Per chi non conosca Frobenius:
Testo nascosto, fai click qui per vederlo
\( \displaystyle {f}_{{{\left({i},{j},{k}\right)}}}{\left({X},{Y},{Z}\right)}=\sum_{{\sigma\in{S}{\left({\left\lbrace{i},{j},{k}\right\rbrace}\right)}}}{{X}}^{{\sigma{\left({i}\right)}}}{{Y}}^{{\sigma{\left({j}\right)}}}{{Z}}^{{\sigma{\left({k}\right)}}}\in\mathbb{Z} \)
\( \displaystyle {0}={{\left({a}+{b}+{c}\right)}}^{{n}}=\sum_{{{\left(\matrix{{0}\leq{i}\leq{j}\leq{k}\leq{n}\\{i}+{j}+{k}={n}}\right)}}}{\frac{{{n}!}}{{{i}!{j}!{k}!}}}{f}_{{{\left({i},{j},{k}\right)}}}{\left({a},{b},{c}\right)} \)
(\( \displaystyle {\frac{{{n}!}}{{{i}!{j}!{k}!}}}={\left(\matrix{{n}\\{i}}\right)} \) . \( \displaystyle {\left(\matrix{{n}-{i}\\{j}}\right)}\in\mathbb{N} \))
\( \displaystyle {n}={p} \) primo:
\( \displaystyle {\frac{{{p}!}}{{{0}!+{0}!+{p}!}}}{f}_{{{\left({0},{0},{p}\right)}}}{\left({a},{b},{c}\right)}={{a}}^{{p}}+{{b}}^{{p}}+{{c}}^{{p}} \)
\( \displaystyle \forall{\left({i},{j},{k}\right)}\ne{\left({0},{0},{p}\right)},{p} \) | \( \displaystyle {\frac{{{p}!}}{{{i}!{j}!{k}!}}} \)
(\( \displaystyle {0}\leq\cdots\leq{i}\leq\cdots\leq{j}\leq\cdots\leq{k}\lt{p} \) e \( \displaystyle {p} \) è primo, quindi \( \displaystyle {i}!{j}!{k}! \) | \( \displaystyle {\left({p}-{1}\right)}! \))
