The Math Factor Podcast

ER. The Great Internet Mersenne Prime Search

November 5, 2008 · numbers, The Mathcast

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In August 2008, the 45th known Mersenne prime, a mere 2^43,112,609-1 was discovered by the Great Internet Mersenne Prime Search! Our puzzle this week is really just to rediscover for yourself proofs that

if a number of the form 2^N-1 is prime, then N must also be prime (Or contrapositively, if N is composite, then 2^N-1 is also composite)
if a number of the form 2^N-1 is prime then the number 2^(N-1) x (2^N-1) is perfect— that is, it is the sum of all its proper divisors.

For example, 2³ – 1 = 7, which happens to be prime. 2² x (2³-1) = 28, which has proper divisors 1, 2, 4, 7, and 14, which sum to (drumroll) 28.

For fun you might look around for numbers of the form 2^{a prime} -1 that are not themselves prime; this shouldn’t take too long since these are far more common than that those that are, the Mersenne Primes.

If you want a little more of a challenge, try to prove that

any even perfect number must be of this form

and if you want to be really famous, settle the conjectures that

this takes care of everything—in other words that there are no odd proper numbers
but that there are in fact infinitely many Mersenne primes and so infinitely many even perfect numbers

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EQ. Ed Pegg Returns

October 22, 2008 · guests, math puzzles, The Mathcast

Ed Pegg, of mathpuzzle.com , Wolfram research and consultant to the TV show Numb3rs, returns to discuss cellular automata and a fiendishly difficult puzzle.

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EP. HIPE

October 13, 2008 · guests, The Mathcast

Peter Winkler discusses the bonus chapter, on the word game HIPE, in his book, Mathematical Mind Benders!

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Follow Up: Loops and the Harmonic Series.

October 13, 2008 · The Mathcast

In an earlier post Spaghetti Loops, we asked several problems that we promised had something to do with the number e.

The first question really has to do more with the famous harmonic series; in this post we showed that the sum

1 + 1/2 + 1/3 + 1/4 + …. + 1/n

adds up to about the natural log of n, plus a small constant, Euler’s γ ≈ .577215664901… In other words, to sum up to, say N, at least e^{(N- γ)} terms are used.

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