Speculations on finding "smooth" X^2+1: [What was wrong with "round"? That's what Hardy & Wright use.] You can generate ordinary integers in order, along with their prime factorizations, using a priority-queue-like scheme. [Suppose the current integer is 6, with divisors 2 and 3. Place queue entries for "2 divides 8" and "3 divides 9". Minor adjustments for prime powers.] A similar method should work for X+i, using complex primes. Unclear if it's any faster than just sieving large blocks of X+i. Both methods are faster than factoring N (or X^2+1) one-at-a-time. Both methods adapt to ignoring large prime divisors, just keeping those up to some limit. If you choose your prime divisor limit ahead of time, you can generate all numbers < 10^12 that factor into just those primes. There's an obvious method that just starts with a list {1}, multiplies in powers of 2 to get a longer list, then multiplies that list with powers of 5, then 13, etc. This would have to be filtered by "is the number an X^2+1?", but that's cheap. The list length could be a problem, and the numbers aren't exactly generated in order. There's a two-list variation that builds two lists from the first and second halves of the prime-set, and then generates the cross product of the two lists in numerical order (and applies the X^2+1 filter). I believe that one can design an algorithm that will directly generate numbers with all prime factors in a given list, in order of size, and not much memory, by keeping a list of "useful steps" along with a recent history. With list = {2,3}, starting from 1, the first steps are 2 and 3, updated to 3/2 and 4/3, then 9/8, 32/27, and so on. The two-prime case looks a lot like finding continued fraction approximants to log3/log2. I'm not sure if the number of steps required in the step-list for the more-prime case is too big. Another approach is to use LLL on the arctangents, directly looking for near-relations among the arctangents of small complex primes. Apparently LLL can be run on matrices of a few hundred rows. Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com on behalf of Joerg Arndt Sent: Tue 4/11/2006 10:38 AM To: math-fun Subject: [math-fun] Indoor sports: arctan relations During the last week I improved on my 1993 computations finding arctan relations for Pi. here is the data: http://www.jjj.de/arctan/arctanpage.html I am pretty confident the relations up to 11-terms are the best possible. Improving on the 12...17-term relations may be possible but will require a good amount of computation. The 18..22-term samples are likely not optimal, but I could not resists to add the 21-term sample +360698976*arctan(1/1290312057) -218005568*arctan(1/1721451206) ... -234213467*arctan(1/14033378718) == -1 * Pi/4 This is the smallest relation so far that has a least convergent term < 1/10^9. A future computation would require a better algorithm to compute all numbers X so that (X^2+1) is smooth. I used the table of all such numbers smaller than M:=10^10 and the first 64 primes 4*k+1 (761 is the 64th). If someone knows a better algorithm than simply testing the partial factorizations of all X^2+1 kindly let me know. The search took 37 hours, a better table should cover the data up to, say, M:=10^12. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun