MathLabs

Problem 3

Find all positive integers nn that can be written as n=a2+b2n=a^2+b^2, where a,ba,b are relatively prime positive integers and every prime p<=sqrt(n)p <= sqrt(n) divides abab.
Step 1 of 4: Handle equal summands
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a=b=1;n=2a=b=1; n=2
Detailed analysis

By symmetry assume a>=ba>=b. If a=ba=b, relative primality forces a=b=1a=b=1, so n=2n=2. This value satisfies the prime condition because no prime is at most sqrt(2)sqrt(2).