Let $T$ be the set of pythagorean triples, that is, triples of integers (a,b,c) satisfying a2 + b2 = c2. We think of $T$ as the set of right angles triangles with integer lengths. And let $f : T \rightarrow \mathbb{Z}$ be the function $(a,b,c) \mapsto \frac{ab}{12}$ which computes the area of a triangle (divided by 6, which seems to always be a factor for some reason). 

I was wondering: what are the number theoretic propertires of $f$? It seems to produce numbers with few prime factors. What is the reason for this? For instance, $f(3,4,5) = 1$, $f(36,77,85) = 3 * 11 * 7$, and $f(65,72,97)=39*5*2$. Can we put a bound on the number of prime factors in the numbers that $f$ spits out? Or at least, can we give a 'generic' statement such as 'The output of $f$ almost always spits out numbers with less than 8 factors' or something?


Question: Question about thermodynamic conjugate quantities
Answer: The fundamental quantity in thermodynamics is entropy, which is a function of $n$-variables $S=S(x_1, x_2,...,x_n)$. For instance, for a simple mono-component system $S=S(U,V,N)$ where $U$ is internal energy, $V$ is volume, and $N$ composition.

Taking the differential

$$\mathrm{d}S = \sum_i \left( \frac{\partial S}{\partial x_i}\right)_{j \neq i} \mathrm{d}x_i = \sum_i F_i \mathrm{d}x_i$$

The quantities $F_i \equiv ({\partial S}/{\partial x_i})_{j \neq i} $ are intensive entropic parameters and measure the change in entropy when variables change. For instance the intensive entropic parameter $(1/T)$ gives the change on entropy due to a change in the energy $U$.

Using the thermodynamic theory of fluctuations it can be shown that

$$F_i = k \left( \frac{\partial \ln P}{\partial x_i}\right)$$

where $P$ is the probability of a fluctuation in the variables near an equilibrium state.

Using the definition of average

$$\langle A \rangle = \int A P \mathrm{d}x_1  \mathrm{d}x_2 \cdots \mathrm{d}x_n$$

the demonstration of the central result of linear nonequilibrium thermodynamics

$$\langle F_i \cdot x_j \rangle = -k \delta_{ij} $$

is direct although it needs first the use of $(\partial \ln P / \partial x_i)P = \partial P / \partial x_i$ in the integrand and next integration by parts.

What Wikipedia makes is to rewrite this central result using the new quantities $X_i \equiv - F_i / k$ but the physically important quantities are the $F_i$ often also called in this context thermodynamic forces or affinities.