This is what Theoni Pappas writes on page 115 of her Magic of Mathematics:
"It seems the further out one ventures in the field of whole numbers, the primes become rarer and rarer. One might think that because they appear less and less frequently, that perhaps they end somewhere. As early as about 300 B.C. Euclid provided the first proof that the prime numbers are infinite."
Euclid -- hey, that name sounds familiar to readers of this Geometry blog! Yes, the same Euclid after whom Euclidean geometry is named is also the one who proved the infinitude of primes.
Pappas continues, "He used indirect reasoning..." -- that's right, as in indirect proof. It's interesting that just as we cover the U of Chicago's Chapter 13 on indirect reasoning, we read about so many indirect proofs in Pappas.
Notice the similarity between Cantor's Diagonal Argument and Euclid's infinitude of primes. Cantor assumes that the reals are countable and derives a contradiction, so the reals are uncountable. And so Euclid assumes that the primes are finite and derives a contradiction, so the primes are infinite.
Pappas assumes that the finite list 2, 3, 5, 7, and 11 contains all the primes. She then multiplies these primes and adds 1 to obtain 2311. But this number can't be divisible by 2, 3, 5, 7, or 11 since each leaves a remainder of 1. Therefore 2311 must be a new prime! Pappas then generalizes this to the case where n is the largest prime. In each case we generate a new prime not on the list, and so n is not the largest prime -- a contradiction, therefore there is no largest prime. The primes are infinite. QED
By the way, tonight's the first episode of Genius, about Albert Einstein, on National Geographic. I'm a science teacher, so I'll watch the miniseries, but I may or may not write about it on the blog.
Notice that Euclid's proof generates a much larger prime that's not on the list, in this case 2311. It doesn't necessarily give the actual next prime, which is 13. Oh, and speaking of 13...:
This is what I wrote two years ago about today's test. As it turned out, I originally posted this test in 2015 on Friday the 13th, and I mentioned this fact throughout the test. Unfortunately, this year today is neither Friday nor the 13th. But if it's any consolation, this is my 13th post blog post in April:
Here's an answer key for the test:
1. a. 90 degrees. I could have made this one more difficult by choosing a heptagon, or even a triskaidecagon, but I just stuck with the easy square.
b. Here is the Logo program:
REPEAT 4 [FORWARD 13 RIGHT 90]
Notice that the side length is 13. I'll still find a way to sneak 13, if possible, into each problem.
2. a. If a person is not a Rhode Islander, then that person doesn't live in the U.S.
b. If a person doesn't live in the U.S., then that person isn't a Rhode Islander.
c. The inverse is false, while the contrapositive is true.
Notice that Rhode Island is the thirteenth state.
3. y = 10.
4. There is a line MN. (M is the thirteenth letter of the alphabet.)
5. Every name in this list is melodious.
6. The equation has no solution. (This question references 13, as 13x appears in the expansion.)
7. a. 13, 11, 9 (descending odds).
b. 13, 17, 19 (increasing primes -- of course, Euclid proved that this sequence is infinite).
8. a = 2, b = 1, c = 3.
10. I discussed this problem earlier this week. It is the same as the problem from the Glencoe text, except that I only drew half of the figure -- the part where a contradiction appears.
Assume that the figure is possible. Then ABC is isosceles, therefore angles A and C are each 40 degrees (as the third angle of the triangle is 100). Then ABO is isosceles (as it has two 40 degree angles), so AO = BO = 3. Then by the Triangle Inequality, 3 + 3 > 8, a contradiction.
11. Through any two points, there is exactly one line. (This is part of the Point-Line-Plane Postulate.)
12. a. KML measures 13 degrees.
b. K measures less than 167 degrees.
c. L measures less than 167 degrees. (This is the TEAI, Exterior Angle Inequality.)
13. a. Law of Ruling Out Possibilities.
b. You forgot to rule out another possibility -- that nothing bad will happen to you today. Hopefully, this will be true for you.