114³ = 18³ + 60³ + 108³ = 57³ + 76³ + 95³: Resolution of the last remaining case below 1000 in the sum-of-three-cubes problem

We report the discovery of two positive integer solutions to the equationx³ + y³ + z³ = 114³,namely114³ = 18³ + 60³ + 108³and114³ = 57³ + 76³ + 95³. This resolves the final remaining unsolved case below k = 1000 in the century-old problem of determining which integers can be expressed as the sum of three cubes. After the solutions for k = 33 and k = 42 in 2019 and k = 165 in 2023, the number 114 was the very last outstanding case. Both solutions were discovered on 31 January, 2026 using a single consumer laptop in under four hours via an exhaustive O(n²) search with precomputed cubes and hash-table lookup up to n = 10⁶. No other solutions exist in this range.