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According to this site : https://www.nitrxgen.net/collatz [update 8'2023: the site seems to be down. An archived version of the site can be found at webarchive: https://web.archive.org/web/20160319021630/https://www.nitrxgen.net/collatz ]

The maximum steps registered so far for a given number (with less than 500 digits) is 62,118, do we know what number it is?

I tried to contact the admin to give this result but I haven't got any response yet, maybe this is a known number in the community?

Thanks

The greatest number of steps I've seen from a single submitted number so far is 62,118 (abiding by the 500 digit restriction) and was found on 18/01/2015.

Update - I got an answer from the admin :

The number that generated 62,118 steps is a 500 digit number, the form limit. The highest value of any step occurs at step 132.

This number was found by an unknown user in January 2015, the same month the tool was released which is kind of weird (this number was the 47th ever number to be submitted), there has been no other number submitted that has beaten that record of steps and there's been 2.6 million submissions at the time of me writing this email.

The number is seeminly random with no obvious pattern:

81710902424862408424 <460 digits omitted> 82927666062150148665.

If you would still like the number then let me know but all I ask is that you don't submit it unnecessarily to my site. Thanks for your understanding!

Gottfried Helms
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toto
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  • Since we know so few generalities about the Collatz map, my immediate guess would be that if anyone knows that there is a number which takes 62,118 steps to get to 1, then that's because they have found such a number explicitly and checked. – Arthur Aug 31 '18 at 12:47
  • @Arthur yes, that's what I'm trying to find out, if anyone knows the value of this number ? – toto Aug 31 '18 at 13:46
  • I ran a random search overnight in the range $10^{500}-1\geq x\leq 10^{499}-1$ testing numbers with a density of one in $2^{1599}$ in the odd integers and the longest sequence found was that of $10^{500}-1$ with $16336$ steps. What this underlines is that the outliers are relatively sporadic. A better strategy may be to select numbers high in the range whose modular properties guarantee that they will initially climb significantly. I certainly tested the same number many times and there will be more efficient methods – it's a hire car baby Sep 06 '18 at 07:53

1 Answers1

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This is not the answer, but an attempt to find it.

It is easy to find a number that takes $k$ Collatz steps to reach $1$, namely $2^k$. This is of course gets infeasibly large, and exceeds the 500 digit limit of that site when $k>1661$.

You can of course do better by not just inverting the 'divide by $2$' step $k$ times to get a number that needs $k$ steps, but also but using an inverse $3x+1$ step when it is possible and useful, because that keeps the number small.

I tried doing this in a slightly naive way, by repeating:

If $x \equiv 4\ or\ 16 \mod 18$ then do $x\to (x-1)/3$ else do $x\to 2x$.

I started at $x=8$ so that it wasn't stuck in the $1,2,4$ loop from the start.

BTW, the reason for the mod is that the $3x+1$ rule with $x$ odd always results in a number that is $4 \mod 6$ so the inverse of that rule is only allowed if we have such a number. We also want to avoid getting an $x$ that is a multiple of $3$ (which happens when the input $x$ is $1 \mod 9$) because otherwise all further inverted steps will be doublings.

The best result for a number that still lies within the 500 digit limit was one that takes $7449$ steps.

Clearly the person who found a number within that limit that takes so many more steps must have been doing something much more clever.

FYI, the $7449$-step number I found is:

$99128,42552,45070,58214,49794,21433,09617,71434,\\ 35751,13169,69228,12299,48208,17398,14999,66674,\\ 07909,73211,57857,75169,98703,66806,51365,17271,\\ 44964,80888,82391,81744,25916,60261,13042,03866,\\ 66103,49056,80916,51374,86393,63112,68646,00014,\\ 51265,78980,63819,41418,78232,36960,28854,91873,\\ 32045,03658,35551,66924,74284,74756,27331,65465,\\ 06917,90508,40100,42755,07304,51692,27781,51879,\\ 28347,37220,21499,69801,81717,10793,96853,00241,\\ 92286,70480,20097,55644,06511,04039,72081,20501,\\ 67137,12978,12761,35222,27067,25690,90233,93366,\\ 78269,76657,53601,98939,20113,98389,35981,59681,\\ 41626,53623,05243,67777$

Edit:

A different but equally naive method is to just try various random numbers, but preferably those whose binary expansion ends in lots of ones because those will cause the $3x+1$ rule to be chosen many times at the start of the Collatz sequence, making for a bigger number.

The best 500-digit number I found used $25858$ steps, and is:
$16910 \cdot 2^{1646} - 1=$

$52907,40618,32017,54152,63867,78240,01344,94071,\\ 10672,87955,14970,95443,32426,37432,94723,15343,\\ 69589,15667,94197,77289,84251,30595,66986,13736,\\ 01999,30240,57853,88392,15872,66353,93648,16021,\\ 76528,02318,47325,39772,11544,04475,63943,10264,\\ 65156,58431,35122,30262,85026,21429,61128,92928,\\ 64147,40168,11368,14718,14844,43105,54823,58463,\\ 32284,16963,01649,12670,32815,41060,21677,81365,\\ 13557,54204,91127,46553,29004,21933,21888,08910,\\ 76179,94524,51718,32033,27205,55270,66366,57606,\\ 00582,53081,68267,14042,35673,44288,21454,35891,\\ 44482,68697,66804,20865,40074,26362,70517,10800,\\ 03191,26585,23809,38239$

Jaap Scherphuis
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    Just by inputting random 500 digits numbers, it's easy to find steps around 10k/15k (max I've done is 16k), which is far lower from 62k... – toto Aug 31 '18 at 14:28
  • I'm not sure how many steps it will take, but the 500 digit number that will take the most steps, should be the number with 1660 binary 1 digits. – Dennis Estenson Dec 13 '21 at 02:45