From Mathologer.
The third video in a trilogy of Mathologer videos dealing with sum-equals-product identities and equations. The other two are:
Way beyond the golden ratio The power of AB=A+B (Mathologer masterclass)
Heron’s formula: What is the hidden meaning of 1 + 2 + 3 = 1 x 2 x 3 ?
Today’s video is all about how 2+2=2×2 and 1+2+3=1x2x3 first extends to an infinite sequence of similar identities and then further to a whole world of sum-equals-product identities. And we chase down a new connection to the so-called Sophie Germain primes.
00:00 Intro
06:27 How many?
11:29 New from old
19:53 Sophie’s primes
27:47 Beyond Sophie
30:15 Thank you!!
Currently the following paper is the best introduction to the sum-equals-product problem circle of ideas:
Maciej Zakarczemny, On the equal sum and product problem
https://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/1662
Very nice translation of this circle of ideas into a set of problems for the Michigan Lemma 2020 maths competition (starts with Problem 2)
https://websites.umich.edu/~michiganlemma/material/2020/PowerRound_Solutions.pdf
Check out the Encyclopedia and integer sequences pages for
2,3,4,6,24,114,174,444 and
7,8,9,10,12,14,15,16,18,20,22,30
https://oeis.org
Some other papers to check out with links:
Michael W. Ecker, When Does a Sum of Positive Integers Equal Their Product?
https://www.tandfonline.com/doi/abs/10.1080/0025570X.2002.11953100
M. A. Nyblom, Sophie Germain primes and the exceptional values of the equal-sum-and-product problem
https://www.fq.math.ca/Papers1/50-1/Nyblom.pdf
Maciej Zakarczemny, Equal-Sum-Product problem II
https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/equalsumproduct-problem-ii/5A5B79FDCB28A1E83D96DD6BA9147FE2
M. A. Nyblom and C. D. Evans, An Algorithm to Solve the Equal-Sum-Product Problem
https://arxiv.org/abs/1311.3874
Leo Kurlandchik and Andrzej Nowicki, When the sum equals the product
https://web.archive.org/web/20230122055244/https://www-users.mat.umk.pl/~anow/ps-dvi/si-krl-a.pdf
Combo class video on the same question https://youtu.be/aCa5sGEtUCs?si=MxNDotKtCu17RIKS
Check out the list of updates in my comment that’s pinned to the top of the comment section.
Music: Just jump by Ian Post
T-shirt: Forgot where I got this one from 🙁
Enjoy!
Burkard
