[math-fun] Playing with c = (a*b) mod (a+b)
Hello MathFun & SeqFans, Start S = 2, 9,... Next term is (2*9)mod(2+9) 18 mod 11 = 7 S = 2, 9, 7,... Next term is (9*7)mod(9+7) 63 mod 16 = 15 S = 2, 9, 7, 15,... Next term is (7*15)mod(7+15) 105 mod 22 = 17 S = 2, 9, 7, 15, 17,... Next term is (15*17)mod(15+17) 255 mod 32 = 31 S = 2, 9, 7, 15, 17, 31,... Next term is (17*31)mod(17+31) 527 mod 48 = 47 S = 2, 9, 7, 15, 17, 31, 47,... Does this end in a loop? If yes, is there another start which doesn't? Best, É.
That one ends in a loop fairly quickly: 2, 9, 7, 15, 17, 31, 47, 53, 91, 71, 143, 95, 19, 95, 95, 95, ... Playing around with different starting pairs, it looks like all starting pairs may end up in a loop, or by going to the pair (0,0) which is either a loop or a singularity, depending on whether you take 0 mod 0 as being 0 or being undefined. Furthermore, you can go an arbitrarily long time without looping. Take for large a: .., 6*a+6, 6*a, ... The next term is 6*a-6, and progressively it goes down the ladder by steps of 6 until it terminates at ..., 18, 12, 6, 0, 0. In a few minutes of searching, the longest sequence I could find which eventually loops without going to zero was the sequence starting with (29,574), which hits 855 at the 79th term and then stays at 855. Jack Eric Angelini wrote:
Hello MathFun & SeqFans,
Start S = 2, 9,...
Next term is (2*9)mod(2+9) 18 mod 11 = 7 S = 2, 9, 7,...
Next term is (9*7)mod(9+7) 63 mod 16 = 15 S = 2, 9, 7, 15,...
Next term is (7*15)mod(7+15) 105 mod 22 = 17 S = 2, 9, 7, 15, 17,...
Next term is (15*17)mod(15+17) 255 mod 32 = 31 S = 2, 9, 7, 15, 17, 31,...
Next term is (17*31)mod(17+31) 527 mod 48 = 47 S = 2, 9, 7, 15, 17, 31, 47,...
Does this end in a loop? If yes, is there another start which doesn't?
Best, Ã.
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 -------------------------------------------------
participants (3)
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Eric Angelini -
Jack Brennen -
Jacques Tramu