# Birthday Card – Infinite of Primes

A new birthday card:

Then: “How about the infiniti of the primes?”

And this proof:

Theorem.

Infinitely many prime numbers exist.

Proof.

Suppose p1=2 < p2 = 3 < … < pr were all the primes that exist || Then imagine a number P = [(p1 * p2.. .pr – 1 *  pr) + 1]. If it is a prime, we’ve proved our theorem. So for now, we’ll assume it isn’t. || Then let pa be a prime that evenly divides (ie: divides without a remainder) P.  [All non-primes can be broken down into primes; so if P is not a prime, it has to be evenly divisible by prime numbers.] ||  Now pa can’t be any of the primes p1, p2, …, pr. [To see why: Suppose, for example, pa = p2  Then—remembering that P = [(p1 * p2.. .pr – 1 *  pr) + 1]—, [(p1 * p2.. .pr-21 *  pr) + 1 ]/p2 =   [(p1 * .. .pr- – 1 *  pr) + 1, which means that there’s a remainder of 1, so  pa didn’t evenly divide P after all.]

Therefore, pa would have to be a prime number not included in p1, p2, …, pr; from which it follows p1, p2, …, pr wouldn’t be all of the primes. So there indeed infinitely many prime numbers.

And on the back, these credits:

The image is “The Ladies Waldegrave” (1780-1) by Joshua Reynolds.

The proof was built from versions from Chris Caldewell of UT Martin (https://primes.utm.edu/notes/proofs/infinite/euclids.html) and from Peter Alfred of the Unversity of Utah (https://www.math.utah.edu/~pa/math/q2.html)

This card’s webpage is https://www.pureloveshop.com/2018/12/17/birthday-card-infinite-of-primes

It’s author is Bartleby Willard

Bartleby’s sites: From-Bartleby.com (upcoming book: “Love at a Reasonable Price, Vol 1: First Loves”; PureLoveShop.com (business of Pure Love); LanguagesAndLiteratue.com (literary studies