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kripkrip420
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Hello there. I have been reading G.H. Hardy's book "A Course of Pure Mathematics". It is a fantastic introduction to Analysis. I have no problems with the book so far, however, it does assume some knowledge in number theory. I just want to make sure that the following proofs for properties of numbers are done correctly and that there is no apparent logical flaw in them. Thank you for your critisism (if any).
Proposition 1:
If α divides β, then α divides β[itex]^{η}[/itex], where α and β are integers and η is a positive integer.
Proof:
If α divides β then (by definition) there exists some integer κ such that ακ=β.
Raising both sides of the above equation to η yields the following;
(ακ)[itex]^{η}[/itex]=β[itex]^{η}[/itex].
By use of the laws of exponents, we get
α[itex]^{η}[/itex]κ[itex]^{η}[/itex]=β[itex]^{η}[/itex].
We can then express α[itex]^{η}[/itex] as α*(α[itex]^{η-1}[/itex]).
Thus,
α*(α[itex]^{η-1}[/itex])=β[itex]^{η}[/itex]
and we have shown that α divides β[itex]^{η}[/itex].
Proposition 2:
This proposition was the opposite of the above propostion. It states that if α divides β[itex]^{η}[/itex] then α divides β.
The proof was done as an extremely simple contrapositive I'm sure you all can figure out.
Proposition 3:
Show that if α and β are coprime, then so are α and β[itex]^{η}[/itex], where α and β are integers and η is a positive integer.
Proof:
Assume there exists some common factor of α and β[itex]^{η}[/itex], call it ω.
Let α=ωc and let β[itex]^{η}[/itex]=ωd where c and d are integers.
α=ωc→ω=α[itex]/c[/itex].
Thus,
β[itex]^{η}[/itex]=(α[itex]/c[/itex])*d.
The above can be expressed as
β[itex]^{η}[/itex]=αdc[itex]^{-1}[/itex]
and it is clear that α[itex]/β^{η}[/itex].
But, from the preceding proofs, if α divides β[itex]^{η}[/itex], then α divides β. But this contradicts our supposition, since α and β are coprime. Hence, by contradiction, there exists no factor ω shared by both α and β[itex]^{η}[/itex] if α and β are coprime, i.e. α and β[itex]^{η}[/itex] are coprime.
Also, can somebody tell me how to display the "divides" symbol please? Thank you all for your help!
Proposition 1:
If α divides β, then α divides β[itex]^{η}[/itex], where α and β are integers and η is a positive integer.
Proof:
If α divides β then (by definition) there exists some integer κ such that ακ=β.
Raising both sides of the above equation to η yields the following;
(ακ)[itex]^{η}[/itex]=β[itex]^{η}[/itex].
By use of the laws of exponents, we get
α[itex]^{η}[/itex]κ[itex]^{η}[/itex]=β[itex]^{η}[/itex].
We can then express α[itex]^{η}[/itex] as α*(α[itex]^{η-1}[/itex]).
Thus,
α*(α[itex]^{η-1}[/itex])=β[itex]^{η}[/itex]
and we have shown that α divides β[itex]^{η}[/itex].
Proposition 2:
This proposition was the opposite of the above propostion. It states that if α divides β[itex]^{η}[/itex] then α divides β.
The proof was done as an extremely simple contrapositive I'm sure you all can figure out.
Proposition 3:
Show that if α and β are coprime, then so are α and β[itex]^{η}[/itex], where α and β are integers and η is a positive integer.
Proof:
Assume there exists some common factor of α and β[itex]^{η}[/itex], call it ω.
Let α=ωc and let β[itex]^{η}[/itex]=ωd where c and d are integers.
α=ωc→ω=α[itex]/c[/itex].
Thus,
β[itex]^{η}[/itex]=(α[itex]/c[/itex])*d.
The above can be expressed as
β[itex]^{η}[/itex]=αdc[itex]^{-1}[/itex]
and it is clear that α[itex]/β^{η}[/itex].
But, from the preceding proofs, if α divides β[itex]^{η}[/itex], then α divides β. But this contradicts our supposition, since α and β are coprime. Hence, by contradiction, there exists no factor ω shared by both α and β[itex]^{η}[/itex] if α and β are coprime, i.e. α and β[itex]^{η}[/itex] are coprime.
Also, can somebody tell me how to display the "divides" symbol please? Thank you all for your help!