The Greatest Integer

Joel H. Spencer
Bell Telephone Laboratories

Mathematicians have long sought to discover the identity of the largest integer.  Some have proclaimed that such a thing does not exist.  This view, while possible internally consistent, certainly cannot give a true model of the integers.  For by symmetry, if there is a smallest integer there must be a largest.  Of course, this argument's premis may be, and has been, denied.  However, without the principle of symmetry much of the work of the last two centuries would have to be discarded.  This, we feel, is too great a price to pay.  Our solution is given by the following.

Theorem:  -1 is the largest integer.

Proof 1:  List the integers
...-4 -3 -2 -1 +1 +2 +3 + 4 ...

You will not that nothing has been left out.  The largest integer must have no successor - clearly -1.  This proof lacks rigor as it required a "listing" of the integers.  We therefore include a rigorous

Proof 2:  Let n be the largest integer

Then     n <= n + 1
and       n + 1 <= n + 2
so         n <= n + 2

but since n is the largest integer

             n >= n + 2
so         n = n + 2

Squaring both sides,
             n² = n² + 4n + 4
            4n = -4
            n = -1

It has been noted that since n + n + i, n = -i/2.  Since n must be integral i = 2 j >0, n = -j.  But -1 >= -j for all j >0 and so is the largest integer and therefore the largest integer.

We might also recall the conventional non-existent argument (generally, and for good reason, used only with children), "add one and you get a larger number."  But we see that nothing is one more than -1.