Show that if p is an odd prime, then n = 2º – 1 passes Miller's test base 2 -

Q: For which odd primes p is 3 a square mod p?

A:

Q: Let p + 7 be an odd prime. Determine the value of the Legendre symbol (7/p).

A: Let p be an odd prime number. An integer a is a quadratic residue modulo p if it is congruent to a…

Q: Show that if p is an odd prime, then every divisor of the Mersenne number 2p − 1 is of the form 2kp…

A:

Q: Is it true that (3 x EN) (x²-3x +20 is a prime)? O True False

A: Is it true that ( there exists x in N)(x^2-3x+20  is a prime ) ?   True False

Q: 2. Let p be an odd prime not equal to 5. Prove that either p² – 1 or p? +1 is divisible by 10. 3.…

A:

Q: Exercise 7.18. Suppose HG and |G| = 2p, where p is an odd prime number. Prove that either |H| = 2 or…

A:

Q: For which odd primes p is 5 a square mod p?

A:

Q: Let p be a prime number and a be a positive integer. Prove that p | (a^p - (p-2)!a).

A:

Q: There are infinitely many primes of the form 4k + 3, where k E N. False O This option True This…

A: The given problem is to check whether the given statement is True or False. Given statement There…

Q: Let p25 be a prime. Prove that p(p+1) (p-1)! +p+1.

A:

Q: Show that: Ifp is odd prime of the form 4k+1, then find a solution of x +1 0 mod p

A: We shall use Euler’s Criterion here. Euler’s criterion states as follows: “Let p be a prime. The…

Q: If p > 2 is a prime of the form 2m +1, prove that m is a power of 2.

A: Let p be a prime and p > 2. And, p = 2m +1 Suppose 't' is an odd prime factor of m. That is, m =…

Q: 8. (a) Prove that if p and q are odd primes and q | a² - 1, then either q |a - 1 or else q=2kp + 1…

A: a) Given that p and q are odd primes and qap-1. So, ap≡1mod q If p=1 then  a≡1mod q So, qa-1 If p≠1…

Q: (i) Prove that if p is a prime, then (p - 1)! = -1 (mod p). (ii ) Show that 18! = -1 (mod 437).

A:

Q: Let o (n) be the sum of the divisors of n and let (n) be the Euler phi functions. Let n = = pq be…

A:

Q: Let p be an odd prime not equal to 5. Prove that either p² – 1 or p? + 1 is divisible by 10.

A: We Shall solve one question only un per the company guidelines as you have asked more than one Que.…

Q: .If p 5 is an odd prime, prove that either p? – 1 or p² + 1 is divisible by 10.

A:

Q: The value of a prime p such that 19p +1 is a square is a 17 Ob 23 Oc None Od 19 Oe e 17 or 23

A: A detailed solution is given below.

Q: Question 5. Prove that o(n) = o(n+2) is satisfied by n = 2(2p-1) whenever p and 2p-1 are both odd…

A:

Q: Show that 2(p − 3)! + 1 ≡p 0, for all odd primes p.

A: Consider the given statement as follows. Show that 2p-3!+1≡0mod p, for all odd primes p. Consider an…

Q: Prove that if a > 1, then there is a prime p so that a < p< a! + 1.

A:

Q: If p is a prime number greater than 3, which of the following represents the sum of the positive…

A: we will check all the options to ensure which is correct.

Q: (#) = 10 Find all primes p so that = 1.

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Prove that there are no 3 consecutive odd prime numbers, that is, for any odd p ≥ 5, at least one of…

A:

Q: Prove that if 2^n+1 is prime then n is a power of 2.

A: Given, 2n+1 is prime To prove that n is a power of 2 if 2n+1 is prime

Q: Show that if p and q are distinct primes, then pª-1 + q®=1 = 1 (mod pq).

A: Given problem is to show the following identity using congruence, we have to consider that…

Q: Prove, that if 2m + 1 is a prime number, then m must be a power of 2 (m = 2n, for some n ∈ N).

A:

Q: describe (a,p)

A: Given, if p is a prime, and a is a positive integer.

Q: a. Suppose p is an odd prime, and let A := a, b € ? (a) Suppose there are ao, bo € Z such that p =…

A: Given p is an odd prime and  A=ab-ba:a,b∈ℤp Suppose, there are a0,b0∈ℤ such that p=a02+b02. Then, by…

Q: Let N=20. Show using the Sieve of Eratosthenes how to find all prime numbers between 10N and 10N+100…

A:

Q: There are infinitely many primes of the form 4k + 1, where k E N. True False This option This option

A: To determine true or false

Q: Let n = pq be the product of two odd primes. Recall that there are no primitive roots modulo n. Show…

A: We use the idea that if primitive root exists then x^2 = 1 mod (m) has only 2 solutions. The…

Q: Show that if n is prime and b is a positive integer with n ̸∣ b, then n passes Miller’s test to the…

A:

Q: (12) What values can be assigned to r so that {4n+r} will include all odd primes?

A:

Q: Let p and q be odd primes. Prove that (p^2) q is not a perfect number. [Hint: Factor 1 + p + q + pq…

A: Let σp2q=σp2σq Then we have  σp2q=σp2σq=1+p+p21+q=p3-1p-1q2-1q-1≤p3p-1q2q-1

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: (2) Let p be an odd prime and suppose that m e Z is not divisible by p. (1) Show that E() \p- = 0.…

A: (1) Given that p is an odd prime. The aim is to show that ∑a=1p-1ap=0. Now take, ∑a=1p-1ap,…

Q: Give an example of a prime p of each of the forms: 6k + 1 6k + 5.

A: We need to give an example of a prime p of each of the given forms. Let us take k=1. So, we get…

Q: Assune p is an odd prime such that p = 1mod4 (that is p 4k + 1) Prove: if a is a primitive root for…

A:

Q: Let p be a prime, a ∈ Z. Prove that ap − a is divisible by p.

A: Fermat's Theorem : If p is any prime and a is any integer then p divides (ap-1-1).

Q: Wh of the following are twin primes. 41 & 643 b) 497 & 499 c) 673 & 675 d) None of these :Select one…

A: Option (a) is correct

Q: 1. It has been conjectured that there are infinitely many primes of the form n2 – 2. Exhibit five…

A: See full solution below

Q: Prove that for all primes p that if p does not divide n2, then p does not divide n.

A:

Q: If p is a prime with p> 2, then 2p + 2 is a multiple of O a. 6 b. 4 Oc. None O d. 3 or 4 O e. 2 ar 5

A:

Q: If p is a prime satisfying n < p < 2n, show that 2n = 0 (mod p)

A:

Q: Show that if a and b are relatively prime then (a+2b,2a+b)=1 or 3.

A:

Q: Let p ≥ 5 be a prime. Prove that p(p+1) | (p − 1)! +p+1. -

A:

Q: 6. Show that if n (a2" - 1)/(a² - 1), where a is an integer, a > 1, and p is an odd prime not…

A: The given question is related with pseudoprimes. We have to show that if n = a2p - 1a2 - 1, where a…

Q: Let N be an integer, and let p be a prime > 7. Suppose p| (N2+7). Show that p is of the form 1+ 7k,…

A: