Prove that there are no 3 consecutive odd prime numbers, that is, for any odd p ≥ 5, at least one of p, p + 2, p + 4 is composite.

Q: let 2^k+1 be a prime number. prove that then k=0 or k=2^n for some n =0

A: We have given that , 2k+1  be a prime number . We need to prove that , k = 0  or  k = 2n When k = 0…

Q: Prove that there are infinitely many primes congruent to 3 mod 8. previous part.

A: We have to show that there are infinitely many primes congruent to 3 mod 8. we will proof this by…

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: 2. Let p be an odd prime not equal to 5. Prove that either p² – 1 or p? +1 is divisible by 10. 3.…

A:

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

A:

Q: 8. Prove that if the integer n has r distinct odd prime factors, then 2" | ¢(n).

A:

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: (a) Show that, for any odd prime p > 5, 1 if p = 1, –5 (mod 12) (주) -1 if p= -1,5 (mod 12)

A:

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: Let p be an odd prime not equal to 5. Prove that either p² – 1 or p? + 1 is divisible by 10.

A: We Shall solve one question only un per the company guidelines as you have asked more than one Que.…

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

A: GIVEN: Prove that there are no 3 consecutive odd prime numbers

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: 1. Prove that the Mersenne number M13 is a prime; hence, the integer n = 2' (213 – 1) is perfect.…

A: handwriting solution

Q: 5. Prove that the equation ø(n) = ¢(n+ 2) is satisfied by n = 2(2p – 1) whenever p and 2p – 1 are…

A: Euler totient function formula: p is prime then ϕ(p)=p-1 ϕ(pk)=pk-pk-1     ∀k∈ℕ   If gcd(a,b)=1…

Q: Show that 2(p − 3)! + 1 ≡p 0, for all odd primes p.

A: Consider the given statement as follows. Show that 2p-3!+1≡0mod p, for all odd primes p. Consider an…

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

A:

Q: 3. Prove the following two theorems about pairs of "twin primes," p and (Recall that "twin primes"…

A:

Q: Suppose p and q are distinct prime numbers. Find the number of positive integers x ≤pq such that…

A:

Q: Prove that n!+1 and (n+1)!+1 integer n. relatively prime for every positive

A: Given, n!+1 and n+1!+1are two numbers and to prove that they are relatively prime.

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

A:

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

A:

Q: Let p be a prime number with p>=5. Prove that p must either be one more or one less than a multiple…

A: We use a fact that any number greater that 5 must be written as one of the form…

Q: Prove that for any odd prime p different from 3, ) = 1 if and only if p = ±1 (mod 12). (hint: Use…

A:

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

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).

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: 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 & p,…

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 it is not possible to find four distinct prime numbers p, q, r, s such that ps − qr = 0.

A:

Q: Prove: in Z12. L12, for every prime number p>iz, [p] = [1], [5], [7) , or [11]

A: Consider the set ℤ12 consist of all equivalence classes of integers obtained by the remainders…

Q: 2. If p > q > 5 and P and q are both primes, prove that 24 p - q² |

A:

Q: infinitely many primes of the form

A:

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

A:

Q: Prove for every pair p, q of distinct primes that

A: To prove that for every pair p, q of distinct primes that pq is irrational.

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

A:

Q: Prove that for all primes p that if p does not divide n2, then p does not divide n.

A:

Q: Prove that the messenger number M19 is a prime hence the integer n = 218 (219 -1) is perfect

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 p is prime, p divides k^2 + m^2 and p divides m^2 + x^2, then p divides k^2 + x^2

A:

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…

Q: Make a table of values of the numbers 10n – 1 and 10n + 1 for 1 < n < 20. Circle any numbers that…

A: Table of values for 10n-1 and 10n+1 for n between 1 and 20. n 10n-1 10n+1 1 9 11…

Q: Let p ≥ 5 be a prime. Prove that p(p+1) | (p − 1)! +p+1. -

A:

Q: is a prime number and that n is an integer such that p|n^2 prove that it follows p|n

A: