While in Southern France, Simon really enjoyed solving this puzzle (he originally saw in a Brilliant.org vid). He was so happy with his solution he kept drawing it out on paper and in digital apps, and later shared the puzzle on Twitter. This sparked quite a few reactions from fellow math lovers, encouraged Brilliant to tweet new puzzles and now Brilliant follows Simon on Twitter, how cool is that!
Simon was showing Dad a graph of how technology is developing exponentially, y = a^x. Dad asked for a specific value of a, and Simon said: “All exponentials are stretched out or squished versions of the same thing.” He then quickly came up with the proof (“a few lines of relatively simple algebra”). “If all exponentials are pretty much the same, that means that all exponentials have proportionately the same derivative.”
Simon has written a code in Python that generates primes using the finite list from Euclid’s proof that there are infinitely many primes. “Starting with one prime (2) the code uses the finite list to generate a couple more numbers that aren’t in the list but are primes. It may not even get to all the primes in the long run!” There is only one problem with Simon’s algorithm…
Simon has written down Euclid’s proof in his own words first https://imgur.com/ML2tI6n
and then decided to program it in Python.
Simon explains that the Van Eck Sequence is and shows the patterns he has discovered in the sequence by programming it in Python and plotting it in Wolfram Mathematica. Simon’s project in Wolfram is online at: https://www.wolframcloud.com/objects/4066d93a-893b-4a99-9fdc-54e265d27888
He also shows Neil Sloane’s proof of why the sequence is not periodic and adds an extra bit to make the proof more complete.
This video is inspired by the Numberphile video about the Van Eck sequence.
Simon working on his proof of the Fundamental Theorem of Arithmetic (he got stuck and then searched for existing proofs online).
Simon shows three false proofs. Can you find the mistake in each proof? Simon reveals the answers to the first two. Try to give your answer to the third one.
And the answer is:
Simon explains why the proof that root 4 is irrational is false and shows a couple more related theorems (he came up with) generalizing the relationship between the exponent and the factor of a number.
Simon explains: “Induction is a mathematical term, type of mathematical proof, if you have a couple of base cases (n base cases), then the inductive hypothesis implies that for the previous n values the statement holds. It proves that if the inductive hypothesis is true, the next value will also hold”.
Below, Simon used induction to prove that “any Lucas number and Lucas number after that divided by 5 equals the Fibonacci number between the Lucas numbers”:
Simon saw this proof on the Numberphile channel.