Using strong mathematical induction, for each prove that there exist positive integers satisfying .
Let There exists such that , where .
First we will show that P(1) and P(2) are true.
There exists such that .
satisfies the equation and hence P(1) is true.
There exists such that .
satisfies the equation and hence P(2) is true.
Strong Induction Hypothesis: The proposition is true for all
We will show that is true.
By strong induction hypothesis, is true. This means there exists such that . Multiply the equation by to get . So the integer triple satisfies
Hence is true.
Observe that I have proved and initially.
Question: Is it sufficient to prove P(1) alone?