Factorial
Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.
The constant that counts what does not happen
Nothing grows in a shuffled pack of cards, and nothing grows in a factorial. Yet e sits in the middle of both — as the chance that a shuffle leaves nothing in place, and as the base that makes n! nearly a power.
A tail too small to be a whole number
If e were a fraction with denominator q, then q! times e would be a whole number. It splits into a whole part and a tail, the tail is squeezed strictly between nothing and one, and there is no whole number there.
An integral that cannot be a whole number
Niven's proof that π is not a fraction is the same squeeze as the one for e, with a much harder multiplier. A polynomial supplies the whole number; its own smallness supplies the contradiction; and both halves are computable.
Nine thousand four hundred and eight
There are four Latin squares of order four once the first row and column are fixed, fifty-six of order five, and nine thousand four hundred and eight of order six. The exact answer is known for eleven orders and for no more — and yet a half-finished square can always be finished.
A rectangle cut by a curve
Integration by parts is taught as the product rule run backwards. It is also a picture: an increasing curve cuts a rectangle into two pieces, one of them the area under the curve and the other the area beside it, and the formula says only that the pieces fill the rectangle. Run repeatedly, the same cut produces the factorials and Wallis's product for π.
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.
A sum of factorials that converges at every prime
1 + 1 + 2 + 6 + 24 + 120 + … grows faster than any geometric series and has no sum in the real numbers. Measure size by divisibility instead and the same series converges at every prime at once, to a different number each time. Whether any of those numbers ends in zero is a question Đuro Kurepa asked in 1971, and a search through every prime below 100,000 says no while a coin-toss model says it should have said yes about twice.
When Euclid's number is itself prime
Multiply the primes up to p and add one. Euclid needed only that the result has a prime factor beyond p; it is itself prime for p = 2, 3, 5, 7 and 11, and then not again until 31, and then not until 379. Up to 1,100 there are nine such primes of each sign, against thirteen that a careful estimate expects. The estimate predicts infinitely many and grows by about 1.8 every time x is multiplied by e; nobody can prove there is even one more.
Named alongside it
The objects these essays reach for when they reach for this one.
HeuristicIntegralIntegralityIrrationalityPiPrimesProof by contradictionSqueezeAntiderivativeApproximationAreaAsymptotics