let $\displaystyle p, r_1 ,r_2,s_1,s_2$ be positive integers with

$\displaystyle p>1 ; r_1 < r_2 $ and $\displaystyle s_1<s_2$ then

(1) $\displaystyle p^{r_1}+ p^{r_2}= p^{s_1}+ p^{s_2}$

if and only if $\displaystyle r_i = s_i$, $\displaystyle i=1,2$ .

(2) $\displaystyle r_1r_1!+ r_2 r_2!= s_1 s_1!+ s_2s_2!$

if and only if $\displaystyle r_i = s_i$, $\displaystyle i=1,2$.

is it correct ?

the one way in each case is obvious. the other way...?

plz help.