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 gives an even perfect number whenever is prime. Theorizing what other values of 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 is prime.
His list did have errors: he missed n = 61, 89, and 107 (which also make 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
and then followed with the numbers 761838257287 and 193707721 and performed long multiplication, obtaining 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 is generalized for any base, using the formula
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.
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.
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.
Filed under: Mathematics