28 Jun '12 10:40

Choose a positive irrational number r and form two sequences;

n x (1 + r); n = 1, 2, 3, ...

m x (1 + 1/r); m = 1, 2, 3, ...

That gives two sequences of irrational numbers. Round them all down to the nearest integer. Say, with r = pi;

r = 3.141

1/r = 0.318

4.14 8.28 12.43, ... rounds down to 4, 8, 12, 16, 20, 24, 28, 33, ...

1.32 2.64 3.95, ... round down to 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, ... 31, 32, 34, ...

It seems all positive integers feature in one sequence or the other, but never both.

Is that true for pi?

If so, is that true for all other positive irrational numbers too?

n x (1 + r); n = 1, 2, 3, ...

m x (1 + 1/r); m = 1, 2, 3, ...

That gives two sequences of irrational numbers. Round them all down to the nearest integer. Say, with r = pi;

r = 3.141

1/r = 0.318

4.14 8.28 12.43, ... rounds down to 4, 8, 12, 16, 20, 24, 28, 33, ...

1.32 2.64 3.95, ... round down to 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, ... 31, 32, 34, ...

It seems all positive integers feature in one sequence or the other, but never both.

Is that true for pi?

If so, is that true for all other positive irrational numbers too?