Plenitudinous Primes

A musical proof of the infinitude of primes

A mathematical musical collaboration with the supremely talented Hannah Hoffman!

Plenitudinous Primes

Lyrics by Joel David Hamkins, Music by Hannah Hoffman

Let's prove the primes' infinitude
they do exist in multitude

of quite astounding magnitude
we count the primes in plenitude!

'Twas proved by Euclid way back when
in Elements he argued then

He proved it with exactitude
in his Book IX he did conclude

for any given finite list
there will be primes the list has missed

to prove this gem let number N
be when you multiply them then

multiply the list for fun,
multiply, and then add one

If into N we shall divide
the listed primes seen in that guide

remainder one each leaves behind
so none as factors could we find

but every number has some prime factor
with our N take such an actor

since it can't be on the list
we thus deduce more primes exist

No finite list can be complete
and so the claim is in retreat

By iterating this construction
primes do flow in vast deduction

as many as we'd like to see
so infinite the primes must be

Thus proved the primes' infinitude
in quite enormous multitude

of arb'trarly great magnitude
we count the primes in plenitude!

'Twas known by Euclid way back then
he proved it in the Elements when

in modern times we know some more
math's never done, there's new folklore:

For Bertrand, Chebyshev had said
about the puzzle in his head

and Erdős said it once again:
there's always a prime between n and 2n.

We proved the primes' infinitude
they do exist in multitude

of inconceiv'ble magnitude
we count the primes in plentitude!

The primes are plenitudinous
they're truly multitudinous!

This is one of several mathematical/philosophical music videos I have undertaken with Hannah Hoffman, all of which will eventually be posted here on Infinitely More—look for them in the Math Music Videos section.