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Case 2: p is an odd prime This now requires the idea of Lifting the Exponents. Please read here if you don’t know it. It is an advanced technique to deal with Diophantine Equations. Let’s check that the conditions of the LTE are satisfying here. p is an odd prime. gcd(a,b) = 1. p doesn’t divide a or b as we are looking for fundamental solutions. \( a^p = a mod p; b^p = b mod p \). Hence, \( a^p + b^p = a + b mod p \). So, p | a+b, and p don’t divide a or b. Hence, we can apply LTE.
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