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

[srfriggen]

Homework Statement

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

Homework Equations

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?

 

[Mark44]

Homework Statement

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

Homework Equations

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.

 

[SteamKing]

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 :/

 

[Mark44]

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?

 

[Office_Shredder]

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.

 

[Office_Shredder]

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.

 

[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?

 

[Office_Shredder]

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.