**Mathematical Proof Problems for Number Theory**---**Problem:**Suppose $ p $ is a prime congruent to 3 modulo 4.**(a)** Prove that the congruence $ x^2 \equiv -1 \pmod{p} $ has no solutions.*Hint: Use Fermat's Little Theorem.***(b)** Show that there are no elements of order 4 modulo $ p $.---This page presents mathematical proof exercises focusing on properties related to prime numbers and modular arithmetic. The first part of the problem invites you to explore the nonexistence of solutions to a specific quadratic congruence using Fermat's Little Theorem. The second part extends the concept to the elements of a certain order within the given modular system.

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Answered: Suppose p is a prime congruent to 3…

Suppose p is a prime congruent to 3 modulo 4. (a) Prove that the congruence r = -1 (mod p) has no solution Hint: Use Fermat's Little Theorem. (b) Show that there are no clements of order 4 modulo p.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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**Mathematical Proof Problems for Number Theory**

---

**Problem:**

Suppose $ p $ is a prime congruent to 3 modulo 4.

**(a)** Prove that the congruence $ x^2 \equiv -1 \pmod{p} $ has no solutions.

*Hint: Use Fermat's Little Theorem.*

**(b)** Show that there are no elements of order 4 modulo $ p $.

---

This page presents mathematical proof exercises focusing on properties related to prime numbers and modular arithmetic. The first part of the problem invites you to explore the nonexistence of solutions to a specific quadratic congruence using Fermat's Little Theorem. The second part extends the concept to the elements of a certain order within the given modular system.

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