## Carnival of Mathematics #52

Marin Mersenne (1588 – 1648) was a French theologician, philosopher, and music theorist. One of his works, Cogitata Physico-Mathematica, investigates both acoustic and physical phenomena. Here’s a sample picture:

In the introduction to the work he discusses perfect numbers. Perfect numbers are the sum of the proper divisors of the same number; 28 is perfect because 1 + 2 + 4 + 7 + 14 = 28. Mersenne knew Euclid’s proof that $(2^{n-1})(2^{n}-1)$ gives an even perfect number whenever $2^{n}-1$ is prime. Theorizing what other values of $2^{n}-1$ are prime, Mersenne decided on values of n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257, and that these were the only values up to 257 where $2^{n}-1$ is prime.

His list did have errors: he missed n = 61, 89, and 107 (which also make $2^{n}-1$ prime), but he also named two values of n which shouldn’t have been on the list at all.

The first mistaken value (n = 67) was discovered in 1876 by Edouard Lucas. However, the actual factors weren’t known until 1903. In what may be the most bizarre lecture ever presented, Frank Cole stood up at meeting of the American Mathematical Society to deliver a talk entitled On the factorisation of large numbers. Cole never spoke; he simply wrote

$2^{67}-1 = 147573952589676412927$

and then followed with the numbers 761838257287 and 193707721 and performed long multiplication, obtaining $2^{67}-1$ to the acclaim of a standing ovation.

The second error (n = 257) was found by Maurice Kraitchik (of the two envelopes problem) in 1922. If $2^{257}-1$ is generalized for any base, using the formula

$\frac{b^{257}-1}{b-1}$

then the smallest base b for which the number is prime is …

… 52! And with that, we start the Carnival of Mathematics.

Kate Nowak presents an example designed to fool her students of a pattern that breaks down.

Edmund Harriss writes about (with lovely pictures!) the making of a mathematical sculpture.

David Richeson celebrates May Day by devising and investigating the maypole braid group. (Nice pictures on this one too!)

Let’s Play Math has been making fantastic math calendars. Perhaps your students would like to contribute to the next one?

Dick Lipton has been making gobs of good posts lately, but to start I recommend his explanation of Cantor’s Non-Diagonal Proof.

Terry Tao has rescaled the United States federal budget to the scale of 100 million:3, to give a better perspective of the impact of spending. (I’ve already stolen this concept for a new lesson plan.)

Sam Shah asks which books would make good awards for students.

My fellow conspirators at 360 in boosting the math history content of the blogosphere discuss Mayan math and the Dresden Codex, which now has full-color pictures available on the Internet.

David Eppstein poses an intriguing problem involving half-integer triangles and Pick’s theorem.

While that last post is well commented on, try Nick Hamblet’s question on Riemann sums that nobody has (as of this writing) answered.

Here’s a series asking a simple question with complex ramifications: how many quadratic equations with real roots are there? (Part one, Part two)

John Conway is a genius when it comes to devising notation. Exhibit A: neverendingbooks explains Conway’s big picture.

Pat Ballew points out an mathematical error in a science magazine. I recommend reading the post as a puzzle and just looking at the picture and the opening joke to try to figure out what’s wrong before checking the rest of the text.

A tricky probability problem presents itself as Praveen Puri asks: what’s the chance the patient has the disease?

Nathan Bloomfield just finished his first research paper! To celebrate he has written about his experience.

A late submission to sneak in, but a really good one: Maria Anderson has done surveys to summarize mathematical instructional practices.

And finally, a smattering of logic: Andrew Bacon discusses Restall’s Paradox, Kenny Easwaran endorses a particular kind of probabilistic proof, and I consider a slight variation on an old classic.

That’s a wrap! If you’re wanting more math carnival goodness, be sure to check the Math Teachers at Play carnival which alternates Fridays with this one.

### 18 Responses

1. I’ll leave this comment here, since Praveen Puri’s site has neither comments nor any contact information!

The problem he states is a classic in counter-intuitive probability, but I think the counter-intuitive-ness is amplified by the way the question is phrased.

His problem states that a test for a disease has “5% false positives”. I think a lot of people would read that and think (quite reasonably) that 5% of the tests that come back positive are false. Thus the belief that the answer to the question is 95%. I think it takes a little bit of extra knowledge to know that “5% false positives” means “5% of the tests given to those who do not have the disease come back positive.”

• Hi Matt,

Thanks for your comment. I just edited the post. It now reads:

A test for a disease has 5% false positives (This means that 5% of the tests given to those who do not have the disease come back positive – so if 100 people test positive, only 95 actually have the disease)…

Praveen

2. […] of Mathematics #52 By Ξ YES!  It’s Carnival of Mathematics #52, hosted by The Number Warrior.   He starts with a great math problem that had me wondering, […]

3. Matt, I forwarded the comment to Praveen.

I also fixed the Sam Shah link above which was broken.

4. […] I’m in the Carnival! By sumidiot My post on a Riemann sum made the 52nd Carnival of Mathematics! […]

5. […] go to the carnival The 52nd Carnival of Mathematics is underway over at The Number Warrior. I’m glad it was resurrected. Check it […]

6. […] Carnival of Mathematics #52 is up and running at The Number Warrior, with tidbits about perfect numbers and Mersenne primes as […]

7. Thanks for putting together a *great* carnival. I esp. loved the posts about the quadratic equation/real v. complex roots. I’ve asked myself that question and came up with a few solutions that all made sense. I am definitely going to be using that question in my MV calc class next year, fo’ sho’.
Sam

• That’s a new blog (last I checked I was the only subscriber on Google Reader) but the quality bodes well for future posts by the same author.

8. […] comes the actual carnival: almost 20 sharp links. I don’t know if Jason tilted it on purpose, but his carnival has a […]

9. […] while you’re playing around with math, don’t forget the Carnival of Mathematics #52, which went up last week at The Number Warrior. Plenty of fun there, too. Does anyone know where […]

10. Great math carnivale! How about some math carnivals for younger students, though?

• Does Math Teachers at Play (which I have plugged on this blog, but haven’t hosted yet) match what you are describing?

Math Teachers at Play

If you mean not just with links to things teachers can use for younger students, but a carnival *for* younger students … that seems like it’d need to be hand-curated, but it’s possible. Worth a thought. Maybe a compilation of material from the last 5 carnivals of Mathematics and Math Teachers at Play presented in an appropriate way could do the trick.

11. Does anyone know where and when the next carnival (number 53) is?

12. […] 4. Your blog traffic will shoot up like a rocket. You may be familiar with the WordPress fastest growing blogs list. I might bring your attention to #17 on May 9, which happened to be the same day I posted the Carnival of Mathematics #52. […]

13. […] Next Math Teachers at Play Hosted Here Posted on July 21, 2009 by Jason Dyer I have hosted the Carnival of Mathematics three times before: Carnival of Mathematics #30 Carnival of Mathematics #43 Carnival of Mathematics #52 […]

14. […] while you’re playing around with math, don’t forget the Carnival of Mathematics #52, which went up last week at The Number Warrior. Plenty of fun there, too. Does anyone know where […]