From Prime to Square

$(5p+1)$ is a square $\iff 5p+1=(a+1)^2$ for some non-negative integer $a$ .

Then $5p+1=a^2+2a+1 \iff 5p=a(a+2)$ . But $a\neq1$ , so, either $(a,a+2)=(5,p)$ or $(a,a+2)=(p,5)$ .

Therefore, $p=7$ or $3$ .

Hence, $(7+3)=\boxed{10}$ is the answer.

5p + 1 is a perfect square. It means that the value of 5p + 1 must be {1,4,9,16,25,36,49,64,81,100} But, in this question, it stated that it must be primes number. So, p = {2,3,5,7,...} Check it one by one and we find : As example : 5 * 2 + 1 = 11 ( 11 is not a perfect square. So, this statement is false.) But, 5 * 3 +1 = 16 (16 is perfect square) And 5 * 7 +1 = 36 (36 is perfect square) Therefore, p = 3 and 7 Hence( 3 + 7) = 10. So, the answer is 10