Smallest Positive Integer not of form +-4 mod 9 not representable as Sum of Three Cubes

Conjecture

It is possible to express every positive integer, not of the form $\pm 4 \pmod 9$, as the sum of cubes of $3$ integers.

Progress

In $1997$, in his Curious and Interesting Numbers, 2nd ed., David Wells reported that $30$ was the smallest positive integer that had not been so represented.

Such a representation was found in $1999$:

$30 = 2220422932^3 + \paren {-2218888517^3} + \paren {-283059965^3}$

As of $2019$, the smallest positive integer for which such a representation has not been found is $114$.

Sources

• 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.)

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $30$