Pythagorean Triplets

A Pythagorean triplet is a set of three natural numbers, \(a < b < c\), for which \({ a }^{ 2 }+{ b }^{ 2 }={ c }^{ 2 }\). For example, \({ 3 }^{ 2 }+{ 4 }^{ 2 }=9+16=25={ 5 }^{ 2 }\).

There exists exactly one Pythagorean triplet such that \(a + b + c = 1000\). Find the value of \(c\).

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