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.