Number Theory

  • #1
halvizo1031
78
0

Homework Statement



Prove that they are no integers a,b,n>1 such that (a^n - b^n) | (a^n + b^n).

Homework Equations





The Attempt at a Solution


Do I solve this by contradiction? If so, how do I start it?
 

Answers and Replies

  • #2
Mechdude
117
0
I think it is by contradiction (i suppose you could show the gcd of [itex] (a^n - b^n , a^n + b^n ) [/itex] is 1 or 2 ) viz,
let d be the gcd clearly then , d must divide the sum (and the difference) of the two , [tex] d | a^n + b^n + a^n - b^n [/tex]
[tex] d | 2a^n [/tex]
which implies, [itex] d|2, [/itex] [itex] d|a^n [/itex] this last result shows d is either 1 or 2 , thus if the gcd of the two is 2 or 1, no integers a,b, c >1 can exist to satisfy the requirement (there are no numbers a,b, c> 1 that can divide in that manner) though I am only into number theory as a hobby, i might not be quite on the mark, ;-) ,
good luck
 
Last edited:
  • #3
halvizo1031
78
0
I think it is by contradiction (i suppose you could show the gcd of [itex] (a^n - b^n , a^n + b^n ) [/itex] is 1 or 2 ) viz,
let d be the gcd clearly then , d must divide the sum (and the difference) of the two , [tex] d | a^n + b^n + a^n - b^n [/tex]
[tex] d | 2a^n [/tex]
which implies, [itex] d|2, [/itex] [itex] d|a^n [/itex] this last result shows d is either 1 or 2 , thus if the gcd of the two is 2 or 1, no integers a,b, c >1 can exist to satisfy the requirement (there are no numbers a,b, c> 1 that can divide in that manner) though I am only into number theory as a hobby, i might not be quite on the mark, ;-) ,
good luck


I consulted with my study partners and we agree that what you have is correct. But we didn't get exactly what you got so we had to make some corrections. thanks for your help.
 
  • #4
frustr8photon
30
0
(a^n - b^n) | (a^n + b^n)

what does that mean?? does the bar symbol mean "given"
 
  • #5
Dick
Science Advisor
Homework Helper
26,263
621
(a^n - b^n) | (a^n + b^n)

what does that mean?? does the bar symbol mean "given"

The bar symbol means "divides" in the sense of integer divisibility.
 
Last edited:

Suggested for: Number Theory

Replies
7
Views
443
Replies
12
Views
402
Replies
7
Views
418
Replies
2
Views
359
Replies
2
Views
361
Replies
4
Views
265
  • Last Post
Replies
5
Views
341
Replies
3
Views
438
  • Last Post
Replies
2
Views
449
Replies
8
Views
920
Top