The taxicab numbers are those integers of the form

$$N=a^3+b^3=c^3+d^3$$

where $a, b, c, d \in \mathbb{N}$ and $\{ a, b \} \neq \{c, d \}$. Famously, the first taxicab number is $1729 = 1^3+12^3 = 9^3 + 10^3$.

Let us consider the list of taxicab numbers in ascending order and keep track of what proportion of them are even. After the first $30,000$ terms, it appears that a bias towards evenness is present:

This seems surprising to me. After $30,000$ terms, we have about $\sim 63%$ of taxicab numbers being even, quite substantially more than half. Why does this appear to be the case?

This apparent bias seems to come from the fact that the numbers are sums of cubes in two different ways. Indeed, the weaker property of a number merely being the sum of two cubes results an even split between even and odd numbers - intuitively this is to be expected, with formal argument can be made along the lines of observing that the sums of cubes modulo $2$ are $0^3+0^3, 0^3+1^3,1^3+0^3$ and $1^3+1^3$, two of which are $0$ and two of which are $1 \mod 2$.

It's possible that this trend does not continue and after some extensive enumeration of the taxicabs, perhaps the proportion drops back down again (similar surprising things can happen). But the consistency of the trend is so consistent$^1$ and remarkable that I thought a question about it is of note regardless, even if it is merely a coincidence.

$^1$ indeed, the only times that we ever have more odd than even taxicabs enumerated in my data is after enumerating the first $1, 35$ and $37$ taxicab numbers.

The taxicab numbers are those integers of the form

$$N=a^3+b^3=c^3+d^3$$

where $a, b, c, d \in \mathbb{N}$ and $\{ a, b \} \neq \{c, d \}$. Famously, the first taxicab number is $1729 = 1^3+12^3 = 9^3 + 10^3$.

Let us consider the list of taxicab numbers in ascending order and keep track of what proportion of them are even. After the first $30,000$ terms, it appears that a bias towards evenness is present:

This seems surprising to me. After $30,000$ terms, we have about $\sim 63\%$ of taxicab numbers being even, quite substantially more than half. Why does this appear to be the case?

This apparent bias seems to come from the fact that the numbers are sums of cubes in two different ways. Indeed, the weaker property of a number merely being the sum of two cubes results an even split between even and odd numbers - intuitively this is to be expected, with formal argument can be made along the lines of observing that the sums of cubes modulo $2$ are $0^3+0^3, 0^3+1^3,1^3+0^3$ and $1^3+1^3$, two of which are $0$ and two of which are $1 \mod 2$.

It's possible that this trend does not continue and after some extensive enumeration of the taxicabs, perhaps the proportion drops back down again (similar surprising things can happen). But the consistency of the trend is so consistent$^1$ and remarkable that I thought a question about it is of note regardless, even if it is merely a coincidence.

$^1$ indeed, the only times that we ever have more odd than even taxicabs enumerated in my data is after enumerating the first $1, 35$ and $37$ taxicab numbers.