3x+1 Delay Records

The table on the right depicts all currently known and confirmed delay records. Numbers with much higher delays are obviously known, but of course a record can only be considered a confirmed record when all smaller numbers have been checked and found to have a lower Delay. Records up to number 106 were already published in 1992 by Leavens and Vermeulen. As far as the author is aware the others were all found (or confirmed) by the distributed project.

From these data a few interesting observations can be made. First of all it is noteworthy that approximately eight records per decade (power of ten) are found, implying that on average a delay record is 1.33 times larger then its predecessor. It should be obvious that the maximum gap that can occur is a factor two, when a new record is exactly twice the previous record. This is no rare occcurrence, and indeed no less than 15 delay records are 'doubles', improving the previous record by just one. Four of these are less than 100, when candidates for new records are rare anyway. Most others occur when a previous record has yielded a big improvement.

Many improvements in delay are just small. In a few cases though a delay record hugely improves the previous record. Record #59, at 63,728,127 improves the previous record by 205, easily the largest gap found so far. The only three other gaps to exceed 100 are 106, by record #96, 158 by record #109 and 163 by record #131. Interestingly these numbers are the lowest numbers of levels -1, -2, -3 and -4 respectively. Also all four are among the extremely rare strength records, of which only five are known with certainty.

The 'residue' column shows that many delay records are in a way 'related', sharing near identical residues. Record #59 shares its residue of 1.183418 with the next 11 delay records, showing that their paths are coalescing well before they reach any numbers below 1,000,000. Similarly 11 delay records share the residue 1.151611. Beyond that another 'family' takes over, all sharing their residue of 1.148153 with strength record number 3. Finally no less than 16 consecutive delay records, among which we find strength record 4, show a residue of 1.146963.

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# Delay N Level Residue Remark