To prove that every integer $n \geq 1$ can be written as a sum of distinct primes using strong induction and Bertrand's postulate (with the assumption that 1 is a prime number), we'll proceed as follows.

Step 1: Base Case

For $n = 1$:

• $n = 1$ is trivially written as a sum of distinct primes because we are pretending that 1 is a prime number. Thus, $n = 1$ can be written as the sum of the distinct prime $1$ alone.

Thus, the base case holds.

Step 2: Inductive Hypothesis

Assume that for all integers $k$, where $1 \leq k \leq n$, every integer $k$ can be written as a sum of distinct primes. This is the strong induction hypothesis.

Step 3: Inductive Step

We need to prove that $n + 1$ can also be written as a sum of distinct primes.

Case 1: $n+1$ is prime

• If $n+1$ is prime, then $n+1$ itself is a valid sum of distinct primes (with only one term). This satisfies the condition that $n+1$ can be written as a sum of distinct primes.

Case 2: $n+1$ is not prime

• If $n+1$ is not prime, then we can use the inductive hypothesis. Since Bertrand’s postulate guarantees that between any number $n/2$ and $n$, there is a prime $p$, we can use this prime number to break down $n+1$ into two smaller distinct primes.

  • Let $p$ be a prime such that $1/2(n+1) < p \leq n+1$ (by Bertrand’s postulate). Now, subtract this prime $p$ from $n+1$. That is: $n+1 - p = m$ where $m$ is a smaller number than $n+1$.

  • By the inductive hypothesis, $m$ can be written as a sum of distinct primes, say: $m = p_1 + p_2 + \dots + p_k$

  • Hence, $n+1$ can be written as: $n+1 = p + p_1 + p_2 + \dots + p_k$ Since $p$ and the primes $p_1, p_2, \dots, p_k$ are distinct, this is a valid sum of distinct primes.

Conclusion:

By strong induction, we have proven that every integer $n \geq 1$ can be written as a sum of distinct primes. Bertrand’s postulate ensures that we can always find a prime within the appropriate range to apply the inductive step.

Would you like any clarifications or more details on this proof?

Related Questions:

1. How does Bertrand's postulate work for large values of $n$?
2. Can you give more examples of writing integers as sums of distinct primes?
3. What would change in this proof if we do not treat 1 as a prime number?
4. How does strong induction differ from simple induction?
5. Can this method be extended to non-prime numbers?

Tip:

When applying strong induction, always carefully verify the base case and ensure that the inductive hypothesis covers all relevant cases up to $n$.