Playing with c = (a*b) mod (a+b)
Jacques Tramu
jacques.tramu at echolalie.com
Mon Jul 14 12:39:35 CEST 2008
>From: "Eric Angelini" <Eric.Angelini at 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.
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