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I have been getting into the Collatz Conjecture and I have noticed something interesting. Consecutive numbers often take the same amount of numbers to get to 1. For example, $84$ goes to $42$ to $21$ to $64$ to $32$ to $16$ to $8$ to $4$ to $2$ to $1$. That's ten numbers. $85$ goes to $256$ to $128$ to $64$ to $32$ to $16$ to $8$ to $4$ to $2$ to $1$. That's also ten numbers. There are many more examples, like $60$ and $61, 76$ and $77,$ and $92$ and $93.$ As the numbers get larger, the number of consecutive integers that take the same amount of numbers to get to one grows. $386, 387, 388, 389, 370,$ and $371$ all take $121$ numbers to get back to $1$! I have thought a lot about this, but I can't figure out why this pattern works. I also cannot find a pattern of when this pattern occurs. It seems to be random, but I might be wrong. Can someone help with these questions?

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    $\begingroup$ The number of steps taken to reach $1$ is tabulated at oeis.org/A006577 along with links, formulas, etc. $\endgroup$ Oct 16, 2020 at 12:10
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    $\begingroup$ The numbers from 8769 to 8781 take 140 steps each. $\endgroup$ Oct 16, 2020 at 12:15
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    $\begingroup$ math.stackexchange.com/questions/470782/… $\endgroup$
    – Collag3n
    Oct 16, 2020 at 12:17
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    $\begingroup$ Thousand people are striving since years. Looking for patterns is a basic weapon. The lengths of the sequences have been investigated without success. $\endgroup$
    – user65203
    Oct 16, 2020 at 12:24
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    $\begingroup$ "why this pattern works": which pattern ? It is no real surprise that there is an increasing number of numbers with the same length. $\endgroup$
    – user65203
    Oct 16, 2020 at 12:54

2 Answers 2

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There are indeed patterns of this form.

If, say, you start with a number of the form $8n+4$ the chain begins

$$8n+4\mapsto 4n+2\mapsto 2n+1\mapsto 6n+4$$

While if you add $1$ to get $8n+5$ you get $$8n+5\mapsto 24n+16\mapsto 12n+8\mapsto 6n+4$$

Thus the consecutive numbers $8n+4,8n+5$ always have the same Collatz length. That explains your pairs $(84,85)$, $(60,61)$, $(76,77)$, $(92,93)$. I expect there are other patterns as well.

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  • $\begingroup$ Thanks! This was very helpful. From similar reasoning, I figured out that $16n+2$ and $16n+3$ also have this pattern. $$16n+2\rightarrow 8n+1\rightarrow 24n+4\rightarrow 12n+2\rightarrow 6n+1\rightarrow 18n+4$$ $$16n+3\rightarrow 48n + 10\rightarrow 24n+5\rightarrow 72n+16\rightarrow 36n+8\rightarrow 18n+4$$ $\endgroup$ Nov 12, 2020 at 11:19
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There are many patterns where consecutive numbers sync up in their Collatz progressions after a certain number of steps. (As of this writing, I’ve found over 8,600, most of which I have to confirm through proofs such as below.) So far, all that I’ve found are based on multiples of powers of 2, just as you see with 8n+4 (2^3+4) and 16n+2 (2^4+2). As you prove each one of these equations, you’ll notice more patterns. For purposes of this analysis, if one of the two numbers reaches 1 before the other, just keep applying the equations until they either sync (usually at 4) or don’t. The extra steps past reaching 1 count toward the number of steps until they sync.

[NOTE: This analysis has nothing to do with predicting how many steps until they reach 1, only with how many steps before consecutive numbers sync up.]


(8n+4,8n+5) or (2^3+4,2^3+5)
  8n+  4→
  4n+  2→  2n+  1→  6n+4

  8n+  5→
 24n+ 16→ 12n+  8→  6n+4


(16n+2,16n+3) or (2^4+2,2^4+3)
 16n+  2→  8n+  1→ 24n+  4→ (This is the same as 8(3n)+4)
 12n+  2→  6n+  1→ 18n+  4

 16n+  3→ 48n+ 10→ 24n+  5→ (This is the same as 8(3n)+5)
 72n+ 16→ 36n+  8→ 18n+  4


(32n+22,32n+23) or (2^5+22,2^+23)
 32n+ 22→ 16n+ 11→ 48n+ 34→ (This is the same as 16(3n+2)+2)
          24n+ 17→ 72n+ 52→ (This is the same as  8(9n+6)+4)
 36n+ 26→ 18n+ 13→ 54n+ 40

 32n+ 23→ 96n+ 70→ 48n+ 35→ (This is the same as 16(3n+2)+3)
         144n+106→ 72n+ 53→ (This is the same as  8(9n+6)+5)
216n+160→108n+ 80→ 54n+ 40


(32n+5,32n+6) or (2^5+5,2^5+6)
 32n+ 5→ 96n+16→
         48n+ 8→ 24n+ 4→ (This is the same as 8(3n)+4)
 12n+ 2→  6n+ 1→ 18n+ 4

 32n+ 6→ 16n+ 3→
         48n+10→ 24n+ 5→ (This is the same as 8(3n)+5)
 72n+16→ 36n+ 8→ 18n+ 4


(64n+14,64n+15) or (2^6+14,2^6+15)
 64n+ 14→ 32n+  7→ 96n+ 22→ (This is the same as 32( 3n  )+22)
          48n+ 11→144n+ 34→ (This is the same as 16( 9n+2)+ 2)
          72n+ 17→216n+ 52→ (This is the same as  8(27n+6)+ 4)
108n+ 26→ 54n+ 13→162n+ 40

 64n+ 15→192n+ 46→ 96n+ 23→ (This is the same as 32( 3n  )+23)
         288n+ 70→144n+ 35→ (This is the same as 16( 9n+2)+ 3)
         432n+106→216n+ 53→ (This is the same as  8(27n+6)+ 5)
648n+160→324n+ 80→162n+ 40


(64n+45,64n+46) or (2^6+45,2^6+46)
          64n+ 45→192n+136→
          96n+ 68→ 48n+ 34→ (This is the same as 16(3n+2)+2)
          24n+ 17→ 72n+ 52→ (This is the same as  8(9n+6)+4)
 36n+ 26→ 18n+ 13→ 54n+ 40

          64n+ 46→ 32n+ 23→
          96n+ 70→ 48n+ 35→ (This is the same as 16(3n+2)+3)
         144n+106→ 72n+ 53→ (This is the same as  8(9n+6)+5)
216n+160→108n+ 80→ 54n+ 40


(128n+29,128n+30) or (2^7+29,2^7+30)
128n+ 29→384n+ 88→
   1      192n+ 44→ 96n+ 22→ (This is the same as 32( 3n  )+22)
          48n+ 11→144n+ 34→ (This is the same as 16( 9n+2)+ 2)
          72n+ 17→216n+ 52→ (This is the same as  8(27n+6)+ 4)
108n+ 26→ 54n+ 13→162n+ 40

128n+ 30→ 64n+ 15→
         192n+ 46→ 96n+ 23→ (This is the same as 32( 3n  )+23)
         288n+ 70→144n+ 35→ (This is the same as 16( 9n+2)+ 2)
         432n+106→216n+ 53→ (This is the same as  8(27n+6)+ 4)
648n+160→324n+ 80→162n+ 40


(128n+45,128n+46) or (2^7+45,2^7+46)
128n+ 45→384n+136→
         192n+ 68→ 96n+ 34→ (This is the same as 16( 6n+2)+ 2)
          48n+ 17→144n+ 52→ (This is the same as  8(18n+6)+ 4)
 72n+ 26→ 36n+ 13→108n+ 40

128n+ 46→ 64n+ 23→
         192n+ 70→ 96n+ 35→ (This is the same as 16( 6n+2)+ 2)
         288n+106→144n+ 53→ (This is the same as  8(18n+6)+ 5)
432n+160→216n+ 80→108n+ 40
 

(128n+94,128n+95) or (2^7+94,2^7+95)
 128n+  94→  64n+  47→ 192n+ 142→ (This is the same as 64(3n+2)+14)
             96n+  71→ 288n+ 214→ (This is the same as 32(9n+6)+22)
            144n+ 107→ 432n+ 322→ (This is the same as 16(27n+20)+2)
            216n+ 161→ 648n+ 484→ (This is the same as  8(81n+60)+4)
 324n+ 242→ 162n+ 121→ 486n+ 364

 128n+  95→ 384n+ 286→ 192n+ 143→ (This is the same as 64(3n+2)+15)
            576n+ 430→ 288n+ 215→ (This is the same as 32(9n+6)+23)
            864n+ 646→ 432n+ 323→ (This is the same as 16(27n+20)+3)
           1296n+ 970→ 648n+ 485→ (This is the same as  8(81n+60)+5)
1944n+1456→ 972n+ 728→ 486n+ 364

As I said, there are thousands of such patterns. Where it starts to get interesting is when three consecutive numbers sync up on the same step (and I’ve notice as many as five consecutive numbers syncing up.) The above pairs sync up in 11 steps or fewer, and there are a couple of others that sync up that quickly. I have not yet determined the patterns where three or five consecutive numbers sync up on the same step, but I’m sure there is one. I may post those when I find them.

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  • $\begingroup$ Wayne - have you seen this older discussion math.stackexchange.com/a/1253988/1714 ? It's much similar to your analysis - perhaps you can take something from this. $\endgroup$ Nov 20, 2021 at 21:34
  • $\begingroup$ Thank you. I didn't imagine for a moment that I was the first to observe this, but this is the first place I found that talked about what I noticed. Now that I've gotten this far, I'm going to continue analyzing when three or more consecutive numbers sync up. I just won't be able to state it as mathematically as the experts. Thanks for the link. $\endgroup$ Nov 20, 2021 at 22:35

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