Math geeks only

[Ghoulish Delight]

- OR -

Yes, I am geeky enough to be kept awake at night by number theory


I need some formal proof help. Never my strong suit. I grasp the general concept, but when it comes to being properly formal and pedantic, I get a bit fuzzy.

What I'm struggling with now is whether a certain kind of deductive conclusion is permissable, or if I need more rigor.

What I'm trying to prove is that for any integer value of n > 2, n*7 < 10^(n-1).

It's certainly a true statement, but I'm trying to formalize it. The best I've come up with is to know that for the value n=3 it is true and that the expression on the left grows linearly while the expression on the right grown exponentially.

Is that sufficient proof? That given any 2 functions f and f` such that f(n) < f`(n) AND f grows linearly while f` grows exponentially, f(x) < f`(x) for all values > n?

[Morrigoon]

Not a math geek, but is 2 the absolute top of the non-qualifying values for n in this? I mean, short of testing it myself, is there any value of n=2.x that might also be disqualified?

[Ghoulish Delight]

I specified integer values only.

But to answer your question, yes. There is a non-integer value (that I'm ill equipped to work out at the moment) of n, between 2 and 3, for which n*7 = 10^(n-1). Below that value, the result is > 10^(n-1), above it, it's > 10^(n-1).

[Morrigoon]

Originally Posted by Ghoulish Delight

I specified integer values only.

Ah, that would be me forgetting that integer meant a whole number

(Proof that I'm not a math geek... now ask me the integer value that represents how many times I had to take algebra II in order to pass it. Hint: n>2)

[Strangler Lewis]

Doesn't it also work for zero?

[Ghoulish Delight]

Originally Posted by Morrigoon

Ah, that would be me forgetting that integer meant a whole number

Just because this IS a number theory thread, I should point out that in most mathematic circles "whole numbers" consist of the positive integers (or the very similar "non negative integers" which also includes 0). So while whole numbers are indeed integers, an integer is not necessarily a whole number.

But in this particular case, we do happen to be dealing with only whole numbers.

[Morrigoon]

That wouldn't nullify the equation though. He's stating for all integer values greater than 2, but not addressing values 2 and under. Right?

[Ghoulish Delight]

Yes it does work for zero (7*0 = 0, 10^-1 = .1). However, I'm not concerned with any of the infinite other values of n may or may not work. All I'm interested in proving is that for intergers > 3, it's true. Anything else is irrelevant to the overall problem.

ETA: Thank you Mori, precisely.

[Morrigoon]

How about proving it with a curve? Does that do anything for you? I mean, demonstrating that one increases linearly while the other increases exponentially... a curve would demonstrate that, yes?

(ETA: I can't believe I have even this many posts in a math thread... did I mention n>2?)

[Strangler Lewis]

Then how much of a number theory problem is it?

Is "For all C > -40: 9/5 * C + 32 > C" a number theory problem?