Concept

Twin primes

Pairs of primes differing by two. Nobody knows whether there are infinitely many, and the reciprocals of the ones there are add to a finite number — which rules out the argument that settles the primes themselves.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Five families of primes, counted below 100,000. Five counting curves on logarithmic axes: every prime, the primes one more than a multiple of four, the twin pairs, the primes one more than a square, and the Mersenne primes. Two of the five families are known to be infinite and three are open questions.

Which infinitudes are proved

The primes never stop, and neither — apparently — do the twin pairs, the primes one more than a square, or the Mersenne primes. Three of those four statements are theorems and one is not, and counting the members of each family tells nobody which.

number · Infinitude of primes

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.

Open problem Prime Asymptotic Conjecture Convergence Counting argument Divergence Heuristic Inclusion exclusion Sieve

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