# The Three n Plus One Conjecture

### Contents

#### Function Threenplus1

You might want to download function threenplus1.m from my textbook Numerical Computing with MATLAB. This not only produces the graphs in this post, it also has a pair of`uicontrols`that allow you to increase and decrease the value of

`n`. Here is the core of

`threenplus1`. The generated sequence is collected in a vector

`y`. We don't know ahead of time how long

`y`is going to be. In fact, that's the point of the computation. Does the

`while`loop terminate and, if so, how long is

`y`?. The ability of MATLAB to grow a vector an element at a time comes in very handy here.

dbtype 44:52 threenplus1

44 y = n; 45 while n > 1 46 if rem(n,2)==0 47 n = n/2; 48 else 49 n = 3*n+1; 50 end 51 y = [y n]; 52 end

#### n = 7

A good example is provided by starting with $n = 7$. Here is the sequence collected in`y`. Its

`length`is 17 and its

`max`is 52.

7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1Here is the plot. The

`y`-axis is logarithmic in anticipation of some fairly large values in later runs. Notice that as soon as the sequence hits a power of 2, it plunges to termination.

#### n = 108, 109, 110

It is interesting to see how the graphs produced by`threenplus1`change from one value of $n$ to the next. Here are the first ten elements of

`y`for $n$ = 108, 109, and 110.

108 54 27 82 41 124 62 31 94 47 . . . 109 328 164 82 41 124 62 31 94 47 . . . 110 55 166 83 250 125 376 188 94 47 . . .After the first eight steps all three sequences reach the same value, so they are identical after that. The

`length`is 114 and the

`max`is 9232. Here are the three graphs, superimposed on one another. The similarity between the three sequences is very apparent when you use the increment

`n`buttons provided by

`threenplus1`.

#### n = 2^13-1

Prime numbers of the form $n = 2^p-1$ where $p$ itself is prime are known as*Mersenne primes*. But that's a whole 'nother story. For now, suffice it to say that such numbers are candidates to produce 3n+1 sequences with a high maximum value. Here is $n = 2^{13}-1$. The sequence reaches 6,810,136 before it eventually terminates in 159 steps. I have to admit that this triggers a bug in some versions of MATLAB. If you see specious characters in the formatting of the legend on the

`y`-axis, insert this line near the end of

`threenplus1`.

set(gca,'yticklabel',int2str(get(gca,'ytick')'))

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