Diophantine equation
Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.
How often a number can appear in Pascal's triangle
Every number bigger than one appears in Pascal's triangle at least twice, next to the 1 at each end of its own row. A few appear more often: 120 appears six times, and 3003 eight — as C(3003, 1), C(78, 2), C(15, 5) and C(14, 6), each twice. Counting every entry up to a million million finds seven numbers that appear more than four times and none that appear five, seven or more than eight. Whether any number appears ten times, or whether there is any limit at all, is unknown.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
ConjectureExhaustive searchBinomial coefficientElliptic curveFibonacci numbersHeuristicHilberts tenth problemModular arithmeticPascals triangleSum of cubes