Try this Number Theory Problem based on finding the number of solutions from PRMO - 2016.
Consider all possible integers $n \geq 0$ such that
$$
\left(5 \times 3^{m}\right)+4=n^{2}
$$
holds for some corresponding integer $m \geq 0$. Find the sum of all such $n$.
GCD of Numbers
Power of Primes
Parity of a Number
Elementary Number Theory by David Burton
Excursion in mathematics
PRMO 2016
The required answer is 10
$
\left(5 \times 3^{m}\right)+4=n^{2}
$
$\rightarrow \left(5 \times 3^{m}\right) = n^{2}-4$
$\rightarrow \left(5 \times 3^{m}\right) = (n-2)(n+2)$
Observe that $gcd (n-2,n+2)$ = $1$ or $2$ or $4 $
But $\left(5 \times 3^{m}\right) = (n-2)(n+2)$
Hence both $(n-2)$ , $(n+2)$ are odd.
Hence one possible case:
$(n-2)$ = $5$
$(n+2)$ = $3^{m}$
$\rightarrow$ $4$ = $3^{m}$ - $5$
$\rightarrow$ $m$ = $2$
$\rightarrow$ $n$ = $7$
Similarly find other values of n and add
Try this Number Theory Problem based on finding the number of solutions from PRMO - 2016.
Consider all possible integers $n \geq 0$ such that
$$
\left(5 \times 3^{m}\right)+4=n^{2}
$$
holds for some corresponding integer $m \geq 0$. Find the sum of all such $n$.
GCD of Numbers
Power of Primes
Parity of a Number
Elementary Number Theory by David Burton
Excursion in mathematics
PRMO 2016
The required answer is 10
$
\left(5 \times 3^{m}\right)+4=n^{2}
$
$\rightarrow \left(5 \times 3^{m}\right) = n^{2}-4$
$\rightarrow \left(5 \times 3^{m}\right) = (n-2)(n+2)$
Observe that $gcd (n-2,n+2)$ = $1$ or $2$ or $4 $
But $\left(5 \times 3^{m}\right) = (n-2)(n+2)$
Hence both $(n-2)$ , $(n+2)$ are odd.
Hence one possible case:
$(n-2)$ = $5$
$(n+2)$ = $3^{m}$
$\rightarrow$ $4$ = $3^{m}$ - $5$
$\rightarrow$ $m$ = $2$
$\rightarrow$ $n$ = $7$
Similarly find other values of n and add
