Smooth number
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
The largest prime in a typical number
Count a number's prime factors and the answer is about log log n. Ask instead how big the largest one is, and the answer is a power of n with a law of its own: half of all large numbers have a prime factor above n^0.6065, and the share whose primes all stay below n^(1/u) is Dickman's ρ(u), a function defined by its own past. The same law governs the longest cycle of a random shuffle, and there it arrives at once; for the numbers themselves, at ten million, it is still visibly on its way.
Two neighbours share out their small primes
A number and the next one have no prime in common, and the natural guess is that their largest primes are independent. In the limit that is believed and nearly proved. Below ten million it is visibly false: neighbours are both made of small primes only nine-tenths as often as independence predicts, a run of four only two-thirds as often, and the correlation between their largest primes is negative at distance one and positive at distance six — because a small prime that divides one of two neighbours is a small prime the other cannot have.
Sums with too few primes
Add 2 to 6,436,341 and the sum is 23⁵: three numbers with only four primes between them, and their product 15,042 is far smaller than the sum. The abc conjecture says this can happen only so far. Of 606 such triples up to 200,000, only seven reach quality 1.4, and if the conjecture holds, Fermat's last theorem and much else follows for large exponents in a line.
Named alongside it
The objects these essays reach for when they reach for this one.
Largest prime factorPrime factorisationSieveAsymptoticsCorrelationCycleDifferential equationExhaustive searchFactoringGreatest common divisorIndependenceOpen problem