Exercise · ZW-SUM-OF-TWO-SQUARES-006 · ★★☆ Intermediate
Decide whether 37 is a sum of two squares using Fermat's criterion.
Choose one option.
Show answer
- A. Yes, because \(37 \equiv 1 \pmod{4}\)
Explanation
Since \(37 = 36 + 1\) leaves remainder 1 upon division by 4, 37 is a sum of two squares: \(37 = 1^2 + 6^2\).
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-007 Is 65 a sum of two squares?
- 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?