Concept

Inclusion exclusion

Counting a union by adding the parts, subtracting the overlaps, adding back the triple overlaps, and continuing. It is exact and has one term per subset, so every practical use of it is a truncation — and truncating alternates between over- and under-counting.

Named by 2 essays across 2 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.

analysis · The exponential

Two sums of reciprocals: 2.89 and climbing against 1.71 and level. Two curves against the logarithm of the bound: the sum of reciprocals of all primes, rising steadily, and the sum over the twin primes, flattening towards a limit.

The sieve that cannot finish

Sifting out the composites is the oldest method in the subject and it has a ceiling nobody has raised. The reciprocals of the twin primes add to a finite number, so no argument that measures thickness can reach them — and the inclusion–exclusion every sieve truncates goes wildly wrong before it goes right.

number · Infinitude of primes

Named alongside it

The objects these essays reach for when they reach for this one.

Approximation Convergence Counting argument Derangement e, the number Factorial Limit Open problem Power series Prime Probability Sieve

All concepts