Show directly that for p a prime integer, p can be written as p=(a)^2 - d(b)^2 then d is a perfect square in Z/pZ. Hint: Are a and b elements of Z/pZ?    Are a and b elements of Z/pZ*?

Q: Show that the square roots of two successive natural numbers greater than N2 differ by less than…

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Q: - 1 Use the factorization A = PDP to compute A", where k represents an arbitrary integer. I а 6b-а)…

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Q: Show that for any integer n, n² – 2 is not divisible by 3.

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

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: Prove that if n is odd, then 2 is not a square in GF(5n). In other words, prove that there is no…

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Q: 4. Show that if k is a triangular number, then 8k + 1 is a perfect square.

A: Since you have posted a multiple question according to guildlines I will solve first question for…

Q: Prove that there do not exist three consecutive integer values of nn for which 43n+1443n+14 is a…

A: Given 43n+14 We have to Prove that  ∄  three consecutive integer values of n for which 43n+14 is a…

Q: Use a direct proof to show that every odd integer is a difference of two squares. (For example: 1 =…

A: first we take a odd integer then we write in 2y+1 form then we transfer to difference of two…

Q: Show that for any integer n , n2 – 2 is not divisible by 3.

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Q: Use the factorization theorem to prove that x−c is a factor of x^n−c^nfor any positive integer n.

A: Let's understand the factor theorem before we get into solving the question directly. (x - a) is a…

Q: Use the factorization A PDP -1 to compute A", where k represents an arbitrary positive integer. -5…

A: Use the factorization A=PDP-1 to compute AK, where k represents an arbitrary positive integer.…

Q: (b) Let z = 24695 + 99288 i. Enter a square root of z, in maple syntax, in the box below.

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Q: Show that the cube for any integer is of the form 7k, or 7k ± 1

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Q: Suppose, and no are the smallest of two positive integers so that 300x is a perfect square and 300y…

A: The given problem is to find the values of x and y and sum of value of x+y for which smallest…

Q: Use the Division Algorithm to prove that (a) The square of any integer is either of the form 3k or…

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Q: Use the factorization A = PDP to compute A. where k represents an arbitrary integer. a 2(b- a) 12 a…

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Q: Use the Principie to prove the Ving 3n+1 – 1 is divisible by 2 for all n > 0

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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: Prove that 67n+ 17 cannot be a perfect square for any three consecutive integers n.

A: We will use quadratic residue and proof by contradiction method.

Q: Theorem 2.9. Let n be a positive integer and write it as n = N²m, where m is square-free. If m…

A: n be a postive Integer and wollte it as mn = N ^ 2 * m where m is square furee. fm Contains no…

Q: Prove that if n is odd, then 2 is not a square in GF(5n) In other words, prove that there is no…

A: PROOF :

Q: Use the factorization A = PDP - 1 to compute A", where k represents an arbitrary integer. а 11(b -а)…

A: 1st we will derive a formula for Ak and then find its value using that

Q: Use the factorization A = PDP-1 to compute Ak, where k represents an arbitrary integer. a 4(b- a) a…

A: Image is attached with detailed solution.

Q: (d) Prove by induction that 5 "– 1 is divisible by 5 for all n e Z +.

A: we have to show that 5n-1 is divisible by 5 for all n belongs to Z+ by induction for n =1.

Q: (b) Every integer n> I is the product of a square-free integer and a perfect square. [Hint: If n =…

A: By the fundamental theorem of arithmetic for an integer n>1 n=p1k1·p2k2.....prkr where each pi,…

Q: Prove that the cube of any integer in the form 9k or 9k + 1 or 9k + 8 .

A: Let n be some integer then by the Division Algorithm, n = 3p , 3p+1 or 3p+2 for some integer p

Q: Find the largest positive integer p for which the R computation 2°p yields a finite representable…

A: Given: Determine the biggest positive integer p for which the R calculation 2^p produces a finitely…

Q: Prove or disprove that all chessboards of these shapes can be completely covered using L-shaped…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: Prove that for all primes p2 5, p + 1 cannot be a perfect square. Recall that is a perfect square if…

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Q: 22 Find the PALU factorizations (and check them) for 0 1 1 A = 1 0 1 and A = 2 3 4 1 2 0 24 1 1 1 1

A: Solution :- As per guidelines I submitted only first question. If you want all question the repost .…

Q: 2. ) Show that the cube of any integer is of the form 7k or 7k +1 b) Verify that if an integer is…

A: 2) a. To show that ''The cube of any integer is of the form 7k or 7k±1''

Q: Prove that if n is odd, then 3 is not a square in GF(5n).  In other words, prove that there is no…

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Q: All perfect squares are divisible by 2. A.) True B.) False, counterexample of 16 C.) False,…

A: Numbers that are divisible by 2 are {0,2,4,6,8 ,10 } Perfect sq 1X1=1 2X2=4 3X3=9  4X4=16 5X5=15…

Q: 28. Suppose that [E:Q] = 2. Show that there is an integer d such that E = Q(Vd) where d is not…

A: Given [E:D]=2.

Q: ) Prove that for all positive integers k, 22k – 1 is a multiple of 3.

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Q: Use the factorization A = PDP' to compute A", where k represents an arbitrary positive integer. -2…

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

A: Given that E:Q=2 We know that Being finite E is an algebraic extension. Let α∈E−Q, then minimal…

Q: Let m and n be positive integers. Show that mZ+nZ= (m, n)Z.

A: Given: Let m, n be two positive integers. To prove: mℤ+nℤ=m,nℤ. Proof: Let I=mℤ,J=mℤ Since…

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: Give a direct proof that if m and n are even are both perfect squares, then mn is also a perfect…

A: Given that, m and n are even and both perfect squares.

Q: Attached is the question I'm needing help with. Thanks!

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Q: Let p and q be primes. Prove that pq +1 is a square if and only if p and q are twin primes.

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Q: Use the factorization A= PDP-1 to compute Ak, where k represents an arbitrary positive integer. -5…

A: Given: A=PDP-1 -524-29=34113001-141-3

Q: A. Prove that n3 + 2n is divisible by 3 for all integers 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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Q: T(n) is an odd integer if and only if n is a perfect square. a(n)i on oda intese

A: Since you have asked multiple questions, we will solve the Question Q7 (a) for you. If youwant any…

Q: Use the factorization A = PDP 1 to compute A", where k represents an arbitrary positive integer. 9 -…

A: 9-124-5=3221 1003 -122-3Compare with A=PDP-1Ak=PDkP-1Ak= 32211k003k-122-3     =32211003k-122-3…

Q: IF 9. 36x + O is a perfect square, what is07 A 36 e 2121 EduAide Saftwane ine All riala

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Q: 3. Prove that if n is a triangular number, then so are 9n + 1, 25n + 3, and 49n+ 6. Hint: Recall…

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Q: 3. Prove that if n is a triangular number, then so are 9n +1, 25n + 3, and 49n + 6. Hint: Recall…

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