7. Prove that there are two different primes p and q whose last six decimal digits are the same. Is your proof con- structive? If so, produce the primes.

Q: prove the formula of Mersenne primes

A: A number of the form is a Mersenne prime if and only if p is prime.To prove this we need to show…

Q: 1. Prove or disprove: (a) The sum of two even integers is an even integer. (b) The sum of three odd…

A: To prove the given statement or disprove it :-

Q: If n is a positive integer, explain why each prime in the factorization 'n² must occur an even…

A: (a) Let p be a prime number that occurs in the prime factorization of n2.   This implies that p…

Q: Prove or disprove: There are inline number of primes

A: Known fact:A prime is a natural number whose factors are 1 and itself.

Q: 4c Let m and n be integers. If m +n is even, then either ma even or m and n are both odd. 10 = 2F+1…

A: Consider the following statement: For all integers m and n, if m+n is even then m and n are both…

Q: 2. Make a conjecture about the sum of primes based on the following pattern: 4= 2+2 6=3+3 8=3+5…

A: Given:4=2+26=3+38=3+510=3+712=5+7and so on...

Q: 2. Prove that n + 100 > 3" if n is a positive integer with 1 < n < 3. (Hint: Proof by cases)

A: Concept:

Q: How Many Positive Integers ≤630 are relatively prime to 630?

A: The given number is .We have to find the number of positive integers are relatively prime to .

Q: Characterize the primes p mod 20 such ial

A: We have to characterize p mod 20 such that -5p=1.

Q: What is the converse of the statement "If n is an even integer, then 5n - 3 is an odd integer."?

A: A statement which is  obtained by reversing the given hypothesis and conclusion of a conditional…

Q: c.) There is a conjecture called "Goldbach's weak conjecture" (yes, really!) that every odd number…

A: Given: Odd numbers greater than 5. To prove: Every odd number greater than 5 can be written sum of…

Q: 3. The GCF of any two distinct prime numbers is 1. (Distinct they are not the same number) means

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: Is the following proposition true or false? Justify your conclusion with a proof or a…

A: Check the following proposition, "For each natural number n, if 3 does not divide (n2+2) then n is…

Q: B. Prove that if n > 10, n, n + 6, n+ 12, n + 18, n+24 cannot all be prime prime numbers

A: The possibility of 5 prime numbers in a interval of 25 consecutive numbers is possible only in the…

Q: It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a…

A:

Q: 11. Use you the test for divisibility and test for primality to find a pair of twin primes between…

A: We want to find the twin prime between 100 to 106 We know that twin prime are the number differ by…

Q: Is 2 • 3 • 5•7•11•13•17•19 + 1 prime or composite

A:

Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

A: