Can it be natural?

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GilgameshProject

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Can an expression 1713308017730.png be a natural number, if n is natural? This seems so easy, but I've hit a brick wall for some reason. I understand that the nominator can be divided by 4 due to one of the results of Bézout's theorem, and for the same reason denominator is dividable by 3, but I can't seem to advance any further in solving the problem. Pls help me!
 
GilgameshProject said:
Can an expression View attachment 37670 be a natural number, if n is natural? This seems so easy, but I've hit a brick wall for some reason. I understand that the nominator can be divided by 4 due to one of the results of Bézout's theorem, and for the same reason denominator is dividable by 3, but I can't seem to advance any further in solving the problem. Pls help me!
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when n=2.................natural number

we get

(25-1)/(15-1) = 12/7 .............NOT a natural number
 
Yeah, I have to either prove that it can't be natural or give an example of such n so that the expression takes natural value. I don't understand your response
 
GilgameshProject said:
Can an expression View attachment 37670 be a natural number, if n is natural? This seems so easy, but I've hit a brick wall for some reason. I understand that the nominator can be divided by 4 due to one of the results of Bézout's theorem, and for the same reason denominator is dividable by 3, but I can't seem to advance any further in solving the problem. Pls help me!
Click to expand...
What about it seems easy? Proving that something is impossible is not usually easy. I don't see a quick answer, but I don't know a lot of number theory.

Can you tell us the context of the question? What has been taught that might be relevant? (Besides your reasonable observation, which doesn't seem to help, at least yet.)

By the way, the top of a fraction is called the "numerator", not the "nominator". ("Numerator" means "numberer", giving the number of parts; "denominator" means "namer", naming the parts.) I've seen several people make this mistake recently, so it seems worth mentioning. (I've checked, and don't find that it's standard anywhere. just a fairly common error.)
 
Oh, ok, I'm sorry, thanks for correcting, English isn't my first language. Well it doesn't really have any context, it's an exam problem my colleague encountered. I don't know a lot of number theory myself, just looking for clues, what can I try to do with the expression
 
I'm not sure mod 4 helps, but I think mod 5 might. I haven't taken the time to work it out fully, so I'm probably wrong.

What does it take to be able to simplify a fraction and get an integer? What would make it impossible?
 
khansaheb said:
when n=2.................natural number

we get

(25-1)/(15-1) = 12/7 .............NOT a natural number
Click to expand...
The above calculation is incorrect (due to temporary insanity)...... but the conclusion is correct.

(5^2 - 1)/(4^2 - 1) = 24/15 = 8/5................. NOT a natural number.
 
GilgameshProject said:
I have to either prove that it can't be natural or give an example of such n so that the expression takes natural value
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Your OP did not explicitly say that. I was simply showing that the given expression need not produce natural number, for all 'n' (natural).
 
khansaheb said:
The above calculation is incorrect (due to temporary insanity)...... but the conclusion is correct.

(5^2 - 1)/(4^2 - 1) = 24/15 = 8/5................. NOT a natural number.
Click to expand...
But that has nothing to do with the question, which is to show either that there is some n for which the expression is a natural number, or to show that it is never a natural number. Showing that it can be a non-natural number is irrelevant. That's trivial.
GilgameshProject said:
Can an expression View attachment 37670 be a natural number, if n is natural?
Click to expand...
 
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