Let $1-5^n+5^{2n+1}=k^2,k\in \mathbb{N}$. $1-5^n+5^{2n+1}=k^2\Leftrightarrow 5^n(5^{n+1}-1)=(k-1)(k+1)\,\,(1)$.Since $(k-1,k+1)=2$ we have that $5^n|(k-1)$ or $5^n|(k+1)$ Case 1: $5^n|k+1$.Let $k+1=f5^n\Leftrightarrow k=f5^n-1,f\in \mathbb{N}\,\,\,(*)$. $(1)\overset{(*)}{\Rightarrow}5^n(5^{n+1}-1)=f5^n(f5^n-2)\Leftrightarrow 5\cdot 5^n-1=5^nf^2-2f\Leftrightarrow 2f-1=5^n(f^2-5)$ Since $2f-1>0$ we must have $f^2-5>0$,so $2f-1\geq 5(f^2-5)\Leftrightarrow 5(f+1)(f-\dfrac{12}{5})\leq 0$ so $f\leq 2$,contradiction since $f^2>5$ Therefore in this case we have not solutions Case 2:$5^n|k-1$,let $k=f5^n+1\,\,(**)$. $(1)\overset{(**)}{\Rightarrow}5^n(5^{n+1}-1)=f5^n(f5^n+2)\Leftrightarrow 5\cdot 5^n-1=5^nf^2+2f\Leftrightarrow 5^n(5-f^2)=2f+1$ We must have that $5-f^2>0$ so $f=1,2$ $f=1\Rightarrow 5^n\cdot 4=3$,contradiction. $f=2\Rightarrow 5^n=5\Leftrightarrow n=1$ So the only solution is $\boxed{n=1}\,\,,(1-5+5^3=11^2)$