I guess people may be familiar with the sorts of memes with equations involving fruit that are generally relatively trivial, while claiming something like “95% of people can’t solve this!” Sometimes this leads to amusing parodies. However someone went the whole other way and created an real, solvable instance of the problem using only natural numbers whose smallest solution is … very large.
I like the way this type of problem can actually open up a dialogue of how mathematics is really open ended, creative and unfinished. Framing a fruit equation problem as a filter that makes people either feel stupid for not understanding it (it is basically algebra, after all), or smug and superior for getting it, is extremely unhelpful. But explaining that what can seem like an easy problem actually requires a lot of work, and would stump professional mathematicians (cf Fermat’s last theorem), is a good conversation starter. It can also touch on things like how geometry is brought to bear on algebraic problems (and vice versa!), how the solution can use methods not present in the problem and so on.
So, in honour of this, I thought I take what seems to be, at time of writing, the simplest currently unsolved Diophantine equation (that is: a multivariable polynomial, and looking only for whole-number solutions), and turn it into a fruit equation. We can think of it as trying to count fruit:
Here ‘simplest’ is according to the notion of “size” defined in this MathOverflow question, basically it’s a measure of how large all the powers and coefficients of a multivariable polynomial is. There are only finitely many polynomials of a given size. The polynomial from the picture is , and has size 29. Every polynomial of size 28 and smaller has either been solved or shown to have no solutions. The idea is to see (experimentally) where the rough threshold is between equations than be solved in a more-or-less elementary way, and where really serious techniques or obstructions really kick in, of the sort outlined here.
I’m happy for people to share the above image as widely as they can. If nothing else, maybe someone will actually solve the equation above, and contribute a piece of new knowledge!
EDIT: check out the neat new preprint Fruit Diophantine Equation (arXiv:2108.02640) on this problem