Given an integer $n$, let $P(n)$ denote the set of odd prime divisors of $n$. Let $\Delta$ be the simplicial complex over the set of sets of odd prime numbers which consists of the simplices $S$ such that for every triple $\{A, B, C\} \subseteq S$ there are integers $a$, $b$ and $c$ which satisfy $P(a) = A$, $P(b) = B$, $P(c) = C$ and $a + b = c$.

Question 1: Which is the largest dimension of a simplex in $\Delta$?

Question 2: Does $\Delta$ have nontrivial automorphisms — and if so, which?

Example: It is easy to check that $\{\emptyset, \{3\}, \{5\}, \{7\}, \{11\}, \{13\}\} \in \Delta$. — The largest triples of integers to be looked at for this are $2 \cdot 7 + 2 \cdot 13^2 = 2^5 \cdot 11$, $2 \cdot 3^2 + 2^5 \cdot 7 = 2 \cdot 11^2$, $2^5 \cdot 3 + 2 \cdot 11^2 = 2 \cdot 13^2$ and $2 \cdot 5^3 + 2^3 \cdot 11 = 2 \cdot 13^2$.

Given an integer $n$, let $P(n)$ denote the set of odd prime divisors of $n$. Let $\Delta$ be the simplicial complex over the set of sets of odd prime numbers which consists of the simplices $S$ such that for every triple $\{A, B, C\} \subseteq S$ there are integers $a$, $b$ and $c$ which satisfy $P(a) = A$, $P(b) = B$, $P(c) = C$ and $a + b = c$.

Question 1: Which is the largest dimension of a simplex in $\Delta$?

Question 2: Does $\Delta$ have nontrivial automorphisms — and if so, which?

Example: It is easy to check that $\{\emptyset, \{3\}, \{5\}, \{7\}, \{11\}, \{13\}\} \in \Delta$. — The largest triples of integers to be looked at for this are $7 + 13^2 = 2^4 \cdot 11$, $3^2 + 2^4 \cdot 7 = 11^2$, $2^4 \cdot 3 + 11^2 = 13^2$ and $5^3 + 2^2 \cdot 11 = 13^2$.