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Let $h(x)$ be the number of steps^ needed for $x$ to reach $1$ in the Collatz/3n+1 problem. I found that

$$h(238!+n)=h(238!+1), \;\; \forall 1 < n \leq 690,000,000$$ Here "!" is the standard factorial.

This is a lot of consecutive terms with the same height and beats the current record by far. Now I am wondering:

What is the smallest $m > 1$, such that $h(238!+m) \neq h(238!+1)$?

I don't know of an efficient way of finding it. We know that $h(2\cdot(238!+1))=h(238!+1)+1$, so $m \leq 238!+1$, but that's a rather large upper bound.

UPDATE 16/08/2021: Martin Ehrenstein found that $10^9 < m < 10^{94}$. See A346775. Later user mjqxxxx improved the upper bound to $m \leq 2^{64}$.

UPDATE 21/08/2021: Martin Ehrenstein improved the upper bound to $m < 11442739136455298475$.

^ $3x+1$ is considered one step and $x/2$ is one step.

Dmitry Kamenetsky
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    Thank you for the edits! – Dmitry Kamenetsky Aug 12 '21 at 08:58
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    I think you should call a mathematician. There may actually be more interest in HOW you found this. – DanielWainfleet Aug 12 '21 at 10:32
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    @DanielWainfleet my method is not very exciting. I used a simple brute force program that iterates through all the factorials. I got stuck at $238!$ and realized I need help. I am hoping that this information could be useful for someone working on the Collatz problem. – Dmitry Kamenetsky Aug 12 '21 at 12:04
  • $2^{232}x+r$ might help. – Roddy MacPhee Aug 12 '21 at 22:16
  • @RoddyMacPhee can you provide more information please? – Dmitry Kamenetsky Aug 13 '21 at 01:09
  • Parity falls to $r$ so it mostly follows the sequence of $r$ are remainders until you use up the exponent on $2$ , and every multiply by $3$ and add $1$ step, you add a factor of $3$ in the lead coeffient. – Roddy MacPhee Aug 13 '21 at 01:22
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    I’m going to conjecture that $m=2^{64}$ is the smallest. (Certainly it’s a much improved upper bound.) – mjqxxxx Aug 16 '21 at 03:18
  • @mjqxxxx that's a great find! – Dmitry Kamenetsky Aug 16 '21 at 14:05
  • Are factorials more likely than other integers to begin a sequence of consecutive integers which all have the same number of Collatz steps? If so, why? – SeekingAnswers Aug 17 '21 at 04:00
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    @SeekingAnswers great question. So far I have only investigated powers of two (https://oeis.org/A277109) and factorials (https://oeis.org/A346775). Powers of two seem to have more patterns and are perhaps more predictable, while factorials look more chaotic. Both give many consecutive terms with the same height. – Dmitry Kamenetsky Aug 17 '21 at 04:48
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    Factorials are rich in powers of $2$ letting $m$ decide parity. – Roddy MacPhee Aug 18 '21 at 11:56
  • @RoddyMacPhee hmmm but powers of 2 are also rich in powers of 2 :) so I am not convinced that factorials will give more terms. – Dmitry Kamenetsky Aug 19 '21 at 07:54
  • Please add explicite information, whether you use counting by "$(3x+1)/2$ is one step" or by "$3x+1$ is one step" – Gottfried Helms Nov 03 '21 at 04:14
  • @GottfriedHelms $3x+1$ is one step and $x/2$ is one step. I will add this in. – Dmitry Kamenetsky Nov 03 '21 at 06:07
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    What container did you use for your numbers? Python/numpy? Also, for an efficient way to find the max: Check the number $238!+2^p$ for increasing $p$ until you find a counterexample $n_1$. Then check $\frac{n+n_1}2$ and go to the mean below if it fails and to the mean above if it succeeds and repeat. This will give you a candidate to check downwards by brute force. – Robert Frost Jul 11 '24 at 10:50

1 Answers1

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Based on an exhaustive search, $m = 107150589646$.

Here are the first few values of $m$ where $h(238! + m) \neq h(238! + 1)$:

m =            107,150,589,646 / 0x00000018f2ac3ece  steps = 12067
m =            107,150,589,647 / 0x00000018f2ac3ecf  steps = 12067
m =            107,150,590,270 / 0x00000018f2ac413e  steps = 12067
m =            107,150,590,271 / 0x00000018f2ac413f  steps = 12067
m =            107,150,590,508 / 0x00000018f2ac422c  steps = 12067
m =            107,150,590,509 / 0x00000018f2ac422d  steps = 12067
m =            107,150,590,510 / 0x00000018f2ac422e  steps = 12067
m =            107,150,590,513 / 0x00000018f2ac4231  steps = 12067
m =            107,150,590,514 / 0x00000018f2ac4232  steps = 12067
m =            107,150,590,515 / 0x00000018f2ac4233  steps = 12067
m =            107,150,590,521 / 0x00000018f2ac4239  steps = 12067
m =            209,160,603,721 / 0x00000030b2f18849  steps = 12067
m =            425,073,520,521 / 0x00000062f85b2f89  steps = 12067
m =            425,073,520,551 / 0x00000062f85b2fa7  steps = 12067
m =            425,073,520,619 / 0x00000062f85b2feb  steps = 12067
m =          2,677,964,409,337 / 0x0000026f831e09f9  steps = 12067
m =          3,311,176,950,660 / 0x00000302f186a384  steps = 12067
m =          3,311,176,950,661 / 0x00000302f186a385  steps = 12067
m =          3,311,176,950,662 / 0x00000302f186a386  steps = 12067
m =          3,311,176,950,667 / 0x00000302f186a38b  steps = 12067
m =          3,403,245,468,542 / 0x00000318613cbb7e  steps = 12067
m =          3,403,245,468,543 / 0x00000318613cbb7f  steps = 12067

Note that there are longer sequences with identical step counts starting at values other than $238! + 1$, for example:

$$h(238!+n)=h(238!+1), \;\; \forall 425,073,520,620 \leq n \leq 2,677,964,409,336$$

I also spent some time trying random values of $m < 2^{64}$. Below are the first $100$ results found. They appear to be distributed rather unevenly. For example, $14$ of the $100$ values have the form $11,442,739,xxx,xxx,xxx,xxx$, which may help to explain the upper bound found by Martin Ehrenstein.

m =    289,157,651,973,531,131 / 0x04034b5a39f7e5fb  steps = 12067
m =  1,512,384,269,071,043,038 / 0x14fd117ad1de49de  steps = 12067
m =  4,794,524,866,141,075,010 / 0x428992f6991db642  steps = 12067
m =  5,025,390,884,248,844,399 / 0x45bdc6623123786f  steps = 12067
m =  5,227,290,433,584,055,957 / 0x488b10f437934695  steps = 12067
m =  5,424,813,733,282,473,439 / 0x4b48cf5938f665df  steps = 12067
m =  5,458,152,087,510,481,714 / 0x4bbf4067b21e7b32  steps = 12067
m =  5,830,659,013,938,846,348 / 0x50eaa97b24411a8c  steps = 12067
m =  5,975,988,740,763,890,078 / 0x52eefa18faf7319e  steps = 12067
m =  5,977,645,376,814,668,499 / 0x52f4dccc9e4b5ad3  steps = 12067
m =  5,986,804,479,539,848,938 / 0x531566f466b0a6ea  steps = 12067
m =  6,028,912,560,581,966,761 / 0x53ab00080360fba9  steps = 12067
m =  6,049,014,397,250,805,447 / 0x53f26a8b955b9ec7  steps = 12067
m =  6,058,721,270,254,614,339 / 0x5414e6e512a13743  steps = 12067
m =  6,058,968,369,229,845,609 / 0x5415c7a14945d469  steps = 12067
m =  6,059,306,858,516,293,353 / 0x5416fb7bf6f3a6e9  steps = 12067
m =  6,117,985,319,424,462,290 / 0x54e7733bc9598dd2  steps = 12067
m =  6,179,234,051,900,558,495 / 0x55c10ca19787cc9f  steps = 12067
m =  6,461,152,029,596,821,363 / 0x59aa9f89cbe9a773  steps = 12067
m =  7,368,892,372,999,181,087 / 0x6643909238b3931f  steps = 12067
m =  7,507,841,641,408,400,254 / 0x68313631d2088b7e  steps = 12067
m =  7,507,841,691,645,510,478 / 0x6831363d8466034e  steps = 12067
m =  7,646,877,045,461,628,611 / 0x6a1f2a286fd436c3  steps = 12067
m =  7,943,274,652,744,571,847 / 0x6e3c2e361e345bc7  steps = 12067
m =  8,330,424,227,929,012,561 / 0x739b9cb2e6043951  steps = 12067
m =  8,814,794,176,125,127,968 / 0x7a547099b41ecd20  steps = 12067
m =  9,177,512,912,124,176,263 / 0x7f5d135e78b8e787  steps = 12067
m =  9,468,512,367,376,637,146 / 0x8366e9d4f0e09cda  steps = 12067
m =  9,747,755,035,171,931,260 / 0x8746fb8f04f2c47c  steps = 12067
m = 10,398,405,693,811,129,640 / 0x904e8ee2aa02fd28  steps = 12067
m = 10,399,113,004,131,063,529 / 0x9051122e2e6096e9  steps = 12067
m = 10,842,610,996,537,801,423 / 0x9678b14136613acf  steps = 12067
m = 11,402,726,701,639,877,502 / 0x9e3e9f855a1b437e  steps = 12067
m = 11,411,386,603,021,500,907 / 0x9e5d63a7c3bf81eb  steps = 12067
m = 11,442,724,751,195,810,502 / 0x9eccb988f998aac6  steps = 12067
m = 11,442,739,112,197,027,071 / 0x9eccc698a7f8f4ff  steps = 12067
m = 11,442,739,118,599,330,760 / 0x9eccc69a25945bc8  steps = 12067
m = 11,442,739,118,863,829,982 / 0x9eccc69a35584bde  steps = 12067
m = 11,442,739,118,978,355,960 / 0x9eccc69a3c2bd2f8  steps = 12067
m = 11,442,739,136,567,267,031 / 0x9eccc69e548d4ed7  steps = 12067
m = 11,442,739,158,525,242,975 / 0x9eccc6a371596e5f  steps = 12067
m = 11,442,739,160,309,135,299 / 0x9eccc6a3dbad77c3  steps = 12067
m = 11,442,739,181,493,773,424 / 0x9eccc6a8ca616470  steps = 12067
m = 11,442,739,182,016,899,407 / 0x9eccc6a8e98fa94f  steps = 12067
m = 11,442,739,185,378,213,063 / 0x9eccc6a9b1e93cc7  steps = 12067
m = 11,442,739,211,633,913,245 / 0x9eccc6afcedf6d9d  steps = 12067
m = 11,442,739,212,709,640,023 / 0x9eccc6b00efdb757  steps = 12067
m = 11,442,739,216,869,165,919 / 0x9eccc6b106eb0b5f  steps = 12067
m = 11,442,739,236,349,097,374 / 0x9eccc6b59003359e  steps = 12067
m = 11,445,449,891,804,874,219 / 0x9ed66809379005eb  steps = 12067
m = 11,449,815,137,723,182,491 / 0x9ee5ea343be16d9b  steps = 12067
m = 11,457,651,587,717,461,721 / 0x9f01c169ee16f6d9  steps = 12067
m = 11,493,082,078,672,145,651 / 0x9f7fa141f1a94cf3  steps = 12067
m = 11,929,568,657,320,238,444 / 0xa58e577cff34dd6c  steps = 12067
m = 12,097,606,391,485,285,241 / 0xa7e354eaccf9c379  steps = 12067
m = 12,148,976,269,924,272,160 / 0xa899d58ca97c8820  steps = 12067
m = 12,695,310,437,805,610,306 / 0xb02ecd949eb65542  steps = 12067
m = 13,033,980,731,820,178,767 / 0xb4e2006b296d7d4f  steps = 12067
m = 13,339,645,316,487,902,101 / 0xb91ff0bd26f6f395  steps = 12067
m = 13,498,714,295,605,379,965 / 0xbb551121f20bc37d  steps = 12067
m = 13,515,030,658,979,709,679 / 0xbb8f08c754ffc2ef  steps = 12067
m = 13,567,482,973,873,116,761 / 0xbc4961e191de1659  steps = 12067
m = 13,845,820,254,485,450,011 / 0xc0263c29c4838d1b  steps = 12067
m = 13,870,134,414,975,573,207 / 0xc07c9dc3678a1cd7  steps = 12067
m = 14,022,505,580,600,926,108 / 0xc299f287f8a69f9c  steps = 12067
m = 14,307,055,145,588,420,612 / 0xc68cdeda56b8f004  steps = 12067
m = 14,307,370,231,888,806,555 / 0xc68dfd6c15bb029b  steps = 12067
m = 14,307,468,293,412,140,093 / 0xc68e569bcffeb43d  steps = 12067
m = 14,307,469,041,334,513,544 / 0xc68e5749f3a46f88  steps = 12067
m = 14,307,516,155,262,834,907 / 0xc68e8223849a98db  steps = 12067
m = 14,307,980,852,535,304,068 / 0xc69028c748751f84  steps = 12067
m = 14,308,071,900,487,589,986 / 0xc6907b96094fa062  steps = 12067
m = 14,308,164,734,798,855,725 / 0xc690d004b574fe2d  steps = 12067
m = 14,308,667,168,790,725,067 / 0xc69298fabd10c1cb  steps = 12067
m = 14,308,707,196,740,229,033 / 0xc692bd6278c6d7a9  steps = 12067
m = 14,439,394,294,886,822,589 / 0xc863089ba687f2bd  steps = 12067
m = 14,507,263,241,271,540,278 / 0xc954270e1f127a36  steps = 12067
m = 14,523,705,563,181,607,840 / 0xc98e914283cdeba0  steps = 12067
m = 14,647,376,007,957,761,776 / 0xcb45eedfc5e6cef0  steps = 12067
m = 14,665,285,465,589,235,453 / 0xcb858f6e52253efd  steps = 12809
m = 14,694,158,714,254,031,334 / 0xcbec237f645279e6  steps = 12809
m = 14,816,372,153,724,898,887 / 0xcd9e53f92ac30e47  steps = 12067
m = 14,846,901,848,346,489,406 / 0xce0aca919dc8723e  steps = 12067
m = 15,177,225,550,845,423,092 / 0xd2a05639e21fadf4  steps = 12067
m = 16,443,275,672,513,659,879 / 0xe432401a9bffc3e7  steps = 12067
m = 17,025,188,759,948,589,089 / 0xec459ef94eb57821  steps = 12067
m = 17,417,269,601,730,143,222 / 0xf1b6926c1055a3f6  steps = 12067
m = 17,666,248,327,907,728,957 / 0xf52b1f4122c75e3d  steps = 12067
m = 17,697,185,528,543,798,078 / 0xf59908797909633e  steps = 12067
m = 17,720,428,549,573,982,981 / 0xf5eb9be1050c3b05  steps = 12067
m = 17,727,505,707,775,828,267 / 0xf604c08451b0212b  steps = 12067
m = 17,740,277,533,567,276,813 / 0xf632206cbd1ed70d  steps = 12067
m = 17,757,546,399,658,394,050 / 0xf66f7a5df227c1c2  steps = 12067
m = 17,758,244,925,087,637,566 / 0xf671f5ac123b603e  steps = 12067
m = 17,956,472,411,176,655,626 / 0xf93234853f495b0a  steps = 12067
m = 18,064,708,869,238,803,042 / 0xfab2bd0171d3ca62  steps = 12067
m = 18,082,881,890,716,661,479 / 0xfaf34d45bb82b6e7  steps = 12067
m = 18,125,628,766,673,208,847 / 0xfb8b2b546028860f  steps = 12067
m = 18,387,333,447,583,723,915 / 0xff2cee59ade2f18b  steps = 12067
m = 18,408,020,495,035,728,693 / 0xff766d1c40b70335  steps = 12067
Jeremy Sawicki
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