Add a square to $853$ to form another square Given that $853$ is a prime number, find the square number $S$ such that $S + 853$ forms another square number.
I have no idea how you would find $S$, and trial and error doesn't help.
Is there a way of finding $S$?
 A: Since $S$ is a square number, we can write it as $S=s^2$.  Then suppose that $T$ is another square number which can be written as $T=t^2$.  We then have the relation $$T=S+853\implies t^2=s^2+853\implies t^2-s^2=(t-s)(t+s)=853.$$
Since $853$ is a prime number, its only factors are $1$ and $853$, meaning that we have the linear system
\begin{align}
t-s&=1\\
t+s&=853
\end{align}
Can you take it from here?
A: Any odd natural number $2n+1$ is the difference between two (consecutive) squares, $(n+1)^2-n^2$. Apply this with $2n+1=853$.
(If the problem had asked for the smallest $S$ of the desired sort, then you'd have to take into account that $853$ is prime, because otherwise there could be smaller solutions. But as long as nobody cares about minimizing $S$, it doesn't matter that $853$ is prime. The same method works for all odd natural numbers.)
A: Let $S=s^2$ and $S+853=T=t^2$. Then $T-S=(t+s)(t-s)=853$. But $853$ is a prime, so $t+s=853$ and $t-s=1$, yielding $s=(853-1)/2=426$ and $S=426^2=181476$.
A: Just to algebra
Let $S = n^2$ and let $S +853 = m^2$ so 
$n^2 + 853 = m^2$.
Now what?
....
Well, $m^2 - n^2 = 853$ and $(m+n)(m-n) = 853$.
Now remember... $853$ is prime and $m,n$ are natural numbers so ....
$m+n$ and $m-n$ are factors of the prime number $853$.
So $m+n = 853$ and $m-n=1$.  
Two equations, two unknowns.
A: Lets see :


*

*knowing $(2x)^2=4x^2$ and $(2x+1)^2=4x^2+4x+1$ plus $853=4(213)+1$ we can conclude $S$ is of form $2x$ (if we substitute $x+1$ for $x$ and expand yhe difference is of wrong form)

*knowing $(3y)^2=9y^2$ and $(3y+1)^2=9y^2+6y+1$ and finally $(3y+2)^2=9y^2+12y+4$ plus $853=3(284)+1$  means $S$ is of form $3y$  ... 


Together these limit $S$ to a multiple of 6 . It also has to be greater than 29 as $29^2<853$ and lower than 428 as $428^2-427^2= 428+427=856$
