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

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: In the symbol  n√a , the integer n is called the ____________ .

A: Given: In the symbol  an,  the integer n is called the _____. To fill: In the blanks.

Q: For what values of n is 3" > n³ ? Use induction to prove your answer.

A: We have the given statement is 3n>n3 For n=1 we see that 31>(1)3 3>1 This is true. For n=2…

Q: Determine for which positive integer values of n, 3n3 +2<n* and prove your claim by mathematical…

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Q: Consider a product of an odd prime with itself and represent it as a sum of two squares. 1. 13^2 =…

A: According to question given:Product of an odd prime with itself and represent it as a sum of Two…

Q: a) Find the population after 7 years. (Round to the nearest integer as nee

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Q: Find the canonical prime factorization and solve for No. of positive divisor and sum of positive of…

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Q: Use induction to show that n4 + 15 is a multiple of 16 for every positive odd integer n.

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Q: For every positive integer a, find a composite number n such that n|a^n - a.

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Q: Which of the following are twin primes. a) 841 & 843 b) 857 & 859 c) 701 & 703 d) None of these

A: Twin primes: Two prime numbers are called twin primes if there is present only one composite number…

Q: The prime factorization of 7007 is B.11.13 O11.13 V.11.13 011³.13

A: First option is correct.

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: Wh of the following are twin primes. 41 & 643 b) 497 & 499 e) 673 & 675 d) None

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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: For each odd prime p< 31, identify all of the squares and non-squares in Fp.

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Q: Prore the only prime of the form n°-4 is 5.

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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: Determine the largest prime you need to consider to be sure that you have excluded, in the sieve of…

A: (a) According to the sieve of Eratosthenes, first calculate the square root of the given number. The…

Q: Do you thinkthat there are infinitely many primes of the n -1

A: there are infinitely many primes of the n -1 Explain the above statement

Q: Before arriving at his famous conjecture, Goldbach had conjectured that every odd positive integer…

A: Goldbach conjecture says that every integer greater than 5 is the sum of three primes.

Q: FIND THE SMALLEST n So THAT I

A: In this question, we have to find n for given |Rn|&lt;0.0001.

Q: Evaluate (37603|48611), by regarding it as a Legendre symbol, after first checking by trial division…

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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: Find the proportion of primes p in the interval from 10^12 and 10^12 + 1000 such that p-1 is B= 105…

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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: Find ged(92928, 123552) and lem(92928, 123552) and verify that the ged(92928, 123552) x lem(92928,…

A: Prime factorization of 92928 is  92928=2×2×2×2×2×2×2×2×3×11×11=28×3×112 and…

Q: Do you think it is worthwhile, or useful, to solve the puzzles proposed in Euclid's theorem of…

A: The objective is to solve the puzzles proposed in Euclid's theorem of primes There are infinitely…

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: how many pairs of primes p and q are there such that q-p=3? Prove your answer.

A: Here, we need to find all the possible pairs of prime numbers p and q such that the difference (q-p)…

Q: Find the canonical prime factorization of this using Fermat’s Method and solve for the Number of…

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Q: 2. For which odd primes p is 11 a

A: If p is a prime, then by Fermat's Little Theorem

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

A: Twin prime numbers: Two prime numbers are called twin primes if there is only one composite number…

Q: Give an example of a prime p of each of the forms: (a) 4k +1 (b) 4k + 3 (c) 6k + 1 (d) 6k + 5.

A: Que Ans -  prime number - A number Is divisible only by itself and number 1 is called prime number…

Q: Use Dirichlet's theorem to show that there are infinitely many primes whose dicimal expansion ends…

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Q: Use Fermat's theorem to show that for any two distinct odd primes P.9, and any integer C: CM-c-c +c…

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Q: Find all prime numbers pand of for which 2p²q + 45 pp ² = y² 2 де

A: Given:

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

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Q: Evaluate |P(A)| if A={n: n is a prime number from 1 to 100}

A: To evaluate : PA if A=n : n is a prime number from 1 to 100

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: 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: Infer a formula for Ak using factorization where A =

A: Calculate eigenvalues: 1-λ-62-6-λ=0-6+6λ-λ+λ2+12=0λ2+5λ+6=0λ=-2,-3

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: 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…

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Q: If p2q25 is a prime number, prove that p-q² is divisible by 8.

A: Given, p≥q≥5 are prime numbers, Now, p2-q2=p+qp-q. Since, p and q are prime which are grater than…

Q: Let p and q be odd primes and let m and n be positive integers. Explain why U(p") O U(q") is not…

A: given, let p and q be odd primes. and n be a positive integer.

Q: Find all values of primes p for which (¹¹) = 1 Р

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Q: Let pand g n=p8ge, be distinct primes. het show that v (^)n.

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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.…

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.