1 Why to generalize an apparently intractable problem?

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The problem ax + b: the most natural generalization of Collatz Problem

Walter Carnielli

CLE and Department of Philosophy State University of Campinas

Campinas, Brazil

Abstract

I define here the simplest natural generalization of Collatz Prob- lem, dubbed the “Problem ax+b”, stating the corresponding conjec- tures and noting some first interesting facts.

1 Why to generalize an apparently intractable problem?

The Collatz Conjecture or 3x+ 1 Conjecture, extremely simple to state and awfully hard to solve, is perhaps one of the most perplexing unsolved mathematical problems, challenging equally mathematicians, logicians and even philosophers. One of its generalizations is even undecidable (cf. [3], more on this below).

Lothar Collatz (1910-1990)¹ proposed the problem in 1928, originally stated as follows: consider the function which inputs a non-zero integer x and outputs 3x+ 1 ifx is odd, and x/2 if x is even. The 3x+ 1 Conjecture asserts that, starting from any positive integer x, repeated iteration of this function eventually produces the value 1. In a more appropriate notation, the conjecture is usually rephrased by considering the function:

T₂(x) =x/2 if x≡0 (mod 2) and T₂(x) = (3x+ 1)/2 ifx≡1 (mod 2)

The (rephrased) conjecture states that every trajectory starting from a non-zero integer will end in an elements of one of the four cycles:

(1,2);

(-1);

1Collatz died while attending an international Symposium on Computer Arithmetic, Scientific Computation, and Mathematical Modelling (SCAN-90) in Varna, at the Black Sea in the Bulgaria coast.

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(-5,-7,-10);

(-17,-25,-37,-55,-82,-41,-61,-91,-136,-68,-34).

Of course, the cycle is unique, i.e., (1,2), if inputs are restricted to positive integers x.

There is an extensive literature on the many attempts to settle the conjecture, as well as related questions, touching from number theory to Markov chains (see specially [4]) and dynamical systems, and there is a 47-page an- notated bibliography in [6] and an excellent survey up to 1985 (cf. [5])

The problem is also known as the Syracuse problem, Kakutani’s problem, Hasse’s algorithm, Ulam’s problem, Thwaites’s problem and Hailstone Algorithm, and it is not known whether it is provable in Peano Arithmetic.

But even if it is intractable, Lagarias in [5] offers a good reason to keep try- ing: “No problem is so intractable that something interesting cannot be said about it.”

I present in the next section what I consider to be the most natural generalization of Collatz Problem.

2 A natural generalization

One of the most natural reasons why the original Collatz algorithm keeps running smoothly (whether or not it stops is another matter) is that it es- tablishes a parity equilibrium in eh sense that x/2 reaches 1 (in which case it enters a cycle) or an odd number greater than 1, in which case 3x+ 1 is even again. The present generalization simply starts form widening the scope of such a parity equilibrium to general congruences.

Define:

T_d(x) =x/d if x≡0 (mod d)

T_d(x) = ((d+ 1)x+d−i)/d if x≡i(mod d), 1≤i≤d−1.

It is easy to see that x≡i (mod d) implies Td(x)≡0 (mod d).

For instance, d = 2 gives the original 3x+1 mapping in the form T₂(n above.

The conjecture is that the sequence of iterates x, T_d(x), T_d²(x), ..., T_d^k(x), ...

for each d always eventually enters a cycle, for finite k, and that there are only finitely many such cycles.

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2.1 Interesting cycles

It is easy to prove² that (i) T_d(x) = x for x =−1, ...,−(d−1) and that (ii) 1,2, ..., d is always a cycle; those are called elementary cycles (respectively, positive and negative.

There are many non-elementary cycles, as for instance (for the first values of d and x≤50000):

• For d= 3:

(7, 10, 14, 19, 26, 35, 47, 63, 21) (cycle length 9) (-22, -29, -38, -50, -66) (cycle length 5)

• For d = 4: (23, 29, 37, 47, 59, 74, 93, 117, 147, 184, 46, 58, 73, 92) (cycle-length 14)

(-18, -22, -27, -33, -41, -51, -63, -78, -97, -121, -151, -188, -47, -58, -72) (cycle-length 15)

• For d= 5:

(-57, -68, -81, -97, -116, -139, -166, -199, -238, -285) (cycle length 10)

• For d= 6:

(23, 27, 32, 38, 45, 53, 62, 73, 86, 101, 118, 138) (cycle length 12) (88, 103, 121, 142, 166, 194, 227, 265, 310, 362, 423, 494, 577, 674, 787, 919, 1073, 1252, 1461, 1705, 1990, 2322, 387, 452, 528) (cycle-length 25)

The first d for which apparently there no cycles other than the elementary ones is d = 7; the same ford = 14, d= 18 and d= 21. An interesting point is that, differently from the original Collatz problem, several positive cycles arise.

A webpage and a CALC number theory program related to the present conjecture, developed by Keith Matthews, can be found at http://www.

numbertheory.org/php/carnielli.html. The here exposed notion of “bal- ancing parity” has been used by Keith Matthews to generalize upon a sug- gestion by Lu Pei (Nanjing University), see http://www.numbertheory.

org/php/Lu_Pei.html. A concise information on cycles with d ≤ 150 and

2This observation is due to Keith R. Matthews, personal communication.

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x ≤ 50000 provided by Keith Matthews can be found in Appendix. It is notorious in the studied cases that cycles are getting bigger and rarer, which itself suggests an obvious conjecture.

It has been shown by Kurtz and Simon in [3] that a certain generalization of the Collatz problem is undecidable, building on previous work by J.H.

Conway in [2]. It is easy top see that the definition of Collatz function given by Conway (definition 1.2. of [3] is a restriction of the functionsT_d(x) defined above: indeed, a functiong is called aCollatz function if there is an integern together with rational numbers {ai :i < n},{bi :i < n} such that whenever x≡i (mod p), then g(x) =a_ix+b_i is an integer.

The above mappings T_d(x) define of course an infinite family of Collatz functions:

g(x) = ¹_dx+ 0 if x≡0 (mod d)

g(x) = ^d+1_d x+ ^d−i_d if x≡i (mod d), 1 ≤i≤d−1.

and {ai :i < n}={¹_d,^d+1_d },{bi :i < n}={0,^d−1_d , ...,¹_d}

Conway had proved in [2], and the results is simplified in [3] (Theorem 1.4) that given a Collatz function g, it is undecidable whether or not for all integers x there exists an k such that g(j)(x) = 1.

The undecidability result does not affect (at least directly) neither Collatz original problem nor our generalizedax+b problems, but of course it gives a hint on the expected difficulties. It is encouraging, however, that the general ax+bproblems may have many positive cycles, and they woud not necessarily fall under intractable even if such undecidability result could be enhanced to values other than 1.

From a more philosophical standpoint, let me point out here that I com- pletely disagree from van Bendegen [7] when he says at page 10 (albeit rec- ognizing the conceptual importance and interest of Collatz Conjecture) that:

Firstly, it is quite easy to “invent” similar problems, so why should this particular case attract our attention?

Not only it is not easy to “invent“ sufficiently attractive problems to fur- ther research, but also beauty is an important fuel in many sciences, and especially in Mathematics. Erd˝os is reported to have offered US$ 500,00 for a solution of the original Collatz Conjecture, but he certainly was underesti- mating.

Acknowledgements: I am indebted to Keith R. Matthews (University of Queensland, Australia) for discussions, for his suggestions and for the com-

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puter programs testing the conjecture. I am also indebted to Jo˜ao Fernando Schwarz (State University of Campinas, Brazil) for the first tests and for discussions. This research was financed by FAPESP (Brazil), Thematic Project ConsRel 2004/1407-2 and by individual research grants from The National Council for Scientific and Technological Development (CNPq), Brazil.

References

[1] Marc Chamberland. An update on the 3x+ 1 problem. Online athttp:

//www.math.grinnell.edu/~chamberl/papers/survey.ps

[2] John H. Conway. Unpredictable Iterations . In: Proceedings 1972 Num- ber Theory Conference. University of Colorado, Boulder, CO, 1972, pp.

4952

[3] Stuart A. Kurtz and Janos Simon. The Undecidability of the Generalized Collatz Problem. In Theory and Applications of Models of Computation Lecture Notes in Computer Science, 2007, Volume 4484/2007, 542-553.

[4] Keith R. Matthews. Generalized 3x+ 1 Mappings: Markov Chains and Ergodic Theory. In The Ultimate Challenge: The 3x+ 1 Problem (ed.

J.F. Lagarias), American Mathematical Society, 2011, in print.

[5] Jeff Lagarias. The 3x+ 1 problem and its generalizations. American Mathematical Monthly Volume 92 (1985) , 3-23.

[6] Jeff Lagarias. The 3x+1 Problem Annoted Bibliography. Online at http://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/

node16.html#Ter79

[7] Jean Paul Van Bendegem. The Collatz Conjecture: A Case Study in Mathematical Problem Solving. Logic and Logical Philosophy Volume 14 (2005), 7-23.

[8] J. R. Whiteman. In memoriam : Lothar Collatz, Internat. J. Numer.

Methods Eng.. 31 (8) (1991), 1475-1476.

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3 Appendix: cyles for T

_d

(x), d ≤ 150 and x ≤ 50000

Note: “ No extra cycle” means no other cyles than the elementary ones.

U = 15 R = 50000

the lower value of d is m = 2 the upper value of d is n = 150

d=2:cycle 4: minumum element -5 cycle-length 3 cycle 5: minumum element -17 cycle-length 11 d=3:cycle 5: minumum element 7 cycle-length 9 cycle 6: minumum element -22 cycle-length 5 d=4:cycle 6: minumum element 23 cycle-length 14 cycle 7: minumum element -18 cycle-length 15 d=5:cycle 7: minumum element -57 cycle-length 10 d=6:cycle 8: minumum element 23 cycle-length 12 cycle 9: minumum element 88 cycle-length 25 d=7:No extra cycles

d=8:cycle 10: minumum element -92 cycle-length 19 d=9:cycle 11: minumum element 35 cycle-length 21 cycle 12: minumum element -272 cycle-length 22 d=10:cycle 12: minumum element 42 cycle-length 49 d=11:cycle 13: minumum element 642 cycle-length 57 d=12:cycle 14: minumum element -25 cycle-length 35 cycle 15: minumum element -63 cycle-length 33 cycle 16: minumum element 1348 cycle-length 32 cycle 17: minumum element 1010 cycle-length 96 d=13:cycle 15: minumum element -147 cycle-length 36 d=14:No extra cycles

d=15:cycle 17: minumum element 53 cycle-length 41 d=16:cycle 18: minumum element 178 cycle-length 46 d=17:cycle 19: minumum element 79 cycle-length 49 d=18:No extra cycles

d=19:cycle 21: minumum element -954 cycle-length 117 d=20:cycle 22: minumum element 71 cycle-length 60 cycle 23: minumum element 141 cycle-length 61 d=21:No extra cycles

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d=22:cycle 24: minumum element -426 cycle-length 71 d=23:cycle 25: minumum element 82 cycle-length 72 cycle 26: minumum element -800 cycle-length 75 d=24:cycle 26: minumum element 335 cycle-length 78 cycle 27: minumum element 513 cycle-length 157 d=25:cycle 27: minumum element 4064 cycle-length 83 d=26:cycle 28: minumum element -453 cycle-length 88 d=27:cycle 29: minumum element 545 cycle-length 91 d=28:No extra cycles

d=29:cycle 31: minumum element 111 cycle-length 97 d=30:No extra cycles

d=33:No extra cycles d=34:No extra cycles d=35:No extra cycles d=36:No extra cycles d=37:No extra cycles d=38:No extra cycles d=39:No extra cycles d=40:No extra cycles d=41:No extra cycles d=42:No extra cycles

d=43:cycle 45: minumum element -634 cycle-length 166 d=44:No extra cycles

d=45:No extra cycles d=46:No extra cycles d=47:No extra cycles

d=48:cycle 50: minumum element 136 cycle-length 181 cycle 51: minumum element -137 cycle-length 198 d=49:No extra cycles

d=50:No extra cycles

d=54:cycle 56: minumum element 3406 cycle-length 218 cycle 57: minumum element 3647 cycle-length 218 d=55:No extra cycles

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d=56:cycle 58: minumum element -157 cycle-length 240 d=57:cycle 59: minumum element 3416 cycle-length 466 d=58:No extra cycles

d=59:No extra cycles d=60:No extra cycles d=61:No extra cycles d=62:No extra cycles d=63:No extra cycles d=64:No extra cycles d=65:No extra cycles

d=68:No extra cycles d=69:No extra cycles d=70:No extra cycles d=71:No extra cycles d=72:No extra cycles d=73:No extra cycles d=74:No extra cycles d=75:No extra cycles d=76:No extra cycles d=77:No extra cycles

d=80:No extra cycles d=81:No extra cycles d=82:No extra cycles d=83:No extra cycles

d=84:cycle 86: minumum element -2119 cycle-length 377 d=85:cycle 87: minumum element 4435 cycle-length 380 d=86:No extra cycles

d=87:No extra cycles d=88:No extra cycles d=89:No extra cycles d=90:No extra cycles d=91:No extra cycles d=92:No extra cycles d=93:No extra cycles

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d=96:No extra cycles d=97:No extra cycles d=98:No extra cycles d=99:No extra cycles d=100:No extra cycles d=101:No extra cycles d=102:No extra cycles d=103:No extra cycles d=104:No extra cycles d=105:No extra cycles d=106:No extra cycles d=107:No extra cycles d=108:No extra cycles

d=111:cycle 113: minumum element 2838 cycle-length 524 d=112:cycle 114: minumum element -34727 cycle-length 532 d=113:No extra cycles

d=120:No extra cycles d=121:No extra cycles d=122:No extra cycles

d=125:No extra cycles d=126:No extra cycles

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d=132:No extra cycles d=133:No extra cycles

d=138:No extra cycles d=139:No extra cycles d=140:No extra cycles d=141:No extra cycles d=142:No extra cycles d=143:No extra cycles d=144:No extra cycles d=145:No extra cycles d=146:No extra cycles d=147:No extra cycles d=148:No extra cycles d=149:No extra cycles d=150:No extra cycles