# Prove or disprove: Is 10^1,000 - 9 Prime?

srfriggen

## Homework Statement

Prove or disprove: Is 10^1,000 - 9 Prime?

## The Attempt at a Solution

10^1,000 = 999...91.

Is there a way to logically argue to drop the first nine hundred ninety eight 9's and just look at 91 as being a prime?

Mentor

## Homework Statement

Prove or disprove: Is 10^1,000 - 9 Prime?

## The Attempt at a Solution

10^1,000 = 999...91.

Is there a way to logically argue to drop the first nine hundred ninety eight 9's and just look at 91 as being a prime?
No.
Notice that 101000 is a perfect square, and so is 9.

Staff Emeritus
Homework Helper
You would think that might work.
However:
91 is prime
991 is prime
9991 = 103*97, but both of these are prime factors
99991 is prime
999991 = ?

srfriggen
No.
Notice that 101000 is a perfect square, and so is 9.

But the difference between two perfect squares isn't always prime. For example, 25-16=9.

Not following the logic yet :/

Mentor
But the difference between two perfect squares isn't always prime. For example, 25-16=9.

Not following the logic yet :/

What can you do with the difference of two squares?

Staff Emeritus
Gold Member
2021 Award
But the difference between two perfect squares isn't always prime. For example, 25-16=9.

Not following the logic yet :/

That's why he answered "No" to the question of whether it's prime!

srfriggen
What can you do with the difference of two squares?

LOL he gave away the answer then!

I've gotten to this point now: "Call 10^1,000 x^2 and 9=3^2. x2-32=(x+3)(x-3)."

Thinking a proof by contradiction technique may work but mulling it over I can't see how (x+3)(x-3)=p, where p is prime, would lead to a contradiction. If I'm on the right path let me know and I'll try to work it out some more.

Staff Emeritus
Gold Member
2021 Award
LOL he gave away the answer then!

I've gotten to this point now: "Call 10^1,000 x^2 and 9=3^2. x2-32=(x+3)(x-3)."

Thinking a proof by contradiction technique may work but mulling it over I can't see how (x+3)(x-3)=p, where p is prime, would lead to a contradiction. If I'm on the right path let me know and I'll try to work it out some more.

You should just start writing down what you know about prime numbers, you should write down the relevant point fairly quickly.

• 1 person
srfriggen
You should just start writing down what you know about prime numbers, you should write down the relevant point fairly quickly.

"...(x+3)(x-3)=N, and N is divisible by (x+3) OR (x-3). It cannot be prime since a prime is only divisible by itself and the number 1."

that work?

Staff Emeritus
• 