Hi,

I hope someone can help. I'm trying to determine whether the following statements can be proven merely by an example? It doesn't matter whether the statements are actually true, let's just assume for this exercise that they are true. As follows:

Statement 1: "Powers of primes increase without bond"

It seems that this would not be proven solely with just an example since the idea of "without bond" suggests an infinite series and not just a single value. Sounds vague what I'm saying, but hopefully I'm on the right track with my thinking. Please let me know!

Statement 2: "There are prime numbers of the form 3^k + 1"

I would think that you could prove this by example (assuming that the statement is true, even though I secretly know it is false) by plugging in whatever value forkresults in the expression being a prime number. Is this right for me to suggest?

Statement 3: “Not all integers q which cause a^(q−1) to have remainder 1 after dividing by q, are prime”

I would think that you could prove by example for the same reasons that I provided for Statement 2.

Statement 4: “There are infinitely many prime numbers”

It appears that this could not be proven by example since the idea of infinity cannot be proven in just an example, and therefore neither can you prove infinity in the context of prime numbers. Right?

Really really would appreciate critique and feedback on my thinking! I look forward to a response.

- Olivia