I'd like to encourage posters to present a little bit more background when they mention a problem. In the case of Mrs. Perkins' Quilt, the puzzle is to dissect an integer sized square into smaller integer squares, with non-integer-square pieces forbidden. (And the gcd of the piece sizes must be 1, to exclude dissecting a 6^2 into four 3^2.) In the case of "Lens Spaces", there's probably no short definition, so we're stuck. But the recreational problems are usually easy to state. It's easy for those of us with high-speed net connections to regard giving a URL as equivalent, even better, than summarizing background material. But to some folks, those URLs represent minutes of download time each, often loading useless decorations of no information value. (I've been on both sides of this fence, so I have some sympathy for the bandwidth challenged.) Rich rcs@cs.arizona.edu --- In other news, John Cosgrave &al have a new Fermat Composite. -- Rich Date: Sun, 12 Oct 2003 11:59:12 -0400 Sender: Number Theory List <NMBRTHRY@LISTSERV.NODAK.EDU> From: John Cosgrave <john.cosgrave@spd.dcu.ie> Subject: F[2478782], a new largest known composite Fermat number To: NMBRTHRY@LISTSERV.NODAK.EDU Dear colleagues, In Ferbuary this year I announced the discovery of a new largest known composite Fermat number: the 645817-digit prime (3*2^2145353 + 1) is a factor of F[2145352] = 2^(2^2145352) + 1, only [then] the second prime factor - with more than 100,000 digits - of a Fermat number. See www.spd.dcu.ie/johnbcos for complete details and further links. Now - in a slight state of shock, and delight - I wish to announce a further computational advance: the 746190-digit prime (3*2^2478785 + 1) is a factor of F[2478782] = 2^(2^2478782) + 1, and is now the fourth more-than-100,000 digit factor of a Fermat number (a 'smaller' one - now the third in ranking - was found by Takahiro Nohara on August 4th this year; for details see Wilfrid Keller's Fermat numbers site at http://www.prothsearch.net/fermat.html). There will most likely be some GFN results following, which I will put up at my web site over the next several days. (3*2^2478785 + 1) was discovered on M?ire N? Bhaoill's computer (a 2.6GHz Pentium IV) - pre-sieved with Paul Jobling's newpgen and PRP-ed with George Woltman's program - and then subjected to Yves Gallot's Proth (primality testing) and ProthF (generalised Fermat number testing). She (M?ire) is one of my supportive St Patricks College Proth-Gallot group members (http://www.spd.dcu.ie/johnbcos/proth-gallot_group_(spd).htm), and the new number actually appeared as a PRP-probable prime on her machine on June 22nd of this year. I didn't notice it until Tues 7th October... The Fermat divisibility result - "3*2^2478785+1 Divides GF(2478782, 2)" - appeared on my office screen at 11:02 today (the independent primality test was done at home), minutes before my double 2nd year number theory lecture was due to begin. Of course I had to tell my students, and one of them suggested we should all go across the road for coffee and cake, which is exactly what we did. We were there for almost two and a half hours, during which time they came to know of Euclid (and perfect numbers), ... , Mersenne, Fermat, Euler, Landry, ... , Brillhart-Morrison, CRAY, Wilfrid Keller, Jeffrey Young, Richard Crandall, Richard Brent, Proth, Yves Gallot, George Woltman, Paul Jobling, ... I will put up digital photos of computer screens sometime next week on my home page. John Cosgrave