What's the next freak identity? A new deep connection with Sophie Germain primes

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) https://youtu.be/cCXRUHUgvLI Heron’s formula: What is the hidden meaning of 1 + 2 + 3 = 1 x 2 x 3 ? https://youtu.be/IguNXoCjBEk Today's video is all about how 2+2=2x2 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