Gamma function
Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.
Only one curve through the factorials bends the right way
Infinitely many smooth curves pass through 1, 1, 2, 6, 24, 120, and some of them even obey the factorial's own rule, f(x + 1) = (x + 1)·f(x), at every x. Ask that the logarithm of the curve bend upwards everywhere and exactly one survives — Euler's integral. A wiggle of any size breaks the condition somewhere, the smaller the wiggle the further out, and the proof that nothing else survives is a squeeze that computes √π along the way.
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.
ConvexityDensityExhaustive searchFactorialFunctional equationHeuristicIntegralInterpolationModular arithmeticSum of cubesUniqueness