The discussion revolves around the question of whether the expression 10^1,000 - 9 is a prime number. Participants are exploring the properties of large numbers and prime determination within the context of number theory.
The conversation is ongoing, with participants sharing insights about the properties of prime numbers and the structure of the expression. There is a mix of attempts to reason through the problem and clarifications about mathematical concepts, but no consensus has been reached.
Participants are grappling with the implications of the difference of two squares and how it relates to primality, while also noting specific examples that illustrate their points. The discussion reflects a range of interpretations and approaches to the problem.
No.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?
Mark44 said:No.
Notice that 101000 is a perfect square, and so is 9.
srfriggen said:But the difference between two perfect squares isn't always prime. For example, 25-16=9.
Not following the logic yet :/
srfriggen said:But the difference between two perfect squares isn't always prime. For example, 25-16=9.
Not following the logic yet :/
Mark44 said:What can you do with the difference of two squares?
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.
Office_Shredder said:You should just start writing down what you know about prime numbers, you should write down the relevant point fairly quickly.