Question

Little brain teaser:
Among the natural numbers between 1 and 1,000,000, which are there more of: numbers that can be expressed as the sum of a perfect square and a positive perfect cube, or numbers that cannot be?

Answer 1

My immediate reactions is that it has to be the second thing, but reactions tend to be iffy with these kinds of questions =/

Answer 2

Eh.

Answer 3

42!!! IT'S DEFINITELY 42!!!

Answer 4

Can i write a programming solution?

Answer 5

I probably wouldn't understand it. :(

Answer 6

So, I'm going to write a programming solution then.

Answer 7

Wait, I dunno how to.

Answer 8

is 0 a perfec square?

Answer 9

Yeah.

Answer 10

So, we have all the perfect cubes.

Answer 11

that's 100 numbers.. Adding time.

Answer 12

Then, all the perfect cubes plus 1.

Answer 13

Then, all the perfect cubes plus 4.

Answer 14

100+99+99+boring. I'm out.

Answer 15

Forgive my posting so late, but I believe there are more numbers that can't be expressed as a sum of a square and a cube.

We know there are 1001 numbers that are perfect squares less than 1000000 (including 0), and 101 numbers that are perfect cubes less than 1000000 (including 0). Thus, the obvious upper bound on the most numbers we could possibly have that are a sum of a square and a cube is \(1001*101=101101\) which is less than 500000. Note that this isn't including any repeated numbers, and many of these combinations will result in numbers over 1000000.

This means that there are far more numbers that cannot be expressed as the sum of a perfect square and a positive perfect cube between 1, 1000000.