From: "Eric Angelini" <Eric.Angelini@kntv.be> Subject: Playing with c = (a*b) mod (a+b)
Start S = 2, 9,...
Next term is (2*9)mod(2+9) 18 mod 11 = 7 ...
The following describes the behaviour of the sequence u(n) = (u(n-2)*u(n-1)) mod(u(n-1) + u(n-2)) for large integers: 10^1000 3.7125500000 4706 43 57 We take 100 random pairs of starters (u(0) =a, u(1)= b) < 10^1000 (1000 digits) , and see that - the average sequence length (AVL) is 1000* 3.71 - the largest sequence found has length 4706, - 43% of sequences end in (0,0), - 57 % end in a loop of fixed point. The same for 10^p, 200 <= p <= 5000 , each time 100 random pairs of starters. One can see that the ratio R= AVL/p is close to 3.6 for all p in this sample. 10^200 3.7440500000 1009 50 50 10^400 3.7919000000 1960 43 57 10^600 3.6428000000 2863 52 48 10^800 3.6871250000 3787 47 53 10^1000 3.7125500000 4706 43 57 10^1200 3.5687666667 5463 54 46 10^1400 3.6217214286 6151 48 52 10^1600 3.5595187500 7125 52 48 10^1800 3.5865500000 7852 50 50 10^2000 3.5276800000 8875 55 45 10^2200 3.4119636364 9566 63 37 10^2400 3.5224416667 10469 55 45 10^2600 3.5431000000 11399 52 48 10^2800 3.6714142857 12160 42 58 10^3000 3.5366266667 12997 52 48 10^3200 3.4938656250 13873 56 44 10^3400 3.4834970588 14798 56 44 10^3600 3.6956305556 15528 39 61 10^3800 3.6228578947 16425 45 55 10^4000 3.4844000000 16984 55 45 10^4200 3.4705285714 18003 57 43 10^5000 3.5578060000 21355 49 51 Question : does the ration R has a limit, has upper and lower bounds ? My friend Georges Brougnard conjectured that lim(R) = PI/2 * log(10) = 3.616892206, but offered no proof. Regards, JT No divergent sequence was found, No animal was harmed in the making of this experiment. -------------------------------------------------- http://www.echolalie.com/gbnums -------------------------------------------------