Birthday Card – Infinite of Primes

[Logbook Chapters]

A new birthday card:

Three 1700s 20-something sisters in gray wigs

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

And this proof:


Infinitely many prime numbers exist.


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 ( and from Peter Alfred of the Unversity of Utah (

This card’s webpage is

It’s author is Bartleby Willard

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

Copyright on the copyrightable aspects: AM Watson

Card available in the Birthday Card Dept of our Zazzle Store.

[Logbook Chapters]

Our first book, “Love at a Reasonable Price Volume 1: First Loves” will be available Spring 2017 at