Exercise · ZW-SUM-OF-TWO-SQUARES-007 · ★★☆ Intermediate
Is 65 a sum of two squares?
Choose one option.
Show answer
- D. Yes, because \(65 = 1^2 + 8^2\)
Explanation
Since \(65 = 1 + 64 = 1^2 + 8^2\), it is a sum of two squares; its primes 5 and 13 are both \(\equiv 1 \pmod{4}\).
Review the concept: Sum of two squares
An integer \(n\) is a sum of two squares if it can be written as the sum of the squares of two integers.
\[n = x^{2} + y^{2}\]
More practice problems: sum of two squares
- ZW-SUM-OF-TWO-SQUARES-001 According to Fermat's criterion, when is an odd prime p a sum of two squares?
- ZW-SUM-OF-TWO-SQUARES-002 Which of these equalities is correct?
- ZW-SUM-OF-TWO-SQUARES-003 Which of these numbers is NOT a sum of two squares?
- ZW-SUM-OF-TWO-SQUARES-004 What is the first prime of the form 4k+1?
- ZW-SUM-OF-TWO-SQUARES-005 Is the number 2 a sum of two squares?
- ZW-SUM-OF-TWO-SQUARES-006 Decide whether 37 is a sum of two squares using Fermat's criterion.
- ZW-SUM-OF-TWO-SQUARES-008 Is 63 a sum of two squares? (Recall that 63 = 3^2 · 7.)
- ZW-SUM-OF-TWO-SQUARES-009 Let n = 2 · 5^2 · 11. Is n a sum of two squares?