Show that there are infinitely many primes of the form 4k + 3, where k is a nonnegative integer.

Q: Find all primes that are the difference of the fourth powers of two integers.

A: Let the two integers are x and y and n is prime so we have  n=(x4-y4)       (suppose x>y)…

Q: Prove that for any 9 consecutive natural numbers, there exists one number that is prime to the other…

A:

Q: Prove that 3n+2 and 5n+3 are relatively prime for every integer n.

A: 3n+2 and 5n+3 are relatively prime for every integer n.

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

A:

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: Show that there are infinitely many primes of the form 4k + 3.

A: We have to show that there are  infinitely many primes of the form 4k+3.

Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

A:

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

A:

Q: Prove that every positive integer n>1 has a prime factorization.

A: Prime factorization is a way of expressing a number as a product of its prime factors.

Q: Prove that there are infinitely many primes.

A:

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

A:

Q: Use the ertended pigeonhole principle to show that there is at least 15 ways to choose 4 integers…

A: Extended Pigeonhole Principle:      If there are n pigeons and m pigeonholes and n>m. Then, there…

Q: For the first 5 odd primes, show that Wilsn's theorem holds to be true.

A:

Q: Find all integers n such that (n^2) - 1 is prime.

A: n² - 1 = (n-1)(n+1)  Now, if (n² - 1) needs to be a prime, then, either n-1 is equal to 1 or, n+1 is…

Q: Prove: If x is a positive prime integer, then x+7 is a composite integer. (i.e., not prime)

A: See solution below

Q: Prove there are infinitely many prime numbers

A:

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: Every even integer greater than 2 is sum of two primes? Give your views on this?

A: We know that 2 is even prime. And all other primes are odd.

Q: Show that there are infinitely many primes to prove that there are infinitely many primes of the…

A: This question is related to Number Theory. We solve the result by contradiction.

Q: Number theory.

A:

Q: Let n be an integer with n ≥ 2. Suppose that for every prime               p ≤√n, p does not divide…

A: given,                                                  p≤n p doesn't divide n. let n is not a prime…

Q: Prove that there are infinitely many prime numbers

A: Let us assume that there exist finitely many prime numbers (Let n). Here, n is a positive integer.…

Q: Supposed a and b are relatively prime integers and c is an integer such that a | c and b | c. Prove…

A:

Q: 21 divides 4n+1 + 52n-1

A:

Q: 6. Show that ifn = pq, a product of distinct primes, then a (n)+1 (mod n) for all a. = a

A:

Q: Show that there are infinitely many primes of the form 6k + 5.

A:

Q: 6. Prove that 2n +5 and 3n+7 are relatively prime for every integer n.

A: What is Relatively Prime Numbers: Relatively prime or coprime numbers are two such numbers which…

Q: If n > 1 is an integer not of the form 6k + 3, prove that n + 2" is composite. [Hint: Show that…

A: Congruence modulo relation on integers: Let a, b∈Z and let m be a fixed positive integer. Then we…

Q: A positive integer n is called superperfect if ơ(ơ(n)) = 2n Show that if n = 2q, where 2q+1 - 1 is…

A: Given: A positive integer n is called superperfect if σσn=2n and n=2q, where 2q+1-1 is prime. To…

Q: (b) Prove that there are infinitely many prime numbers of the form 6n + 5

A:

Q: Show that 5n + 3 and 7n + 4 are relatively prime for all n.

A:

Q: Prove that every prime greater than 3 can be written in the form 6n + 1 or 6n + 5.

A:

Q: If 2n + 1 is prime for some n ≥ 1, prove that n is a power of 2.

A: We have given 2n +1 is prime for some n ≥ 1 We have to prove that n is a power of 2.

Q: Use a direct proof to show that the sum of two odd integers(k and l) is even.

A: The number n is even if it is expressed as  n=2m , where m is an integer. The number n is odd if it…

Q: show that no integer of the form n^3+1 is a prime other than 2

A:

Q: 9. Let n be an odd number with exactly four distinct prime factors. If (3, n) = 1, prove that n is…

A: We will use knowledge of number theory to answer this question.

Q: Prove that 2n>n2 if n is an integer greater than 4.

A: According to the given information, it is required to prove the given statement.

Q: Let pn be nth prime. prove that the sum 1/p1+1/p2+…..+1/pn is never an integer.

A:

Q: Show that if 2n−1 is prime then n is prime.

A: We have to show that if 2n-1 is prime then n is prime.

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: 4. Prove by induction that 21 divides 4"+1 + 52n-1 for all positive integers n.

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

Q: infinitely many primes of the form

A:

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

A: Let p, p + 2 be twin primes. Then p *(p+1) + 1 = p^2 + 2p + 1 = (p+1)^2 ,  a perfect square.

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: (i) Prove that there are infinitely many primes of the form p = 6n – 1. -

A: Given:    Infinitely many primes of the form p=6n-1     Proof:         Suppose p1, . . . , pn are…

Q: If n and a are prime to 7 then prove that nº – a® divisible by 7?

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 there is not a finite number of primes.

A: To Prove: There is not a finite number of primes. That is , to prove there are infinite number of…