Let p >2 be a prime. Prove that [(p-2) = (-1)/2 (mod p).

Q: Let p be a prime and k a positive integer such that ak mod p = a mod p for all integers a. Prove…

A: Consider the provided that, ak mod p=a mod p,                ....1       where p is prime and k is a…

Q: Let n = pq, where p and q are distinct primes such that p = q = 3 (mod 4). Suppose a E Z+ with a <…

A: We use the properties and definitions of Jacobi symbols to solve this problem. The detailed solution…

Q: Let p be a prime. Show that -2 is a quadratic residue modulo p if and only if p = 1 mod 8 or p = 3…

A: Definition(Quadratic Residue Modulo): In number theory an integer qis called a quadratic residue…

Q: Prove that : If p is a prime number and p does not divide a then a- 1(modp).

A:

Q: Find the smallest integer a > 1 such that a –1 = 2a (mod 11

A: To find:

Q: For each of the following primes, find the smallest positive integer k such that k is a prim root…

A: A primitive root of a prime p is an integer g such that g(modp) has multiplicative order p-1.

Q: Suppose a E Z. Prove that if a³ 1 (mod 5), then a # 1 (mod 5).

A: In this question, we use the method of contradiction to prove the desired result.

Q: Give an inductive proof that the Fibonacci numbers Fn and Fn+1 are relatively prime for all n ≥ 0.…

A:

Q: For any integer a, prove that a = 0 or 1 (mod 3).

A: Every number is either of the forms 3k, 3k−1, 3k+1 where k is an integer

Q: suppose a prime p of the form 8n + 3 and that q = (p-1) / 2 is also prime. Prove that ordp (2) = 8n…

A: Consider the prime p is of the form 8n+3 and q=(p-1)/2 is also prime. Then, objective is to prove…

Q: 13. Compute 41023 (mod 11). a. 14. Prove that for any integer n, n31 = n(mod7).

A:

Q: Let r be a primitive root of some n > 3. Prove that r = -1 (mod n).

A: Solution: Let n be any positive integer. Let Un= r : r is less than n and prime to n , where r=r+kn…

Q: For the first 5 odd primes, show that Wilsn's theorem holds to be true.

A:

Q: Let be a prime such that p = 1 (mod 4). Prove that –1F, is a square in Fp. -

A:

Q: Prove, by induction, that, for n2 5, the nth Fibonacci number satisfies Fn s 2".

A:

Q: 4. Let p be an odd prime, and a, b primitive roots mod p. Show that ab is not a primitive root mod…

A: It is given that, p be an odd prime, and a and b primitive roots mod p.

Q: For any integer a, prove that a=0 or 1 (mod 3).

A: The solution is given as follows

Q: 3. Use the above theorem to prove that if p E N is a prime number such that p = 1 mod 4, then p is…

A:

Q: Let n be a positive integer, and a an integer Prove that if the order of a modulo n is n-1 then n is…

A: Problem 2.  Solving it using the concept of phi function

Q: Let p be a prime and a a quadratic residue of p. Show that if p =1 (mod 4), ais also a quadratic…

A: Let P be an odd prime and Let a be a quadratic residue mod p ∵ P-a≡-a (mod p) ∴(P-a)p-12≡(-a)p-12…

Q: 5. Let p be an odd prime, and suppose that ged(a, p) = 1. Show that the congruence z = a (mod p*)…

A: p is an odd prime and gcd(a,p)=1We have to show the congruence x2=a mod pk has exactly 1+ap…

Q: |State whether the following statements are true or false. Show working. a. For each odd prime p, a…

A: We have to state whether the statements are true or false

Q: 4. The least positive residue of 2^1000000 mod 17 is: * 1 None of these

A: Question 01: By Division algorithm,  10,00,000= 16*62,500 and since 17 is prime and 17 is not…

Q: Suppose that p is a prime such that p = 5 (mod 8). Let k E Z such that p = 8k+5. For any a E Zp,…

A: 1. Let a∈ℝn. Then a is called quadratic residue modulo n if there exists an integer x such that…

Q: Let p be an odd prime, and let g be a primitive root modulo p. Prove that -g is a primitive root…

A: Given : p= odd prime g= primitive root modulo p

Q: For p an odd prime in Z , Prove: If -1 is a perfect square in Z/pZ then p= 1(mod 4)

A: This can be done as follows:

Q: 2. Let p be a prime number such that p> 5. Prove that p2-1 =0 (mod 24).

A: To prove the below statement If p is a prime number and p &gt; 5. p2−1≡0mod24

Q: Establish that if a is an odd integer, then for any n ≥ 1 a 2 n ≡ 1 (mod 2n+2). [Hint: Proceed…

A:

Q: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p is not prime in Z[i]

A: Given that p is odd prime. This implies that p is not equal to 2.

Q: Find all positive primes p such that n = 1001p satisfies that a" = a mod n for all a.

A:

Q: DuppOse that a anа т аге intege Such that a, аre al diпerent IMod- ulo m, with am-1 = 1 mod m. Show…

A:

Q: 7. Let p and q be primes and suppose that p = 2q + 1. Let h E F and assume that h 1. Pre that ² h…

A:

Q: Let p be an odd prime. Suppose p5 mod 8. Show p-1 p? – 1 mod 2. 5

A:

Q: Let p > 3 be a prime. Show that if p2+ p + 1 is also prime, then p =5 mod 6.

A: We prove the result by the method of contradiction.

Q: Show that the Bernoulli number B₁ satisfies B₁ = (-1)" ( 2 + 2d²+1) Σ (mod 1) for any positive…

A:

Q: Let a be an integer and let n e N. (a) Prove that if a = 0 (mod n), then n | a.

A: By the definition if congruence let a, b and n are integer, with n &gt;0, then a is congruent to b…

Q: Let pn be nth prime. prove that the sum 1/p1+1/p2+…..+1/pn is never an integer.

A:

Q: For every integer n &gt; 1, prove that (n - 1)!mod n = n - 1 if and only if n is prime.

A:

Q: Find (with proof) all positive primes p such that 217p is a Carmichael number.

A: Carmichael number - A number n is a Carmichael number if and only if n=p1p2…pj , a product of at…

Q: show that If n is a positive integer and a is any integer prime to n, then a ln) = 1(mod n), where…

A: It's called Euler's generalization to Fermats theorem. I have used integer m instead of n.

Q: Let p be a prime such that p mod 4 = 3. How many elements oforder 4 are in GF(pn)*?

A:

Q: Let p be a prime such that p = 1 (mod 4). Prove that -lF, is a square in Fp.

A:

Q: Let be an odd prime. Prove that for all i with 1 sisp-2 one has p| 1° + 2' + ... + (p – 1)'.

A: Given : p is an odd prime To Prove :          p| 1i+2i+. . .+p-1i for all i with 1≤i≤p-2

Q: Let p be prime. Show that C) mod p = C for 0 <k< p What does this imply about the binomial…

A: See the following below.

Q: For every prime p, show that ( p - 2)! mod p = 1.

A:

Q: 2.1.3. If a, b are integers such that a = b (mod p) for every positive prime p, prove that a = b.

A:

Q: Prove that if n = p1a1...prar, where the pi are prime, then σ(n)ϕ(n) = n2(1-p1-a,-1)...(1-pr-a,-1

A:

Q: (This exercise constructs another proof of the infinitude of pimes) Show that the integer Qn = n! +…

A:

Q: Prove that for every prime p, there exists an integer x, such that x = 16 mod p.

A:

Q: Let n > 2 be a positive integer and r a primitive root of n. Suppose that a E Z is such that ar? 1…

A: