6. For all $n \in \mathbb{Z}$, prove that if there exist $a, b \in \mathbb{Z}$ such that $a^2 + b^2 = n$, then $n \neq 3 \pmod{4}$.
Added by Marissa J.
Close
Step 1
Assume that there exist integers a and b such that a^2 + b^2 = n and n ≡ 3 (mod 4). Since n ≡ 3 (mod 4), we can write n as n = 4k + 3 for some integer k. Now, let's consider the possible values of a^2 and b^2 modulo 4: - If a ≡ 0 (mod 4), then a^2 ≡ 0 (mod Show more…
Show all steps
Your feedback will help us improve your experience
Derrick Danso and 54 other Algebra educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
For any n∈Z prove that 6 divides n^3-n
Lien L.
Shaiju T.
Prove that a3 ≡ a (mod 6) for every integer a.
William S.
Recommended Textbooks
Elementary and Intermediate Algebra
Algebra and Trigonometry
Watch the video solution with this free unlock.
EMAIL
PASSWORD