community Music by Marc Rowland and Cody Leavitt. Thanks to @catpfaff for helpful feedback on an earlier version.