Let 2k +1 be a prime number. Prove that then k = 0 or k = 2" for some n > 0. %3D

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 for any positive integer n, n=floor(n/2) + ceiling(n/2)

A: Given,          n is a positive integer.     To show-                 n=floor(n/2) + ceiling(n/2)…

Q: 7) Let p be an odd prime, find if a) p =1 mod 8

A:

Q: Show that 2 is a primitive root modulo 3" for all n > 1.

A: To prove that 2 is a primitive root modulo 3n for all n≥1.

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

A: Please see the attachment

Q: Use permutations to solve. From a cooler with 6 different kinds of soda, 4 are selected for 4…

A: nPr = n!/(n-r)!

Q: Let p be a prime larger than 3. 1. Show that the remainder, when p is divided by 6, is 1 or 5. Note:…

A:

Q: Show that if n is any integer, then 3 divides n3 − n

A: To prove:  if n is any integer, then 3 divides n3 − n. Concept used: Here we shall use the identity:…

Q: Let be a prime number. Stating any results that you use, deduce that xº-1 = (x – 1)(x – 2) · . · (x…

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Q: Show that if n is an even perfect number, then n = 2P-'Mp for some Mersenne prime Mp 2P – 1.

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Q: Prove that 2" >n For all nuatural numbers

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Q: p + 2 is a prime number, prove that p2+3 is divisible by 4. (2.5

A: Prime and divisibility

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: (i) Prove that if p is a prime, then (p - 1)! = -1 (mod p). (ii ) Show that 18! = -1 (mod 437).

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Q: Using Fermat's Little Theorem find 52003 mod 455. Note that 455 is not a prime number and 455…

A: We are asked to use Fermat's Little theorem to find 52003 mod455. Fermat's Little Theorem states…

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: Suppose n ≥ 2 is an integer with the property that whenever a prime p divides n, p 2 also divides n…

A: let n=p1a1p2a2....pmam be the prime factorization of n  Here p1,p2....pm are all distinct primes and…

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

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Q: Prove that a positive integer n is prime number.if no prime p less than or equal to n divides n.

A: Let us assume that the contradiction of the statement , let  n ≥ 0 be a composite number .  Thus , n…

Q: Determine how many ti (o bom is a prime.

A: In this question, the number of times is determined when 1 appears in the multiplication table for…

Q: let s be positive integer, then (2) + (;) = %3D

A: sC5 = s!/5!*(s-5!) And sC6 = s!/6!*(s-6)!

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: Show that — ф+1 n=0 ||

A: Given φ is the golden ration satisfying 1φ=φ-1. To prove ∑n=0∞1φn=φ+1 It is known that φ=1+52.

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: 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: Find all prie 8. If p > q 2 5 and p and q are both primes, prove that 24 p2-q?.

A: Given: p≥q≥5 and p and q are both primes

Q: Use any method to deter (a) 13 The series 8.

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 p be a prime number, n 2 2. Show that 6(pn) = po(n) if and only if p n. %3D

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Q: Let N=20. Show using the Sieve of Eratosthenes how to find all prime numbers between 10N and 10N+100…

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Q: Consider all numbers between 20000 and 70000. Efficiently determine: A) How many of these numbers…

A: Part A - Let’s represent the number as ABCDE. You have to considere the special cases first, since…

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

A: Given:

Q: Let n = pq be the product of two odd primes. Recall that there are no primitive roots modulo n. Show…

A: We use the idea that if primitive root exists then x^2 = 1 mod (m) has only 2 solutions. The…

Q: k is a prime number that has a ones digit of 7 and 0 <k < 100, how many values of k are possible?

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Q: For which natural numbers n is n^3-10n^2+28n-19 a pri For which natural numbers n is…

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Q: Assune p is an odd prime such that p = 1mod4 (that is p 4k + 1) Prove: if a is a primitive root for…

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: Let n > 1 be an integer. Show that if pn for all primes p≤ √n, then n is either prime or the product…

A: Introduction: Proof by contradiction, sometimes referred to as indirect proving or the reductio ad…

Q: yields a true statement when n is replaced by any positive integer.

A: The above statement is true. Step-2 i explained in detailed explanation

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 integers, n, where -15 ≤≤ n ≤≤ 15 which make 15 ≡≡ n mod 4  (4 points) Show all your work:…

A:

Q: If p is a prime satisfying n < p < 2n, show that 2n = 0 (mod p)

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Q: If n is an even integer then 7-n3 is sometimes even and sometimes odd. O is an odd integer is an…

A: We have to state whether the 7-n3 is an even integer, odd integer, or sometimes even and sometimes…

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: 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: Find 3 values of n such that phi(n)= 120

A: According to the given information, it is required to determine the values of n such that:

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

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Q: 5. Find all integers 100 ≤ n ≤ 999 such that the last three digits of n² is same as n.

A: To find: All integers  100≤n≤999 such that the last three digits of n2 is same as n.

Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.