Exercise · ZW-SUM-OF-TWO-SQUARES-008 · ★★☆ Intermediate
Is 63 a sum of two squares? (Recall that \(63 = 3^2 \cdot 7\).)
Choose one option.
Show answer
- C. No, because the prime 7 appears with odd exponent and \(7 \equiv 3 \pmod{4}\)
Explanation
The prime 7 appears with exponent 1 and \(7 \equiv 3 \pmod{4}\), so 63 is not a sum of two squares; the factor \(3^2\) does not compensate for it.
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-007 Is 65 a sum of two squares?
- ZW-SUM-OF-TWO-SQUARES-009 Let n = 2 · 5^2 · 11. Is n a sum of two squares?