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

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: 3. Show by mathematical induction that, for all n2 0, the equality +++.+ = 2 – +2 holds. ... 2Th

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: V. Show for all integers n2 1 that 71(3*n+ – 52n-). а.

A: As you have asked more than one question from different backgrounds, I am solving the first question…

Q: suppose a prime p of the form 8n + 3 and that q = (p-1) / 2 is also prime. Prove that ordp (2) = 8n…

A: Consider the prime p is of the form 8n+3 and q=(p-1)/2 is also prime. Then, objective is to prove…

Q: If is an odd integer, show that n* + 4n2 + 11 is of the form 16k.

A: Firstly, we have to prove that n4+4n2+11 is in the of 16k for an odd integer n. for making the…

Q: What is the smallest positive integer k such that xk = e for all x inU(7 . 17)? Generalize to U(pq)…

A:

Q: Use induction to prove (2k – 1) = n² – 4 k=3 for all positive integers n2 3.

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: Show that n² + 4n + 14 is big-Oh (n²)

A: Take k=4 and C=6

Q: 2. Prove by contraposition: For all integers n, if (3n +4) is even, then n is even.

A: Contrapositive of 'if p then q' is if not q the not p'

Q: Use cases to prove that for all integer n, n2+3n+5 is odd.

A:

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: 2. Prove by induction (n+1) (n+2) - 1 is odd for all positive numbers of n >= 0

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: Show that there is no positive integer n for which In-1+ In+1 is rational.

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: Suppose that m, n are two integers such that m 2n2 = 1. Does it follow that m and n are relatively…

A: Given that m, n are two integers such that m3-2n2=1. To show m, n are relatively primes we have to…

Q: or disprove : [nx] =n [x] %3D for all Positive integers n > 2

A:

Q: 2. Show that for positive integers greater than 1, 2 + 6 + 10+...+(4n – 2) = 2n2 -

A: Given query is to find the sum of the sequence.

Q: 2 1P+ 23+. .. n (nt1) for every integer .3B 2. Proof (by induction)

A: See the detailed solution below.

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: Suppose that m, n are two integers such that m3 – 2n2 = 1. Does it follow that m and n are…

A:

Q: 2. Prove that if n is even, then 3n + 1 is odd

A: The given expression is 3n + 1. Prove that 3n + 1 is odd, if n is even.

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: 4. Argue by contradiction that for all integers n > 2, there is a prime number p such that n < p<…

A: Fact-1: If n is a positive integer and n is greater than or equal to 2, then n must have a prime…

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: 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: 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: 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: Prove that 11 is the only prime number of the form 4n2 - 25.

A:

Q: Show by using construction that n^2+5n+6 is not prime for any n is a natural number.

A:

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: n. Prove that 21|(3n7 + 7n³ + 11n) for all integers

A:

Q: Prove that by mathematically Induction (Marks = 05) 2n < (n + 1)!, for all integers n ≥ 2.

A: Answer

Q: 2. Prove that a positive integer n is even if and only if nis even.

A: We prove that a positive integer n is even if and only if n2 is even.

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

A:

Q: prove n2 + 5 greater than or equal to 4n for all integers n

A:

Q: Suppose that m,n are two integers such that m3−2n2=1. Does it follow that m and n are relatively…

A:

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

A:

Q: b) For which positive integer n is it true that 2"> n³? c) Prove your answer in (b) above using…

A: b. 2n&gt;n3 Now for n=1 21&gt;132&gt;1 For n=2 22&gt;234≯8 For n=3 23&gt;338≯27

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

A:

Q: 8

A:

Q: In an inductive proof that for every positive integer n, n п(п + 1)(2n + 1) Σ 6. j=1 what must be…

A:

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.