Show that the product of two numbers of the form 4n + 1 is still of that form. Hence show that there are infinitely many primes of the form 4n + 3.

Q: Prove by induction that 7 divides 3^(5n+3) + 5^n for every non-negative integer n.

A: We can prove it by using mathematical induction.

Q: 4. Prove by mathematical induction that for every positive integer n, II (G- 1 %3D 3i-1 31 3i+1…

A: We have to prove given expression using mathematical induction.

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: Prove that for a positive integer n we have that  33n+3 - 26n - 27 is divisible by 169.

A: To show  33n+3 - 26n - 27 is divisible by 169.

Q: Prove that the fraction 7n+1/ 20n+3  is irreducible for every natural number .

A: Question:-  Prove that the function 7n+12on +3is irreducible far every natural number  →  Let assume…

Q: Prove that for all integers n.4n² +6n + 1 is not divisible by 4. Be sure to explicitly show how each…

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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 or disprove: For every positive integer n, 22n – 1 is divisible by 3. -

A: I have proved the result using mathematical induction

Q: Prove that there are infinitely many primes.

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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: Prove that 5n-1 is divisible by 4 to any positive integer n

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Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

Q: primes

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Q: prove that n(13n²-1) is divisible by 6 for every n ≥1

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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: 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: Prove that n2-n is divisible by 2 for every positive integer n

A: We have to prove that,  n2-n is divisible by 2 for every positive integer n. We will use…

Q: Theorem 2.9. Let n be a positive integer and write it as n = N²m, where m is square-free. If m…

A: n be a postive Integer and wollte it as mn = N ^ 2 * m where m is square furee. fm Contains no…

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

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Q: Show that 1+3/ -5 is irreducible which is not prime in ZlV-5]

A: We know that ,  ℤ-5 = a + b-5 / a , b ∈ ℤ We need to show that , 1 + 3-5  is irreducible but not…

Q: Show that 3·4n+ 51 is divisible by 3 and 9 for all positive integersn

A: Consider the provided expression,

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: 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: Prove that the cube of any integer in the form 9k or 9k + 1 or 9k + 8 .

A: Let n be some integer then by the Division Algorithm, n = 3p , 3p+1 or 3p+2 for some integer p

Q: Use mathematical induction to prove that 2n+2 + 32n+1 is divisible by 7 for all integers n 2 0.

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

Q: Prove that there is no integer n for such that 5n+3 is divisible by 5.

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

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Q: Prove by induction that (2n+1)3 is an odd number for all positive integers n.

A: First principle mathematically induction : Suppose is a mathematical statement .Here n is a natural…

Q: Show that there are infinitely many integers n such that 4n^2+1 is simultaneously divisible by 13…

A: We need to prove there are infinitely many integers n such that 4n^2+1 issimultaneously divisible by…

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

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

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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: Prove that there exists a positive integer n >1 such that 2n2 + 3n + 11 is prime.

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Q: By using mathematical induction prove that the given equation is true for all positive integers. 9n…

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

Q: Prove or disprove: If three consecutive integers are multiplied together, and the second in order of…

A: We first look out some examples to understand the fact- choose three consecutive numbers be 1,2,3.…

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

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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.

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Q: Prove that n2-n is divisible by 2 for every positive integer n.

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Q: infinitely many primes of the form

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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: Exercise 1. Show that there are infinitely many primes of the form 4n +3. Hint: you might find it…

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Q: 1. It has been conjectured that there are infinitely many primes of the form n2 – 2. Exhibit five…

A: See full solution below

Q: Let n be an odd positive integer. Find all perfect numbers of the form nn +1.

A: n=1,3,5,7,9,11,13,...nn+1=11+1,33+1,55+1,77+1,99+1,... Sequence is: 2, 28, 3126, 823544, 387420490,…

Q: Prove that n^2 is an even number if and only if 3n + 1 is an odd number by proving  If n^2 is even…

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Q: 1) Guess a formula for the sum 1-3+5-7+..+ (-1)*-(2n – 1) =(-1)-(2i – 1) i=1 and prove your fornmula…

A: The given sum is  1-3+5-7+...+-1n-12n-1=∑i=1n-1i-12i-1

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

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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: Prove that there exists exactly one prime of the form n^3 −1. First express the statement…

A: Number

Q: There is an integer n such that 7n2 – 5n + 11 is a prime number. -

A: Let fn=7n2-5n+11for n=1f1=7×1-5×1+11      =13 , which is prime number for n=2f2=722-5×2+11…