I am currently reading through JC Lagarias' "The $3x+1$ Problem and its Generalizations" and have come across some notation reading :

$$\sup_{K \ge 0} T^{(K)}(N)$$

Now I assume that this means "suppose that $K$ is greater than or equal to $0$", however I want to understand any equations I am potentially going to use in a project and thus don't want to rely on assumptions. This may seem a very simple question but I appreciate any help.

  • 1
    $\begingroup$ Did you mean this?$$\sup_{K \ge 0} T^{(K)} (N)$$(That's written like this: $$\sup_{K \ge 0} T^{(K)} (N)$$. The two dollar signs mean that it should be in displayed-math mode—if there was only one dollar sign, it'd be inline-math mode. \ge stands for "Greater-than or Equal".) $\endgroup$ Nov 12, 2014 at 11:27
  • $\begingroup$ That is what I meant yes, thank you $\endgroup$
    – Ben Hortin
    Nov 12, 2014 at 11:32
  • $\begingroup$ This and this are great links on how LaTeX works. For future reference, you can right-click on any math to see how it was written. $(\Delta x)^2\approx0$ $\endgroup$ Nov 12, 2014 at 11:37
  • $\begingroup$ Reading a bit ahead in the article: I'm assuming he's using the symbol $\#$ to mean the number of elements in a set (as in: $\#\{n:n\le x\:and\:\sigma(n)\le k\}$ means the number of elements in the set $\{n:n\le x\:and\:\sigma(n)\le k\}$). $\endgroup$ Nov 12, 2014 at 11:48

3 Answers 3


Sup ("supremum") means, basically, the largest. So this: $$\sup_{k\ge0}T^{(k)}(N)$$ refers to the largest value $T^{(k)}(N)$ could get to as $k$ varies.

It's technically a bit different than the maximum—it's the smallest number that is greater-than-or-equal to every number in the set.

So, for example, the interval $[0,1)$ has no maximum value, but $1$ is the supremum of the interval, because it's greater-than-or-equal-to everything in that interval, and because it's the smallest number with that property. (Note that $(0,1),(0,1],[0,1),[0,1]$ all have the same supremums.)

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    $\begingroup$ It's worth mentioning (for OP's benefit) that the only thing that prevents $1$ from being the maximum of the set $[0,1)$ is the fact that $1$ is not an element of that set. There happens to be a "hole" in the set where the supremum sits, so the supremum doesn't belong to the set. Whenever the supremum belongs the the set, the supremum and the maximum are the same thing. $\endgroup$
    – MPW
    Nov 12, 2014 at 15:29

"$\sup$" and "$\inf$" stand for "supremum" and "infimum". When dealing with real-valued expressions, these are just what you may have seen called the "least upper bound" and "greatest lower bound", respectively.

The stuff below the symbol you wrote describes the set over which $K$ is allowed to roam, and as it does so you are looking to find the least upper bound of the stuff following the "$\sup$" symbol.

So, this expression represents the least upper bound of $T^K(N)$ as $K$ ranges over all nonnegative real numbers.

"$\sup$" and "$\inf$" can often be thought of as generalizations of "$\max$" and "$\min$", respectively.


The abbreviation sup usually stands for supremum. You can read more about that here: http://en.wikipedia.org/wiki/Infimum_and_supremum


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