I did the same calcultation, for the legrange symbol, twice and got two different answers. Could someone please point out where I went wrong for future reference.
1) $\bigg(\frac{79}{81}\bigg)=(-1)^{\frac{79-1}{2}\frac{81-1}{2}}\bigg(\frac{81}{79}\bigg)=+1\bigg(\frac{2}{79}\bigg)=(-1)^{\frac{79-1}{2}}=(-1)$
2) $\bigg(\frac{79}{81}\bigg)=(-1)^{40\cdot39}\bigg(\frac{81}{79}\bigg)=1\bigg(\frac{9}{79}\bigg)\bigg(\frac{9}{79}\bigg)=\bigg(\frac{3}{79}\bigg)^4=(-1)^{39\cdot1\cdot 4}\bigg(\frac{79}{3}\bigg)^4=\bigg[-\bigg(\frac{1}{3}\bigg)\bigg]^4=1 $