Using the Induction hypothesis,
strong induction that every integer greater than 1 can be written as the product of primes
Induction Hypothesis:
This is the assumption that given a general element a[n], if something is true about the elements of an ordered set, we can show that it is also true for a[n+1]. is. This means it must be true for all elements. A classic example is the proof that the sum of the natural numbers is triangular.
Induction Has Strong and Weak Versions
This is a case where strong induction is needed because factoring n gives no information about the factorization of n+1.
For brevity, we say that even prime numbers are products of prime numbers. So the statement is: "Every number n≥2 is a product of primes". So the steps are:
base case, where we prove n=2.
If n>2 and every number m with 2≤m prove that n is also a product of primes. The
base case is because 2 is prime.
Let n>2. If n is prime, we are done. Otherwise, n=ab and 1 products of prime numbers.
Strong Induction: Base Case: n=2 and n has a factor of 1.2 n is prime:
Assume k is prime or can be expressed as the product of sets of prime factors for all k≤n To do.
We need to show that n+1 is prime or that n+1 can be expressed as the product of a set of prime factors.
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