Sum of cubes
Named by 2 essays across one field — each of them below, with the objects they name alongside it.
Thirty-three as three cubes
Which numbers are a sum of three cubes, allowing negatives? Nothing like the rule for two squares is known — only that numbers leaving 4 or 5 on division by 9 are impossible. Every other number below a hundred has a sum with cubes under 5,000, except eight; 33 needed sixteen-digit cubes and was found in 2019.
Nine cubes for two numbers
Every whole number is a sum of at most nine positive cubes, and only two numbers, 23 and 239, need all nine. Fifteen need eight, the last of them 454; 121 need seven, the last 8,042; six are needed for the last time at 1,290,740. The count every number needs is nine, fixed by two small accidents. The count every large number needs is somewhere between four and seven, and the computation says four — a question about how the cubes spread out, not about the cubes themselves.
Named alongside it
The objects these essays reach for when they reach for this one.
Exhaustive searchHeuristicModular arithmeticConjectureDensityDiophantine equationElliptic curveGamma functionHilberts tenth problem