Suppose that p is a prime and p ≠ 2. Let a be a nonsquare inGF(p)—that is, a does not have the form b2 for any b in GF(p). Showthat a is a nonsquare in GF(pn) if n is odd and that a is asquare in GF(pn) if n is even.

Q: m and n be positive integers so that m n. Show for any integer a that the congruence

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Q: By using mathematical induction, show that 62m – 35m – 1 is divisible by 1225, for all positive…

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Q: Every integer n22 either is a prime or can be expressed of a product of primes. This statement is…

A: Every integer n≥2 is a prime or can be expressed as product of primes. This statement is Fundamental…

Q: Prove that for every non-negative integer n the value of n² + n+41 is prime ?

A: The number n² + n + 41 is not prime for every natural number n. Let, f(n) = n² + n + 41

Q: Suppose that [E:Q] = 2. Show that there is an integer d such that E = Q(square root of d) where d is…

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Q: 3. x* + 1 is reducible over Z, for every prime p. a) True b) False

A: We have to find out whether x4+1 is reducible over Zp for every prime p or not a polynomial is said…

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

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Q: Prove by mathematical induction: for all integers n, if a 2n × 2n checkerboard with alternating…

A: Let Cn be a 2n×2n checkerboard with alternating black and white squares. We have to prove that if…

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

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Q: Suppose that the prime factorizations of two positive integers x and y are expressed using a common…

A: The prime factorization is breaking down a number into the product of only prime numbers. These…

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

A: Given that n=2p-1 where, pis odd prime. Since, 2p is an even integer. Then, n=2p-1 is an odd…

Q: Find all the positive prime p such that 9p+1 is a perfect cube. Namely, such that there exists an…

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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: Find all the positive primes p such that 9p+ 1 is a perfect cube. Namely, such that there exists an…

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

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

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Q: Given the polynomials P1 10101 and P2= 11000. Show the result of multiplying P1 P2 in GF(25) using…

A: Given the polynomial P1=10101 and P2=11000. Show the result of multiplying P1⊗P2 in GF25 using the…

Q: For p an odd prime in Z , Prove: If p is not prime in Z[i] then -1 is a perfect square in Z/pZ

A: Given p is an odd prime in Z.Assume that (-1) is a perfect square in Z/pZ.Then, there exists an…

Q: For all n E N,n > 1, there exists a prime p such that p|n. ulgn hi

A: False

Q: ivided by 6.

A: Given, m is odd. So, m = 2k+1 for some integer k  &amp;, n leaves remainder 2 when divided by 3  So,…

Q: How to solve this?

A: According to the given information, let p be a prime number such that p&gt;2. It is required to…

Q: ) Show that if p is a prime such that m +1< p < 2m + 1, then p divides (2m+1).

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Q: Let n be any integer. If n=3k+5 for some integer k. Does 91 6n-3 No, because 6n-3= 18k+30-3 which is…

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Q: Suppose that a, b are two integers such that 3a? – b3 = 1. Does it follow that a and b are…

A: Given that a, b are two integers such that 3a2-b3=1.

Q: Let p and q be distinct primes with p<q and q# 1mod(p). Show that if G|= pq then G Z/pqZ.

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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 for any odd prime p, the Gauss, G(p), can be rewritten as p-1 G(p) = 5². x=0

A: To Show- Show that for any odd prime p, the Gauss, Gp, can be rewritten as Gp = ∑x=0p-1ζpx2.

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

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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: Prove that for every ponitive integer a if aº – 6° is a prime for some positive integer 6 5 = 2 or 5…

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Q: Let p be a prime number such that p> 2. Use the proof by contradiction to show that at least one of…

A: See the solution attentively I think you will understand easily.

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: (a) Express each of the integers 4278 and 71929 as a product of primes. (b) What is gcd(4278,…

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Q: Suppose a is a positive integer and p is a prime. Prove that pla if and only if the prime…

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Q: If |G = 30 and |Z(G)| that |G| 5, what is the structure of G/Z(G)? What is the structure of G/Z(G)…

A: Solution:-

Q: Factorize (i) 1144 in Z (i) 1144 in Z[i] (ii) 1144 in Zp[x] where p is a Z-prime greater than 1144.

A: Given

Q: Let Gn - n! + 1, where n is a positive integer. We will generate a sequence of 5. primes by…

A: Given Gn=n!+1, where n is a positive integer.  We have to follow the sequence, 1) Begin by finding…

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: Suppose that [E:Q] = 2. Show that there is an integer d such thatE = Q(√d) where d is not divisible…

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Q: Use induction on n to prove that 42n – 15n – 1 is divisible by 225 for all non-negative integers.

A: Prove 42n-15n-1 is divisible by 225 for all non-negative integers using mathematical induction.

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

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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: a. For each odd prime p, a residue a in Z*p cannot be both a primitive root and a quadratic residue.…

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

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Q: Show that if 2m +1 is an odd prime, then m 2" for some nonnegative integer n.

A: We solve this question by contradiction

Q: (а) Prove that p(n) p(2n) = if n is odd; 24(n) if n is even. (b) Generalize ç(2n) = 24(n) for all…

A: Here we use the formula φ(n)=n∏p|n1-1p.

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

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Q: Suppose that p is a prime and p does not equal 2. Let a be a nonsquare in GF( p)—that is, a does not…

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