Find a proper factor p of 267 – 1 using the point P = (6,7) on the curve y? = x +6x – 203. State the smallest integer k such that the computation of k!P provides p. Determine the order of P on the curve mod p.

Q: The sum of any two integers of the from 4k + 1is always: Od Zero Even prime

A: Sum of two integers of the form 4k+1

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

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Q: show that for all integers x and y (x-y)3=x3-y3(mod3)

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Q: 1. Use Eq. (3.10) or Algorithm 3.2 to construct interpolating polynomials of degree one, two, and…

A: We have to use Newton's divided difference to construct interpolating polynomial for the following…

Q: Find the least integer n such that (2x^3+x^2logx) is O(x^n).

A: The given expression is 2x3+x2logx. Concept used: Let f and g from the set of integers or the set of…

Q: Find a polynomial p(x) with degree 6 and integer coefficients, such that for all integers x, we have…

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Q: 9. Show that there are infinitely many primes of the form 16k + 1. More generally, show that for any…

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Q: 8. Find the remainder of x2021 + x2020 + 1 modulo x2 – 1 in Q[x].

A: Remainder of the Polynomial

Q: 2. (a) Use Euclid's Algorithm to show that there are integers s and t such that 126s + 1031t = 1.…

A: The detailed solution is as follows below:

Q: Determine the coefficient of at in (1 – 4a²)ó by completing the following given steps.

A: Concept:

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: Let p be a prime divisor of the Fermat number Fn= 22^n+ 1 i. Show that ordp2 = 2n+ 1 ii. Use…

A: Given p be a prime divisor of the number Fn=22n+1. (i) Since p divides Fn=22n+1, therefore: Fn ≅0…

Q: x4 >is ........in Q[x] Not prime ideal ONone of the choices O Prime not maximal

A: Answer is maximal ideal , i.e , last option , and proof is given below.

Q: Y OU (a) Find the least residue of 639 modulo 10. (b) Use Fermat's little theorem to find the least…

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Q: () is even. 3. Prove that for all positive integers n, the number

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Q: Find a number n e N such that 2" > 102000 and 2" has among its 2000 last digits at least 1000…

A: We will use logarithm to solve the problem

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: 111111 is prime. Prove that, in any radix b, The number 11111 (11...1), can be prime only if the…

A: As per the question we have to prove that the number 111...1 containing n number of 1's will be…

Q: Consider the elliptic curve group based on the equation =x' + ax +b mod p where a = 321, b = 152,…

A: For the elliptic curve , E: y2 = x3 + ax + b Addition Law: Let P and Q be 2 distinct points and the…

Q: For m an n non-negative integers with 0 < m <n, give a combinatorial proof for the identity EOC) m…

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Q: The sum of any two integers of the from 4k + 1is always: Even Odd prime Zero Windows b

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Q: Suppose that p≥ 5 is prime. Show that -3 is a quadratic residue (mo p) if p = 1 or 7 (mod 12) and a…

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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: 3. Find integers s and t so that 1 7s + 11t. Show that they are not unique.

A: The division algorithm states that for any integer, a, and any positive integer, b, there exists…

Q: Find k such that, 2 det 4 li0 -8 -5] -1 -7 = 510 k 0 .

A: Given det2-8-54-1-710k0 =510 we have to find k

Q: Let p be a prime divisor of the Fermat number Fn 22" + 1. : 2n+1 (a) Show that ord, 2 (b) Use…

A: As per the question we have the Fermat number Fn = 22ⁿ+1 and its prime divisor p. Here we will use…

Q: (a) Characterize the primes p for which 63 is a quadratic residue.

A: Quadratic residue: Let p be any odd prime member and gcd(a, p)=1. Then a is said to be a quadratic…

Q: Consider the following theorem. Theorem 9.5.1: The number of subsets of size r that can be chosen…

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Q: Find all values of m with 0 ≤ m ≤ 23 such that 10x ≡ m (mod 24) has a solution

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Q: 4. a. Using the Euclidean algorithm, find the greatest common divisor, d, of 55 and 34 and solve the…

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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: the elliptic curve group based on the equation y =+ az + b mod p where a = 8, 6 = 3, and p= 17. This…

A: Given, the equation is, y2=x3+ax+b mod p Where, a=8, b=3 and p=17 Now, according to Hane's theorem,…

Q: 3. Find integers s and t so that 1 7s+ 11t. Show that they are not unique. %3D

A: Apply the division algorithm, we get 11 = 7*1 + 4 7 = 4*1 + 3 4 = 3*1 + 1 Hence, 1 = 4 - 3*1   = 4 -…

Q: Let a be a quadratic residue of the odd prime p. (a) Show that a is not a primitive root modulo p.…

A: An integer a is referred to as quadratic residue modulo p if there exists n∈ℤ such that n2≡amod p.…

Q: If gcd(m,p) = 1 and p is prime, show that gcd(m,pk ) = 1 for all k≥ 1.

A: To show, If gcd(m, p)=1 and p is prime, show that gcd(m, pk) = 1 for all k≥1.

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

A:

Q: Let n and k be integers with 1<k<n. Show that n n+1 Zk=1 (k - - n

A: Given Solution:

Q: Use Euclidean algorithm to find ged, 'd' of 14283 and 6409. Also find the integers x and y such that…

A: Solution by using Euclidean algorithm and extended Euclidean algorithm as follows

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: Use the Euclidean algorithm to obtain integers x, y and z satisfying the following (198, 283, 512)…

A: To find - Use the Euclidean algorithm to obtain integers x, y, z satisfying the following 198, 283,…

Q: List the first 3 positive prime numbers d in Z so that the quadratic integers in Q[Vd] are precisely…

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Q: If |G| = pq, where p and q are primes that are not necessarily distinct,prove that |Z(G)| = 1 or pq.

A: Given: G=p.q where p &amp; q are primes To prove: ZG=1 or pq

Q: Determine a general congruence condition satisfied by all primes p with p = 3(mod4) such that the…

A: As per the question we are given a number theory problem. Where we have to find a congruence…

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

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

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Q: 3. Prove that, Σk() = n2n-¹. k=1

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Q: 1. Find the least no EN such that 10k + 3 no, it has the stated property. and prove that

A: Given: 10k+3&lt;2k, ∀k≥n0, where n0∈ℕ is a least one. To find: Value of n0. To prove: 10k+3&lt;2k,…

Q: Take non-zero positive integers m and n. Show that Z/mZx Z/nZZ/mnZ if and only if m and n are…

A: Given That:To take non-zero integers m and n .   ℤmℤ × ℤnℤ ≅ ℤmnℤ if m and n are relatively prime ,…

Confidence Intervals

When we calculate intervals in probability and statistics, it can be simply termed as finding out a confidence interval. The confidence interval is usually calculated from the data set which is observed. The value of the parameter that needs to be calculated is present here only.