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This question asks if this form of the Collatz Conjecture has been reported or is all ready known. The goal of this question is to determine if I should write a paper on it's discovery or not and whether I can correctly report I made the discovery of.

I am asking if anyone has seen such before.


As someone who has spent many man hours on this I thought to share an observation I made yesterday on my lunch break and to ask if it has been reported before. If not reported then an effort must be made.

We are familiar with the Collatz Conjecture aka Hailstones where we multiply by three then add one for when the state of the integer is odd. 3x+1 and x/2 for when the integer is even.

If we regroup this action and add one to an odd integer then multiply by three the cycle is attracted to the multiplier and not the addend. 3(x+1), x/2

{ 1 ,6 ,3 ,12 ,6 ,3 }

{ 2 ,1 ,6 ,3 ,12 ,6 ,3 }

{ 3 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 4 ,2 ,1 ,6 ,3 ,12 ,6 ,3 }

{ 5 ,18 ,9 ,30 ,15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 6 ,3 ,12 ,6 ,3 }

{ 7 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 8 ,4 ,2 ,1 ,6 ,3 ,12 ,6 ,3 }

{ 9 ,30 ,15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 10 ,5 ,18 ,9 ,30 ,15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 11 ,36 ,18 ,9 ,30 ,15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 12 ,6 ,3 ,12 ,6 ,3 }

{ 13 ,42 ,21 ,66 ,33 ,102 ,51 ,156 ,78 ,39 ,120 ,60 ,30 ,15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 14 ,7 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 16 ,8 ,4 ,2 ,1 ,6 ,3 ,12 ,6 ,3 }

{ 17 ,54 ,27 ,84 ,42 ,21 ,66 ,33 ,102 ,51 ,156 ,78 ,39 ,120 ,60 ,30 ,15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 18 ,9 ,30 ,15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 19 ,60 ,30 ,15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

{ 20 ,10 ,5 ,18 ,9 ,30 ,15 ,48 ,24 ,12 ,6 ,3 ,12 ,6 ,3 }

So: If Odd do x+1 then multiply by three. If Even then divide by two.

Has anyone seen this before? I'll write it up and put it online if not with a short complete C program attached.

I tested this with a 3321928 bit number. It has been tested with multiplier of two and three. The "attractor" then is the multiplier and not the addend as is seen in the Collatz form.

I will try 4,5,... soon.


mpz_set_ui( A, 3 ); // Set A to the multiplier you wish for A > 0

mpz_set_str( Y, "1\0",10);

// Example C language code snippit using GMP bignum laibrary for(x=1;x<18446744073709551615u;x++) { mpz_set_ui(Z,x); w=0; printf("{ "); for(;;) { gmp_printf("%Zu ,",Z); if( mpz_odd_p(Z) ) { mpz_add_ui(Z,Z,1); mpz_mul(Z,Z,A); } else {mpz_divexact_ui(Z,Z,2);} if( mpz_cmp(Z,A) == 0 )w++; if(w==2){ gmp_printf("%Zu }",Z); printf("\n-----------\n");break;} }

}

I have enjoyed the Collatz conjecture since being introduced to it in 1992. I have worked out a great deal about it so I'm open to sharing what I know.

In General I adore the cycle in mathematics. Always have.

Thank You for your time.

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    $\begingroup$ Generally, questions of the form "has anybody seen this?" or "has anybody tried this" are kind of vague. Quite a few mathematicians have probably tried it and not gotten anywhere, so that there would be no literature about it. It is unlikely that some elementary rephrasing of Collatz is going to lead to a resolution of the problem. $\endgroup$ – Thomas Andrews Oct 3 '14 at 19:16
  • $\begingroup$ Thank you for your reply. Mathstackexchange is such a great site because of folks like you and I. I ask so as to not claim works others have done. I can write this up if it has not been reported. That's all I am seeking today. $\endgroup$ – Ernst Oct 3 '14 at 19:20
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    $\begingroup$ If it is any encouragement I have not seen this specific variant of the Collatz iteration. However note that it is equivalent to replacing $3x+1$ with $3x+3$ for the odd-integer rule. Therefore it likely is included in generalized Collatz versions. $\endgroup$ – hardmath Oct 3 '14 at 20:38
  • $\begingroup$ Thank you. I shall investigate further. If the multiplier is 2 then there is a cycle of { 4 2 1 } so you may be on to something here. Multiplier of 4 and { 8 4 2 1 }. Multiplier of 5 and { 1,10,5,30,15,80,40,20,10,5 } Okay then I thank you all for allowing me to ask questions. It seems worthy of further effort on my part. I have not found any prior art yet. $\endgroup$ – Ernst Oct 3 '14 at 21:58

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