Huge thanks to Jane Street! https://www.janestreet.com/join-jane-street/ Check out Ben Sparks's GeoGebra files. Binet formula 2D complex output: https://www.geogebra.org/m/twvvzpga 3D imaginary output of Binet formula: https://www.geogebra.org/m/z6dy9cj5 3D plot of absolute output of Binet formula: https://www.geogebra.org/m/pb7hmxyd My four-part series on Numberphile videos about Fibonacci Numbers (from 2014) starts here. https://www.youtube.com/watch?v=D8ntDpBm6Ok Here is me going on about the square root of five (Numberphile 2018). https://www.youtube.com/watch?v=z1THaBtc5RE This was the Fibonacci puzzle video from Matt Parker's Maths Puzzles. https://www.youtube.com/watch?v=ILrqPpLpwpE Read a whole bunch about "Generalized Fibonacci Sequences and Binet-Fibonacci Curves". https://arxiv.org/pdf/1707.09151.pdf The zero I found was at -9.14202391817 + 2.80064954276i and you can see the exact form here: https://www.wolframalpha.com/input/?i=%2810*pi%29%2F%28pi+%2B+i*%28log%28sqrt%285%29+-+1%29+-+log%281+%2B+sqrt%285%29%29%29%29 Try it for yourself and put the Binet Formula (((1+sqrt(5))/2)^n - ((1-sqrt(5))/2)^n)/sqrt(5) in the Wolfram roots calculator: https://www.wolframalpha.com/widgets/view.jsp?id=b858339e64fa997454dd12f77cb1ece1 This site has everything you'll ever need to know about Fibonacci Numbers. http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibFormula.html Buttercup - The original buttercupchallenge https://www.youtube.com/watch?v=l17l_KTAVJ0 CORRECTIONS This was a long video and in hindsight there are a few things I wish I had phrased better. Here they all are: - I misspoke around 01:13 when I said "negative one, zero" as it is clearly "negative one, one, zero". - At 07:53 I mean the negative values -5 to 0. I said it a weird way. - My language at about the 1D input to 2D plot from 09:17 is a bit sloppy. The real values going into the Binet function are not the horizontal axis shown; the plot onscreen is solely the output. - I say "axis" when I mean "plane" or even "complex plane". The big flat thing. Let me know if you spot anything else! Thanks again, as always, for Jane Street being my principal sponsor. https://www.janestreet.com/ Thanks to my Patreon supports who do support these videos and make them possible. Here is a random subset: Loren Thomas Richard Dickins Barry Salter Susan Moury Sarah Gerweck Ulrich Kempken Piotr Gary Martin Euler Daniel DeJarnatt Support my channel and I can make more videos: https://www.patreon.com/standupmaths Filming and editing by Matt Parker Music by Howard Carter (excluding Buttercup) Design by Simon Wright and Adam Robinson MATT PARKER: Stand-up Mathematician Website: http://standupmaths.com/ US book: https://www.penguinrandomhouse.com/books/610964/humble-pi-by-matt-parker/ UK book: https://mathsgear.co.uk/products/5b9fa76f230ffa140094dc43 Nerdy maths toys: http://mathsgear.co.uk/