So you think you know Pythagorean triplets

Euclid's formula is a fundamental formula for generating Pythagorean triples given an arbitrary pair of positive integers m and n with m > n. The formula states that the integers $a = m^2 - n^2 ,\ \, b = 2mn ,\ \, c = m^2 + n^2$ form a Pythagorean triple. $a = 97 = (m-n)(m+n)$ 97 is a prime number $\Rightarrow m-n = 1 \Rightarrow m = 49 ,\ \,n = 48$ $\Rightarrow b + c = (m+n)^2 = 97^2 = \boxed{9409}$

Only thing needed for solving this problem is knowing the fact that $97$ is a prime.

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Solution:

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$a^2+b^2=c^2 \rightarrow 97\times97 = (c-b)(c+b)$

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Now since $97$ is prime and we know that for positive integral values of $b$ and $c$ , $(c+b) > (c-b)$ hence it is obligatory that $c+b=97^2 \rightarrow c+b=9409$

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There's also a fancier (and definitely much harder solution using the fact that all Pythagorean triplets can be found using the formula below:

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$(x,y,z)=(d(a^2-b^2) , 2abd , d(a^2+b^2))$

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Where $gcd(a,b)=1$