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.