Welcome to the 57th Carnival of Mathematics! This particular carnival is sponsored by the numbers 57 and 2: the first for the obvious reason and the second because it turned out that each contributor has two blog posts (though in some cases that will come as a surprise to the contributors).

The number 57, while not actually prime, is known as a Grothendieck prime in honor of Alexandre Grothendieck. According to legend:

In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.”

This story is not implausible, because Grothendieck didn’t normally think in terms of numbers and actual examples (according to the rest of the article above). Indeed, abstractness was a characteristic of his. Speaking of characteristics, Akhil Mathews at fellow group blog Delta Epsilons has a post on Hensel’s lemma and a classification theorem for complete Discrete Valuation Rings with a residue field of characteristic zero. (It’s actually part of a longer series, which I think starts here and continues over the next few days.)

The number 57 also occurs in the ketchup Heinz 57. When they introduced their catsup, they had roughly 60 products of various sorts on the market. 57 sounded nice, so they called the catchup their 57th product. In this case, they were just using the numbers to count, but another thing that you can do with numbers is to combine and permute them. However, as John D. Cook shares over at The Endeavor, a misunderstanding of those processes can (almost) lead to blows: see Classroom Violence, Combinations, and Permutations for the full story. He also write about Gilbreath’s conjecture in Easy to Guess, Hard to Prove, which provides a great example to share with students about a math problem that seems simple but, as the title suggests, isn’t.

Rod Carvalho wrote two posts on optimal debt allocation over at Reasonable Deviations: Part I is here and Part II is here. The articles, inspired by real-life bill-splitting at dinner, pose some interesting questions that I hope are solved soon. Something else that is solved — well, really, solvable — is any group of order 57. (There are several reasons for this, the simplest being that 57 is the product of two odd primes.)

Both TwoPi and I started our life in California, which has 58 counties. (Yes, that’s not 57. No state has 57 counties, though Montana would win the Price is Right prize with 56). Speaking of life, Nathaniel Johnston shares a post on generating sequences of primes in Conway’s game of life . Also check out today’s post about how primes with millions of digits aren’t useful for cryptography.

And last but not least, Pat Ballew of Pat’sBlog wrote about samuri and mathematics in Pi and the 47 Ronin, with a request for any photos that might be available of the tomb of Matsumura. He also explored an exploration by Leibniz in Limits as *x→*Infinity. Something that isn’t infinite is the list of Idoneal numbers:

An **idoneal number**, also called a **suitable number** or **convenient number**, is a positive integer *D* such that any integer expressible in only one way as (where *x*^{2} is relatively prime to *Dy*^{2}) is a prime, prime power, or twice one of these.

There are only 65 of them, or maybe 66 if the generalized Riemann hypothesis holds. The number 57 is one of those Idoneal numbers. Isn’t that Convenient?

The next Carnival of Mathematics will be hosted by Michael Croucher over at Walking Randomly! See you there in two weeks!