I was having a discussion with a colleague this week about common errors kids make in math.
Here’s one that was used as an example:
Mistaking:
A + B
C + D
for:
A/C + B/D
It’s an elementary mistake — the two expressions are actually very different.
But not always. What if we considered just the class of numbers for which those two statements always product the same results?
That might turn out to be an interesting set of numbers in all sorts of ways.
And what if we were to play the same game with other erroneous equations?
One of those equations could just lead us to some actual math — and maybe that actual math might be very cool.
Mersenne primes are a good example. It was once widely thought that, if p was prime, 2^p-1 was also prime. (This was back in the days when arithmetic was still usually done with Roman numerals, so testing even simple cases was a nontrivial exercise.) Even though this has been known to be false for 500 years, the cases where it is true – the Mersenne primes – have been a fruitful field for mathematicians ever since, and the question of whether the number of Mersenne primes is infinite is still one of the most famous unsolved problems in mathematics.
Yes, that’s a great example!!