4. (i) Using the fauct that 187 = 11 · 17, show that 21к—174 ноd 187. This shows that 187 fails the base 2 primality test. (iї) Соmpute 21911 mod 1911. Use Fermat's little theorem and hand calculations. Does 1911 pass or fail the base 2 primality test? n primo fuotorim Vou ua. of 1011

Q: Prove that there are infinitely many primes congruent to 3 mod 8. previous part.

A: We have to show that there are infinitely many primes congruent to 3 mod 8. we will proof this by…

Q: TION 9 Prove by Induction that for every natural number n, 3"–1 is a multiple of 2? Attach File…

A: We prove by induction

Q: Suppose that p, q are distinct prime numbers, prove that p9-1 + gp-1 = 1 mod pq

A: We use Fermat's Little Theorem, to prove our claim.

Q: 7) Show that there are no primitive roots modulo 21 using EGLF and CRT.

A: Given value of n is 21 . Now, break 21 into its factor, 21=7×3 ,7,3=1 If m>2 and n>2 then the…

Q: As in #3 but now evaluate (37603|48611), again by regarding it as a Legendre symbol, after first…

A: We use the law of quadratic reciprocity to solve this question. The detailed solution typeset in…

Q: Prove 10^47+67 is not a prime(use Fermat's). Thank you :)

A:

Q: Prove that if n is an integer and 7n+2 is even, then n is even. State which method you are using for…

A: Let  n be any integer.  We have given that , 7n + 2 is even. We need to prove that ,  n is even. We…

Q: # 3) Evaluate the Legendre symbol (55|179) using only the definition and multiplicative and mod p…

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Q: c) Using Fermat's Little Theorem, find 6142 mod 11.

A:

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

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Q: Alice and Bob want to use Diffie-Hellman Key Establishment to share a key and they have agreed to…

A: The suitability of these numbers in the order is shown as given below

Q: What is the smallest positive integer k such that xk = e for all x inU(7 . 17)? Generalize to U(pq)…

A:

Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169. - -

A: The given expression is: 33n+3-26n-27 This is to be proved that the expression is divisible by 169.

Q: Prove: in L number psiz 12, for every prime poiz, [p] = [1], [5) , [7) , or [11]

A:

Q: primes

A:

Q: Show that there are infinitely many primes to prove that there are infinitely many primes of the…

A: This question is related to Number Theory. We solve the result by contradiction.

Q: Suppose n is an integer and consider the number n* – 6n³ – 18n² + 6n + 1. a) By expanding the…

A: Explanation: a. Given that,                    Expand the right side and show that both sides are…

Q: 1. Prove that the Mersenne number M13 is a prime; hence, the integer n = 2' (213 – 1) is perfect.…

A: handwriting solution

Q: 5. Use the method of successive squaring to compute 3875 (mod 103) 1

A:

Q: 5-a) How many prime triples of the form (p, 2p + 1, 4p + 1) are there? Describe them all, if…

A: a) The objective is to state the number of prime triplets of the form p,2p+1,4p+1. b) To verify the…

Q: Use Fermat's Little Theorem to find (Show work.) a) 945 mod.23. b) 231002 mod. 41 c) (-9)933 mod. 13

A: Fermat's Little theorem states that " if p is a prime number and a is not divisible by p then…

Q: Suppose n is a natural number greater than 100. Alex and Jerry each compute the gcd(4n, 20n2 + 15)…

A: Given that n is a natural number greater than 100. Alex and Jerry computed the gcd4n,20n2+15 and got…

Q: 3. Prove the following two theorems about pairs of "twin primes," p and (Recall that "twin primes"…

A:

Q: Number 6 please.

A: Problem 6:Recall the following facts.

Q: Least residue of 45^37 mod 19 when using Fermats Little Theorem

A:

Q: Find all prie 8. If p > q 2 5 and p and q are both primes, prove that 24 p2-q?.

A: Given: p≥q≥5 and p and q are both primes

Q: Suppose that m, n are two integers such that m3 – 2n2 = 1. Does it follow that m and n are…

A:

Q: (8) Using that 151 is a prime (so that, in particular, Fermat's Little The- orem can be used; hint,…

A: That's easy. Have a great day!!!

Q: If 2n + 1 is prime for some n ≥ 1, prove that n is a power of 2.

A: We have given 2n +1 is prime for some n ≥ 1 We have to prove that n is a power of 2.

Q: Using Fermat's Little Theorem 62003 (mod 11) Compute the last two digits of…

A:

Q: If possible, let berational. v2 a Then, there exist positive co -primes a andb such that - -

A:

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

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Q: Show that 2047 is a strong pseudoprime to the base 2 by showing that it passes Miller’s test to the…

A:

Q: Verify by mathematical induction Ž (2i – 1)° = n°(2n² – 1) for alln2 1. - i=1

A:

Q: Prove: in Z12. L12, for every prime number p>iz, [p] = [1], [5], [7) , or [11]

A: Consider the set ℤ12 consist of all equivalence classes of integers obtained by the remainders…

Q: Prove that 11 is the only prime number of the form 4n2 - 25.

A:

Q: 8. Show that 18!= -1 (mod 437). 9. Using Euler's Theorem, find the last two digits of 44443333 10.…

A:

Q: duce the expression for u to 2(5 ,Y = (1+5P) +4 (mod 2p - 1) or simply, to 2u, = 1+5P-1(mod 2p - 1).…

A:

Q: In the question that follows, the "units digit" of a number is the last digit in decimal…

A: 41 = 4 42 = 16 43 = 64 44 =256   Conjecture:- For all n>=0, If n is odd then Unit digit of 4n is…

Q: (i) Prove that there are infinitely many primes of the form p = 6n – 1. -

A: Given:    Infinitely many primes of the form p=6n-1     Proof:         Suppose p1, . . . , pn are…

Q: (5n)! 40 n! is natural number for every natural number n. Prove that

A:

Q: If n and a are prime to 7 then prove that nº – a® divisible by 7?

A:

Q: se Euler's Theorem to find a number x between 0 and 28 with x85 congruent to 6 modulo 35. (You…

A: By using the fernet's theorem: If a and n are co prime. That is (a,n)=1 then aϕ(n)≡1modn For n=35…

Q: 3. a) Compute Ek. b) Prove that there are infinitely many prime numbers. c) Assuming modulus 127,…

A: Given: The following conditions. To find: a) Compute ∑k=0nk3 b) Prove that there are infinitely many…

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: In an inductive proof that for every positive integer n, n п(п + 1)(2n + 1) Σ 6. j=1 what must be…

A:

Q: use the "successive squaring" method to check 'by hand and modest calculator' that a^48610 == 1 (mod…

A: Fermat's little theorem says, If n is a prime number, and a is not divisible by n, thenan-1≡1mod n.…

Q: How mony positive întegers are there which are Less thonor equal 6750 which make the ineor…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.