Find all integers z, positive or negatives, such that z2 + z is prime.

Q: 91. Use the Factor Theorem to prove that x - c is a factor of x" – c" for any positive integer n.

A:

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 the statement is true for every positive integer n.

A: -11+-12+-13+---+-1n=-1n-12. We prove this statement by using the Principle of Mathematical…

Q: Prove that 2 is a factor of n2 + 5n for all positive integers n.

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: Determine (with proof) if there exist integers m and x such that 2003m 2 = 2.x(x – 20). If they…

A:

Q: If all entries of A are integers, and det A = 1 or -1, prove that all entries of A -1 are integers.…

A:

Q: Prove by Strong Induction that, 2n-1+2n-2 = 3 x 2n-2 for every positive integer n.

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 that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

A: The objective is to prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

Q: Use the factorization theorem to prove that x−c is a factor of x^n−c^nfor any positive integer n.

A: Let's understand the factor theorem before we get into solving the question directly. (x - a) is a…

Q: Prove: for every positive integer n, 2"> n

A: As per the guidelines, since there are more than one question only one question will be answered.…

Q: The sum of any two integers of the from 4k + 1is always: prime Zero Od Even

A: Given statement, The sum of any two integers of the form 4k + 1 is always……

Q: Find all integers x + 3 such that x – 3 is divisible by x – 3. |

A: To find all integers such that x^3-3 is divisible by x-3

Q: () is even. 3. Prove that for all positive integers n, the number

A:

Q: Determine the smallest subring of Q that contains 2/3.

A:

Q: Prove by mathematical induction that 3n+2n+3 is divisible by 4 for all positive integers

A:

Q: Define: "A positive integer alb). a is divisor or factor of integer b" (Notation: an

A: We Define: "A positive integer a is divisor or factor of integer b "(Notation: a|b).

Q: snip

A: x+c is a factor of xn+cn if n≥1 is an odd integer.

Q: The sum of any two integers of the from 4k + 1is always: Even Odd prime Zero Windows b

A:

Q: Find all integers n such that n| (2n+3). *

A: Given, n|(2n+3)     TO find, All integers n

Q: 3. Find integers s and t so that 1 7s + 11t. Show that they are not unique.

A: The division algorithm states that for any integer, a, and any positive integer, b, there exists…

Q: Find all integer numbers a for which

A:

Q: Prove that for any positive integers m and n there are m consecutive positive intege numbers such…

A: As per the question we have to prove that for any positive integers m, n there exist m consecutive…

Q: Prove that for all positive integers n, 4n-1 is divisible by 3.

A:

Q: Prove that there exists a positive integer n > 2 such that 4n+1 is a prime number.

A:

Q: Prove that the statement is true for every positive integer n.

A: we have to prove that the statement is true for every positive integer n. 4 is a factor of 5n-1

Q: Prove that 6 divides n3-n for all non-negative integers n.

A: See the detailed solution below.

Q: For which positive integers a is the congru

A: Given,               ax4≡2 (mod 13)

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: S Prove by mathematical induction that x 1 is a factor of x" 1 for all integers n > 1.

A:

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: Prove that 4 divides 6 7" - 2.3" for all integers n 20.

A: We need to prove that 4 divides 6.7n-2.3n for all integers n≥0.

Q: 3. Prove by either Well Ordering Principle or induction that for all nonnegative integers,n: + 1) i3…

A: We prove by induction .Case n=1LHS=∑i3i=01=03+13=1RHS=11+122=222=1Induction hypothesis : Assume…

Q: (a) Express each of the integers 4278 and 71929 as a product of primes. (b) What is gcd(4278,…

A:

Q: Use proof by contradiction to show that for all integers m, 7m+4 is not divisible by 7.

A: Recall the following fact about divisibility.

Q: 1) Show that for all positive integers x, x! + 1 has a prime divisor greater than x.

A: “Since you have asked multiple questions in a single request, we will be answering only the 1st…

Q: there are no integers x and y such that x is a prime greater than 5 and x=6y+3

A: Given query is to check the statement whether it's possible or not.

Q: Prove that 3, 5, and 7 are the only three consecutive odd integersthat are prime.

A:

Q: Prove that 2 is a factor of n2 + 5n for every positive integer n.

A: To prove: 2 is a factor of n2+5(n) ∀n.

Q: Prove that, for all integers K and L, there is at least one pair of integers (a, b) for which K2 +…

A:

Q: Vn € Zt [obeb(n, n + 1) = 1]

A: l  let n,n+1 are two consecutive positive integers let  the greatest common divisor i.e…

Q: Prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a non-negative integer

A: We need to prove by mathematical induction that, 3 divides 17k+1-2*5k+3+11k+3 whenever k is a…

Q: Prove that 3" >n² for all positive integers

A:

Q: List the first 3 positive prime numbers d in Z so that the quadratic integers in Q[Vd] are precisely…

A:

Q: Let a relatively prime positive integers. If there exist integers s and t with as + ht t - 11 then

A: if gcd (a,b) =d then there exists integers m and n such that d = a m + b  n  and d is the smallest…

Q: Prove that A,(3, 2) = q² for any integer q>2. %3D

A: To prove : Aq3, 2=q2 for any integer q≥2 Pre-requisite : 1. Sphere-covering bound : For an integer…

Q: For all integers 1. x and y, if x + y ≥ 9, then either x ≥ 5 or y ≥ 5. Write a paragraph proof of…

A:

Q: Find the positive integers n such that , the sum of positive divisor oln) = 24,

A: Sum of Divisors: Let the prime factorization of n be n=p1a1×p2a2×...×pkak Then the sum of divisors…