PRMO 2019 question 14

14. Find the smallest positive integer n≥10n \ge 10 such that n+6n + 6 is a prime and 9n+79n + 7 is a perfect square.

 Solution:
p = n + 6
9n + 7 = k^2
Solving we get
k^2 = 9p - 47
Given n >= 10
So starting value of p is 17(16 is not prime).
Each time we increase p by 1, k^2 increases by 9.
p = 17, k^2 = 106 (not valid)
keep trying for p = 23,29,31,37,41,43,47,53,59
at p = 59, k^2 = 484 => k = 22
n = 53 answer.

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