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

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In summary: 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 sounds reasonable, except I'm not sure why you use the word OR when describing what N is divisible by. Why not just say N is divisible by both (x+3) and (x-3)?"It cannot be prime since a prime is only divisible by itself and the number 1."That makes sense.
  • #1
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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?
 
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  • #2
srfriggen said:

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.
 
  • #3
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
Mark44 said:
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 :/
 
  • #5
srfriggen said:
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?
 
  • #6
srfriggen said:
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!
 
  • #7
Mark44 said:
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.
 
  • #8
srfriggen said:
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.
 
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  • #9
Office_Shredder said:
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?
 
  • #10
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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1. What is the value of 10^1,000 - 9?

The value of 10^1,000 - 9 is an incredibly large number, approximately equal to 1 followed by 1,000 zeros. This value is commonly known as a googol, and it is far too large to be accurately written out or comprehended.

2. Is 10^1,000 - 9 a prime number?

No, 10^1,000 - 9 is not a prime number. A prime number is a number that is only divisible by itself and 1. However, this number is divisible by both 10 and 9, making it a composite number.

3. Can 10^1,000 - 9 be factored into smaller numbers?

Yes, 10^1,000 - 9 can be factored into smaller numbers. One possible way to factor it is 10^1,000 - 9 = (10^500 - 3)(10^500 + 3). However, there are infinitely many ways to factor this number, as it is a composite number with an infinite number of factors.

4. Is there a specific method or formula to determine if 10^1,000 - 9 is prime?

There is no specific method or formula to determine if 10^1,000 - 9 is prime. However, there are many known methods for determining the primality of numbers, such as the Sieve of Eratosthenes or the Miller-Rabin primality test. These methods can be applied to this number, but due to its extremely large size, it would be very computationally intensive.

5. Why is there a debate about whether 10^1,000 - 9 is prime or not?

The debate about whether 10^1,000 - 9 is prime or not stems from a disagreement about the definition of a prime number. Some mathematicians argue that this number should be considered prime, as it is only divisible by two small numbers (10 and 9), while others argue that it should be considered composite due to its ability to be factored into smaller numbers. Ultimately, this debate is a matter of perspective and personal interpretation of the definition of a prime number.

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