Paper 3
A comparison of 3x+1 and 3x-1 leading to a proof
1. 3x+1 loops
| 1.1 | The following goes into pedantic detail to better enable comparison with 3x-1. | |||||||||||||||||||||||||
| 1.2 | There is one known looping thread using 3x+1. Using the step up functions listed in Section 3.1 in Paper 1 above and starting with 1 produces the following pivot sequence as we step from L0, (it takes zero steps to reach 1 from 1):
At L0 1 has format 3e+1 where e = 0, and 3e+1<< e so that the 1chain at L1 has pivot 0. 0 has format 3e, where e = 0, and 3e<<4e, so: the 2chain at L2 has pivot 0 and so does the nchain at Ln for every value of n. |
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| 1.3 | Such a sequence of pivots is a spine.
Using this terminology, only one spine is known for 3x+1; i.e. 1, 0, 0, 0, 0, 0, . . . There is one nchain at at each Ln and this chain has its pivot on the spine. |
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| 1.4 | Every pivot and link in an nchain is the pivot of an (n-1) chain, every pivot and link of the infinite number of such (n-1)chains is the pivot of an (n-2)chain, and so on until the set of 1chains is reached. Thus 0 is not only the pivot of the nchain at Ln, but is also the pivot of a 1chain, 2chain, etc. at Ln. | |||||||||||||||||||||||||
| 1.5 | There is an nchain whose pivot is 0 for every n.
There must be a pivot format for every nchain that can equal 0. From the calculations in the Appendices, such formats do exist: 1chain pivot formats include 2i all of which equate to 0 when i = 0. |
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| 1.6 | The L5 5chain whose pivot is 0 is : 0, 7, 267, 268, 69036, . . . .
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| Example
128 is a 5chain pivot, there is a 5chain : 128, 1031, 33035, 33036, 8457644 which is calculated using the same cycle of functions as the 5chain with pivot 0.
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| The L5 4chain whose pivot is 0 is : 0, 11, 100, 50, 12944, . . . .
This is identical to the L4 4chain whose pivot is 0.
The L5 3chain whose pivot is 0 is : 0, 4, 75, 151, 9708, . . . . This is identical to the L3 3chain whose pivot is 0.
The L5 2chain whose pivot is 0 is : 0, 3, 28, 227, 1820, . . . . This is identical to the L2 2chain whose pivot is 0 .
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| 1.7 | A thread starting at L5 must pass through values at L4, L3, etc., as the previous section shows. As the only integer to loop on itself, 1 is a special case in that it exists at every level. | |||||||||||||||||||||||||
| 1.8 | If a pivot can equate to 0, it must step up to a pivot that also equates to 0.
Consider the pivots listed in 1.5: Substituting 3i for i in pivot 2i, 6i = 3(2i) << 4(2i) = 8i and in general, 2k.3i = 3(2ki) << 4(2ki) = 2k+2.i |
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| 1.9 | From 1.8, if a 0 occurs as a pivot in a chain, there is an infinite spine of pivots equalling 0.
Starting with any other integer and stepping up using 3x+1, produces a spine of rapidly increasing numbers. There are no pivot formats which equate to identical values for levels higher than that at which the constant in the format is a link in a chain. For example, 27 may exist as a pivot at every level up to L40, but does not appear in any nchain for n>40. There can be no other integer than 0 which appears as a pivot in a 3x+1 nchain for every value of n. |
| 2.1 | Since 1 has format 3o-2 << 2o-1 where o=1 and 2*1-1 = 1, the graph has a spine: 1, 1, 1, 1, 1, 1, . . .
0 cannot be a pivot as 3*0-1 is negative, 0 does not appear in any 3x-1 graph. Note: All the following calculations related to 3x-1 use the tables in the Appendices but with every plus and minus sign reversed. |
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| 2.2 | The L5 5chain pivot 32i-31 gives 5chain : 1, 18, 276, 2160, . . .
The L4 4chain pivot 16i-15 gives 4chain : 1, 53, 52, 1620, . . .
The L3 3chain pivot 8i-7 gives 3chain : 1, 20, 10, 304, . . .
The L2 2chain pivot 4i-3 gives 2chain : 1, 4, 29, 228, . . .
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| 2.3 | This is similar to the 3x+1 graphe in that if a pivot format can equate to 1, it must step up to a format which also equates to 1:
substituting 3i-2 for i in 4i-3 gives 12i-11 = 3(4i-3)-2 << 2(4i-3) -1 = 8i-7, etc. |
| 3.1 | 7 is not a pivot as 7 = 4*2-1, hence 7 is the first link in a 1chain whose pivot is 2.
2 has format 3o-1<<8o-3 where o = 1 and 8*1-3 = 5 5 has format 3e-1<< e where e = 2 Integers in this graph step down to either 5 or to 7 and then loop. |
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| Examples
35 >> 13 >> 19 >> 7 >> 5 36 >> 107 >> 5 >>7 |
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| Thus there are effectively two versions for a spine: 2, 5, 2, 5, . . . and 5, 2, 5, 2, . . .
This implies that there must be two formats of pivot for each nchain, one which equates to 2 and one which equates to 5. |
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| 3.2 | The 5chain pivot that can equate to 2 is 32i-30; the 5chain is : 2, 340, 342, 43668, . . .
The 5chain pivot that can equate to 5 is 128i-123; the 5chain is : 5, 16, 15124, 473, . . .
The 4chain pivot that can equate to 2 is 16i-14; the 4chain is : 12, 2836, 709, . . .
The 4chain pivot that can equate to 5 is 32i-27; the 4chain is : 5, 64, 1013, 8092, . . .
The 3chain pivot that can equate to 2 is 4i-2; the 3chain is : 2, 48, 380, 6069, . . .
The 3chain pivot that can equate to 5 is 16i-11; the 3chain is : 5, 9, 532, 266, . . .
The 2chain pivot that can equate to 2 is 6i-4; the 2chain is : 2, 13, 100, 797, . . .
The 2chain pivot that can equate to 5 is 12i-7; the 2chain is : 5, 36, 285, 2276, . . .
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| 3.3 | Substituting 3i-2 for i in 12i-7 gives 36i-31 = 3(12i-10)-1 << 12i-10 which can equate to 2
12i-10 = 4(3i-2) - 2 , which has format 4i - 2
Substituting 3i-2 for i in 6i-4 gives 18i -16 = 3(6i-5)-1 << 8(6i-5)-3= 48i-43 which can equate to 5 48i-43 = 16(3i-2) -11 , which has format 16i - 11
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This section is concerned with the 3x-1 tree that loops on on the thread:
17 >> 25 >> 37 >> 55 >> 41 >> 61 >> 91 >> 17 >> . . .
| 4.1 | There are seven different values to which integers converge, implying the existence of seven spines. | ||||||||||
Examples
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| 4.2 | Two of the convergence values have format 4i-1 and cannot be pivots : 56 == 14 ; 91 == 23 == 6.
The spines created by stepping up from the various possibilities are: 6 << 8 << 21 << 28 << 148 << 788 << 2101 << 1401 << . . . 23 << 8 << 21 << . . . These do not relate to the other converging values, however, 91 << 61 << 41 << 14 << 37 << 25 << 17 << 6 << 8 This appears to be the only spine. No integer steps up to 91, so 91 only exists as a link on the 1chain whose pivot is 6. The 2chain pivot that can equate to 61 is 12i-11; the 2chain is: 61, 484, 3869 , . . .
The 3chain pivot that can equate to 41 is 8i-7; the 3chain is: 41, 2580, 1290, . . .
41 must also be the pivot of a 2chain;12i-7gives 41, the 2chain is: 41, 324, 2589, . . .
The 4chain pivot that can equate to 14 is 8i-2; the 4chain is : 14, 3440, 1720, . . .
The 3chain pivot that can equate to 14 is 4i-2; the 3chain is : 14, 432, 3452, . . .
The 2chain pivot that can equate to 14 is 6i-4; the 2chain is : 14, 109, 868, . . .
The 5chain pivot that can equate to 37 is 64i-27; the 5chain is : 37, 9173, 9172, . . .
The 4chain pivot that can equate to 37 is 32i-27; the 4chain is : 37, 576, 9205, . . .
The 3chain pivot that can equate to 37 is 16i-11; the 3chain is : 37, 73, 580, . . .
The 2chain pivot that can equate to 37 is 12i-11; the 2chain is : 37, 292, 2333, . . .
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| 3.3 | The existence of the anticipated pivots at the relevant levels has been demonstrated.
Substituting 3i-2 for i in 64i-27 gives 192i-155= 3(64i-51)-21 << 2(64i-51)-1 = 128i-103 128i-103 can equate to 25 Substituting 3i-1 for i in 8i-2 gives 24i-10= 3(8i-3)- 1 << 8(8i-3)-3 = 64i-27 Substituting 3i for i in 8i-7 gives 24i-7= 3(8i-2)- 1 << 8i-2 |
| 5.1 | The Collatz conjecture seems higly unlikely to be false
The looping when using 3x-1 is only possible because the pivots have format 2k.i - c. If c increases at a similar rate to 2k, then it is possible to step up to pivots with the same value or which cycle through a set of similar values. This is not possible when using 3x+1 as then the pivots have format 2k.i + c and only when c and i are 0 is it possible to loop using 3x+1. |
| 5.2 | It is simple to prove that the only positive integer that steps down to itself when using 3x+1 is 1.
This means that there must be a spine of at least two different values in any 3x+1 loop. Stepping up from any integer using 3x+1 produces a spine of rapidly increasing values, negating the possibility of the same value being repeated. |