# 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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