Show that the sum of twin primes p and p + 2 is divisible by 12, provided that p> 3.

Q: Determine whether there are infinitely many primes that have a remainder of 12 when divided by 25.…

A: Given question is Determine whether there are infinitely many primes that have a remainder of 12…

Q: Show that there are infinitely many primes.

A:

Q: 21. Find two different factorizations of the number 21 into primes of the form a + b v-2, where a…

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Q: The number of factors of prime numbers are None of the mentioned O 30 Depends on the prime number

A: No. Of factor of prime no. Means how many divisor consist of a prime no.

Q: Prove that pqr is a deficient number, where p, q, r are three distinct primes with p <q < r

A: We have to prove that pqr is deficient number where p, q and r dustinct primes  Such that p &lt;  q…

Q: Find the greatest natural number n such that n + 10 divides nº + 100.

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Q: he sum of any two integers of the from 4k + 1isalways: COdd prime Zero even

A: The sum of two consecutive integers can always be written as 2n+1, but it can be written as 4n+1…

Q: If n has a prime factorization n = pi*p2² ... p" then the sum Eajn d³P(d) is equal to: a1. az

A: Euler's phi function is widely used in number theory and algebra. This function represents the…

Q: Give an odd prime p congruent to ±1 modulo 12. Show that (3) = 1.

A: Please see the attachment

Q: Use ibn al-Haytham's procedure to derive the formula for the sum of the fifth powers of the…

A: We have to derive the formula 15+25+......+n5=n66+n52+5n412−n212 by using ibn al-Haytham's method…

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: For which prime p,  p^2+2 is also prime?

A: Put p=3 in p2+2 and obtain that, 32+2=9+2=11. Clearly, 11 is a prime number. Thus, p=3 satisfies…

Q: For which odd primes p is 5 a square modulo p?

A: As we know that  a is Quadratic residue mod p if and only if   a(p-1) /2 =1 mod p . If p and q are…

Q: 24. Show that if p is a prime and k is an integer such that 1sksp-1. then p divides ().

A: 24. we have to show  that for a prime p with r≤k≤p-1      p divides  pk  Notation: pk=p!p-k! k!

Q: Prove that p=5 and q=2 are the only pair of prime numbers such that p−q=3.

A: you can solve it by using the solution of Diophantine equation. So , Here we have to prove  p = 5…

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: 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: 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: There exist two prime numbers and such that their product gives 1261

A: We have to find two prime numbers whose product gives the required number

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: A. If p 2q25 is a prime number, prove that p-q is divisible by 3.

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Q: If p is a prime number greater than 3, which of the following represents the sum of the positive…

A: we will check all the options to ensure which is correct.

Q: Find (characterize) all primes p > 3 that satisfy (금) 1.

A:

Q: Prove that there are no 3 consecutive odd prime numbers, that is, for any odd p ≥ 5, at least one of…

A:

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

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Q: 24. Show that if p is a prime and k is an integer such that Isksp-1.then p divides ().

A: See the handwritten solution

Q: 5. Find the canonical prime factorization and solve for No. of positive divisor and sum of positive…

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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: 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 cananical prime factorization dnd solve no.of poritive divisor and anyisad to wns

A: We have to find canonical prime factorization, number of positive divisors and sum  of positive…

Q: If n is a product of twin primes, then a) Ø(n)o(n) = (n – 1)(n – 3) b) ø(n)o(n) = (n + 1)(n – 3) %3D…

A: If n is a product of twin primes, then a)  ϕ(n)σ(n) = (n+1)(n-3) b)  ϕ(n)σ(n) = (n+1)(n-3) c)…

Q: describe (a,p)

A: Given, if p is a prime, and a is a positive integer.

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

A:

Q: Let c = 2^100 and p be the smallest prime such that p >= c. How many composite numbers are between c…

A: Let c=2100 And p=min p is prime:p≥c,p∈ℤ Claim: Total number of composite number between c &amp; p,…

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

A:

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

A: Given:

Q: Prove that 2^(27) - 1 is not a prime number.

A: A prime number is a natural number greater than 1 that is not a product of two smaller natural…

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: 15/If n and n +2 are a pair of twin primes, establish that o (n +2) = 0 (n) +2; this also holds for…

A: σ (n) = sum of all the positive divisors of n

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: Show that in Zm there is no nilpotent element except 0 by letting m be the product of three distant…

A: Statement : In Zm, there is no nilpotent element except 0 We need to prove the  above statement by…

Q: Suppose that p is a prime number greater than 30. Prove that when p is divided by 30, the remainder…

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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: How many twin primes are there where both primes are less than or equal to 31? Number of twin…

A: Twin prime: If there is only one composite number between two prime numbers, then that two numbers…

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

A:

Q: Prove that there are infinitely many primes of the form p = 4n + 3.

A:

Q: show that for every positive integer n there is a prime whose decimal expansion contains n…

A: so we have to prove , for every positive integer n there is a prime whose decimal expansion contains…

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: Let p2 q be primes greater or equal to 5. Show disprove that 24 divides p² – q². or

A: Consider the provided question,