The Square-Sum Problem (extra footage) - Numberphile
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it as a follow-up on how the square-sum sequences were found computationally, when discussing the Square-Sum Problem and Hamiltonian paths.
Recommended by 1 people across 1 different months, 2018 to 2021.
What it is
Bonus footage from Numberphile in which Matt Parker and Charlie Turner explain how they used computer code (written in SageMath) to show that the numbers up to 91 can be arranged so every adjacent pair sums to a perfect square. It follows the main Square-Sum Problem video and focuses on the graph and Hamiltonian path approach.
What people said
Recommends Mr Pie · 2018-05 · Mathematics Stack Exchange · on “Integers and Number-theory”
It is conjectured that there is not a maximum value of $n$. See this post [LINK] . It is a question of mine regarding exactly this, and today, it still remains an open problem such that most people I know refer to it as the "Square-Sum Problem." An Australian mathematician by the name of Matt Parker perhaps introduced this problem or popularised it in one of his books, Things to Make and Do in the Fourth Dimension $(2014)$. In the book, he conjectured that there always exists such a sequence for $n > 89$, and then $n > 91$. However, in a Numberphile [LINK] YouTube video, Matt Parker and his friend talks about how they tested this conjecture for more values, and as far as I know, the conjectu
Cites Alexander Day · 2018-01 · Mathematics Stack Exchange · on “constraints to the hamiltonian path: can one tell if a path is hamiltonian by looking at it?”
A few days ago, the channel Numberphile [LINK] released this video [LINK] on the square-sum problem. The show later released this follow-up [this video] video on how they proved that every number from 25 to 91 can have themselves and all the preceding numbers be ordered so the sum of any two adjacent numbers add up to a square. (confused? check out the video!) Anyway, I wanted to see if I could get, say $100$ somehow, but the lines linking the numbers on a graph are way too complex for me to tell where a line goes. So I'm doing it the hard way... First, I made a table from $1-100$ dictating what numbers would be added to said number to get a square number, like $100$ or $4$. To keep this question
Cites Prasun Biswas · 2021-12 · Mathematics Stack Exchange · on “Proving the number circle internet meme, where sum of each adjacent numbers is a perfect square”
A bit of searching informed me that this conjecture was proved in 2018 by Robert Gerbicz (there is a Hamiltonian cycle for the square-sum chain for all $n\ge 32$) (see this Mersenneforum post [LINK] ). Related OEIS sequences: 1 [LINK] , 2 [LINK] and a few others you can find linked there. There's also some Numberphile videos on YouTube on this, see here [LINK] and here [this video] .