Is there a direct relationship between an isotope's neutron count and radioactivity? In my textbook, it lists isotopes of Carbon: C-12, C-13, and C-14.  It noted that C-14 is radioactive (C-12 and C-13 are not).
Is there a direct relationship between the number of neutrons and an element's radioactivity?
In other words, since we know C-14 is radioactive, does that mean we also know that C-15 (if such a thing exists) would also be radioactive?
 A: As @dmckee says the problem is complicated. It is complicated because it is not a solution of a potential describing one force, but a balance between electromagnetic forces and the strong force that is keeping the quarks within the nucleons. (In the nucleus the strong force is like a type of Van der waals potential, a higher order interaction, overflowing from the QCD dynamics of the nucleons). In addition there is the fermi exclusion principle  because both protons and neutrons have spin 1/2.
All these have been approximated with the shell model of the nucleus, and you could maybe spend some time reading the link.

The shell model is partly analogous to the atomic shell model which describes the arrangement of electrons in an atom, in that a filled shell results in greater stability. When adding nucleons (protons or neutrons) to a nucleus, there are certain points where the binding energy of the next nucleon is significantly less than the last one. This observation, that there are certain magic numbers of nucleons: 2, 8, 20, 28, 50, 82, 126 which are more tightly bound than the next higher number, is the origin of the shell model.
Note that the shells exist for both protons and neutrons individually, so that we can speak of "magic nuclei" where one nucleon type is at a magic number, and "doubly magic nuclei", where both are. Due to some variations in orbital filling, the upper magic numbers are 126 and, speculatively, 184 for neutrons but only 114 for protons, playing a role in the search of the so-called island of stability. There have been found some semimagic numbers, notably Z=40.2 16 may also be a magic number.3

So there are stable nuclei and the various models do a good job of predicting them.
There exists a band of instability for the various isotopes  and the island of stability for high Z.
So the answer is no, there is no general rule, except solutions of the shell model, though adding or subtracting neutrons to a stable isotope one  expects a high probability that it will become unstable,as an examination of the table of nuclides shows..
A: Thanks to @dmckee, and the link he suggested: interactive table of the isotopes.  Looking at that table, it seems to me that there is not a reliably direct relationship between number of neutrons and radioactivity.  Using Calcium (Ca) as an example (assuming I'm reading the chart correctly):


*

*Ca-40: stable

*Ca-41: radioactive (with a relatively long half-life)

*Ca-42: stable

*Ca-43: stable

*Ca-44: stable

*Ca-45: radioactive (with a relatively short half-life)

*Ca-46: stable

*Ca-47: radioactive (with a relatively short half-life)

*Ca-48: radioactive (with a relatively long half-life)

