Writing 2026 as a Sum of Two Squares
The problem
The number is prime. Write as a sum of two squares of integers, and explain why the representation is unique once you allow for swapping the two terms and changing signs.
- Both squares must be squares of integers, and zero is allowed as a square.
- You should use the factorisation rather than a blind search.
Take a minute
Reveal hint 1
Fermat's theorem: an odd prime is a sum of two squares exactly when it is modulo . Which residue is ?
Reveal hint 2
Once , use the identity with .
Solution
Answer, and this is the only representation up to order and sign.
Derivation
-
Check the residue of the odd prime.
-
Find that representation.
-
Split the factor .
-
Apply the two-squares product identity.
-
State the answer.
-
Justify uniqueness.