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

1. Oct 10, 2013

### srfriggen

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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?

2. Oct 10, 2013

### Staff: Mentor

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

3. Oct 10, 2013

### SteamKing

Staff Emeritus
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 = ?

4. Oct 10, 2013

### srfriggen

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

Not following the logic yet :/

5. Oct 10, 2013

### Staff: Mentor

What can you do with the difference of two squares?

6. Oct 10, 2013

### Office_Shredder

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

7. Oct 10, 2013

### srfriggen

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.

8. Oct 10, 2013

### Office_Shredder

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

9. Oct 10, 2013

### srfriggen

"...(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?

10. Oct 10, 2013

### Office_Shredder

Staff Emeritus
That sounds reasonable, except I'm not sure why you use the word OR when describing what N is divisible why.... and would be more appropriate.

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