Rational Numbers That Are Hard To Prove?

[Office_Shredder]

There are lots of examples of numbers where "is it a rational number" has been an open question for a while before being proved as not being rational. Pi and e being famous examples. Some of them are still open, like pi+e, or the Euler-Mascharoni constant, but I think the general consensus is that constants like these almost surely are going to be irrational. So I was wondering are there any examples of numbers whose rationality was unknown for an extended period of time (Beyond the first time the question was raised would be interesting, but something on the order of decades would be really nice) before it was discovered that the number is rational?

 

[fresh_42]

This is rather unlikely. The problem comes with the question itself: you have to describe the number somehow. And this description is either within rationals or so complex, that it is almost certainly irrational, if not transcendent. I thought of something like the matrix exponent, the lower bound for the complexity of matrix multiplication, but it is unknown. And so are probably all interesting examples.

 

[mfb]

Legendre's constant was introduced 1808. It was shown to be exactly 1 (and therefore rational) in 1899.

This problem lead to the question what the lower limit on c is (Erdős, 1940). The answer is 0 (shown 2005).

There is some other more famous constant where I don't find the name now that is suspected to be exactly zero.

 

[fresh_42]

Legendre's constant was introduced 1808. It was shown to be exactly 1 (and therefore rational) in 1899.

This problem lead to the question what the lower limit on c is (Erdős, 1940). The answer is 0 (shown 2005).

There is some other more famous constant where I don't find the name now that is suspected to be exactly zero.

1798 (Legendre) 1849 (Chebyshev)