(another)interesting number theory problem

[Mathguy15]

a and b are real numbers such that the sequence{c}n=1--->{infinity} defined by c_n=a^n-b^n contains only integers. Prove that a and b are integers.


Mathguy

 

[checkitagain]

a and b are real numbers such that the sequence{c}n=1--->{infinity}
defined by c_n=a^n-b^n contains only integers. Prove that a and b are integers.


Mathguy

$c_n \ = \ a^n - b^n$



What about any real numbers a and b, such that a = b, so that $c_n = 0 ?$
Here, and b don't have to be integers.


Do I have your problem understood, and/or

are there more restrictions on a and b?

 

[Norwegian]

I assume you mean a≠b.

Since a-b and a²-b²=(a-b)(a+b) are both integers, a+b is rational, and we get a and b are rational.

We can write b=m/t and a=(m+kt)/t with (m,t)=1. Assume t≠1, then there is an integer s such that k is divisible by t^s but not by t^s+1.

Let p be a prime larger than t and 2s+2.

c_p=a^p-b^p=(pktm^p-1+k²t²(...))/t^p

Both the second term and the denominator are divisible by t^2s+2, while the first term is not, so the fraction is not an integer. It follows that t=1 and we are done.

 

[dodo]

a+b is rational, and we get a and b are rational.

Sorry, Norwegian, but why? For example, sqrt(2) and 3-sqrt(2) are both irrational, and they add up to 3.

 

[micromass]

Sorry, Norwegian, but why? For example, sqrt(2) and 3-sqrt(2) are both irrational, and they add up to 3.

Both a+b and a-b are rational. So (a+b)+(a-b)=2a is rational.

 

[dodo]

Ahhh, thanks, Micromass.

 

[dodo]

By the way, this is a beautiful proof, and I'm still trying to figure out how did you come to it, Norwegian.

I presume you started from both ends. At the finishing end, you needed a^n-b^n to be a rational but not an integer. At the starting end, the way you expressed a=b+k suggests the use of the binomial theorem to evaluate powers of b+k (or powers of the numerator of it). If a and b are rational, then a^n and b^n (with a=b+k) were going to end up having a common denominator, so you concentrated in making the numerator of a^n-b^n a non-integer. Then divisibility / factorization issues enter; though I still don't see in which order did (1) finding the largest power of t dividing k, (2) coprimality conditions, and (3) finding a prime p that does not divide most of the things around, in which order these three came to be, and what suggested them.

I always find instructive to see the genesis of proofs; it adds to the inventory of ways of constructing new ones.

 

[Mathguy15]

$c_n \ = \ a^n - b^n$



What about any real numbers a and b, such that a = b, so that $c_n = 0 ?$
Here, and b don't have to be integers.


Do I have your problem understood, and/or

are there more restrictions on a and b?

Oh sorry! Yes, a and b had to be distinct.

 

[Mathguy15]

I assume you mean a≠b.

Since a-b and a²-b²=(a-b)(a+b) are both integers, a+b is rational, and we get a and b are rational.

We can write b=m/t and a=(m+kt)/t with (m,t)=1. Assume t≠1, then there is an integer s such that k is divisible by t^s but not by t^s+1.

Let p be a prime larger than t and 2s+2.

c_p=a^p-b^p=(pktm^p-1+k²t²(...))/t^p

Both the second term and the denominator are divisible by t^2s+2, while the first term is not, so the fraction is not an integer. It follows that t=1 and we are done.

That is an interesting proof, and I will take the time to digest it later! Thanks