Speaking of primes, is it true that
12 = (1-3)(1-5)(1-7)(1-11)(1-13)(1-17)(1-19) …
where the right-side infinite product has terms for all odd primes?
Just asking.
Well, first of all, we’re going to have to introduce some sort of nonstanard convention for evaluating infinite products, as under the standard convention, that product doesn’t converge (it alternates in sign, and grows monotonically in absolute value). The answer to your question, then, might depends on what convention we use for evaluating it.
But there might be some extension of the concept of infinite products where that’s a true statement, similarly to how 1+2+3+4+5+… = -1/12 or 1+2+4+8+16+… = -1.
Indeed, it follows from the two sums at the end of Chronos’s post, in a suitable sense: multiplying (1 + 2 + 3 + 4 + 5 + …) by (1 - 3) amounts to removing all the terms which are multiples of 3; then multiplying by (1 - 5) amounts to removing the remaining terms which are multiples of 5; continuing in this way with each odd prime, we are left with just the terms 1 + 2 + 4 + 8 + 16 + …
Thus, taking 1 + 2 + 3 + 4 + 5 + … to be -1/12 and 1 + 2 + 4 + 8 + 16 + … to be -1 (in senses as explained at length in other threads here on the SDMB), we may correspondingly consider the product of (1 - p) over the odd primes p to be -1/(-1/12) = 12.
http://alumnus.caltech.edu/~chamness/prime.html
Go to this website for an applet that will generate primes for you.