Some years ago, the
Frau was 28 years old: one day she became 29 and I cockily wrote in her birthday greeting "
No longer perfect, but irreducible", and since that date have paid close attention to the primality or otherwise of our respective ages.
She was pleased and impressed when I noted that both before and after our recent birthdays (many thanks for your cards and gifts), both our ages were semiprime: which is to say all four numbers have precisely 2 prime factors (but you knew that already).
Which leads to the obvious questions:
- What is the density of semiprimes?
- What is the density of consecutive semiprimes: in particular is it asymptotically >0?
It took some web-searching to determine that an
answer to the first question is
x log log x / log x
- a result I was unable to prove myself that is predictably due to
Landau, although this is reported as a poor approximation, and better, less quotable, asymptotic formulae exist.
There does not seem to be a clear answer to the second question, which it need not surprise us pre-occupied the ubiquitous
Paul Erdös (did I mention I once
beat him at chess?).
A simple
computer program
suggests there are many consecutives, and indeed triples (clearly, 4 in a row is no-go). Heath-Brown has shown there is an infinite number of such pairs [1] but I am unsure of their density.
Anyway, if I live to be 141-142, the pair of us can enjoy this happy numerological event again.
References
- D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika 31, 141–149, 1984.