Duffin–Schaeffer conjecture
The duffin-schaeffer conjecture is about fractions, which are numbers that represent parts of a whole. For example, half is a fraction that represents one part out of two equal parts.
The conjecture says that if you take any fraction that is not equal to 0 or 1 (so it's somewhere between 0 and 1) and you add up the digits of the numerator and the denominator (which are the top and bottom numbers of the fraction), and then you raise the sum to a power, there will always be an infinite number of other fractions that have the same sum of digits when you raise it to the same power.
This might sound complicated, but it's kind of like a puzzle. Imagine you have a bunch of different fractions, like 1/2, 3/4, 5/6, and so on. If you add up the digits of each fraction's numerator and denominator, you might get different numbers, like 3 for 1/2 (1+2=3), 7 for 3/4 (3+4=7), or 11 for 5/6 (5+6=11).
Now, the duffin-schaeffer conjecture says that if you take any of those sums of digits (like 3 for 1/2) and raise it to a power, you can always find more fractions that also have that same sum of digits when raised to that same power. So if you raised 3 to the power of 3 (which means you multiply it by itself three times), you might find more fractions like 13/19 or 207/299 that also have a sum of digits equal to 27 (which is 3 raised to the power of 3). And this works for any sum of digits and any power you choose!
Even though nobody has been able to prove if the duffin-schaeffer conjecture is true for every single possible fraction, mathematicians have found a lot of evidence to support it, and it's still an interesting problem to study and try to solve.
The conjecture says that if you take any fraction that is not equal to 0 or 1 (so it's somewhere between 0 and 1) and you add up the digits of the numerator and the denominator (which are the top and bottom numbers of the fraction), and then you raise the sum to a power, there will always be an infinite number of other fractions that have the same sum of digits when you raise it to the same power.
This might sound complicated, but it's kind of like a puzzle. Imagine you have a bunch of different fractions, like 1/2, 3/4, 5/6, and so on. If you add up the digits of each fraction's numerator and denominator, you might get different numbers, like 3 for 1/2 (1+2=3), 7 for 3/4 (3+4=7), or 11 for 5/6 (5+6=11).
Now, the duffin-schaeffer conjecture says that if you take any of those sums of digits (like 3 for 1/2) and raise it to a power, you can always find more fractions that also have that same sum of digits when raised to that same power. So if you raised 3 to the power of 3 (which means you multiply it by itself three times), you might find more fractions like 13/19 or 207/299 that also have a sum of digits equal to 27 (which is 3 raised to the power of 3). And this works for any sum of digits and any power you choose!
Even though nobody has been able to prove if the duffin-schaeffer conjecture is true for every single possible fraction, mathematicians have found a lot of evidence to support it, and it's still an interesting problem to study and try to solve.