Determine an integer N for which the following statement is true: given any sequence of N distinct integers, there must exist an increasing subsequence of length 7 or a decreasing subsequence of length 9. Prove your claim.

Assuming every integer is either even or odd, How would I show that for any integer, the difference between it and its cube is even

The famous Goldbach Conjecture says that every even integer n > 4 is a sum of two primes. This conjecture is not yet proved. Prove the weaker statement that there are infinitely many even integers that are sums of two primes. (Hint: There is an infinite number of primes.)

Every positive integer is always divisible by 1 and by itself. That is, for every posi- tive integer n we always have 1|n and n|n. We say an integer p ≥ 2 is a prime number if its only divisors are 1 and p. (By convention, 1 is not a prime number.) (a) List the first ten prime numbers. (b) If a positive integer n is not prime, then it is called a composite number. Show that every positive integer n ≥ 2 can be written as a product of prime numbers. (This is called a prime factorization of n. Notice that some primes may need to be repeated. For example, 180 = 2² × 3² × 5 and the primes 2 and 3 are repeated.) Hint: Consider two separate cases based on whether a given number n is prime or composite. For composite numbers, suppose by strong induction that all smaller integers have a prime factorization.