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halvizo1031
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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?
Mechdude said: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
frustr8photon said:(a^n - b^n) | (a^n + b^n)
what does that mean?? does the bar symbol mean "given"
In mathematics, proving that no integers exist means showing that there are no whole numbers that satisfy a certain condition or equation. In other words, there are no solutions to the equation that involve only whole numbers.
Proving that no integers exist for a specific equation usually involves using techniques from number theory and algebra. One common approach is to use proof by contradiction, assuming that there are integers that satisfy the equation and then showing that this leads to a contradiction.
Proving that no integers exist for a certain equation or condition can have important implications in mathematics. It can help to establish the boundaries of what is possible and impossible in terms of number relationships, and can also lead to important insights and discoveries in related areas of research.
No, it is not possible to prove that no integers exist for all equations. This is because there are an infinite number of equations and it is impossible to test every single one. However, it is possible to prove that no integers exist for a specific equation or a certain class of equations.
While computational methods can be helpful in exploring and testing equations, they are not always sufficient to prove that no integers exist. Proving this requires rigorous mathematical reasoning and proof techniques, which may involve more advanced concepts and theories.