- #1
Analyzing a Proof of Phi-Function Multiplicativity
- Context: Graduate
- Thread starter icystrike
- Start date
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- Proof
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Discussion Overview
The discussion revolves around the proof of the multiplicativity of the phi-function, a concept in number theory. Participants are examining the steps and reasoning involved in the proof, seeking clarification and validation of specific transitions in the argument.
Discussion Character
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant is attempting to present a proof for the multiplicativity of the phi-function and requests feedback on their argument.
- Another participant questions the validity of a specific transition in the proof regarding the sums involving gcd conditions.
- A third participant points out that the sets defined by the gcd conditions may not be disjoint, suggesting an alternative approach using Euler's formula for the phi-function.
- There is a statement affirming the validity of the condition gcd(a,mn) = 1 iff gcd(a,m) = gcd(a,n) = 1, but concerns remain about the clarity of how the sums are split in the proof.
- A participant expresses uncertainty about whether the split sum is valid, indicating ongoing debate about this aspect of the proof.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the validity of the proof's steps, particularly regarding the splitting of sums and the disjoint nature of the sets involved. Multiple competing views remain on how to approach the proof.
Contextual Notes
Limitations include potential misunderstandings about the disjoint nature of sets defined by gcd conditions and the clarity of the proof's transitions. The discussion does not resolve these issues.
- #2
dodo
- 695
- 2
Hi, one question,
how did you step from "the sum where gcd...m and gcd...n" to "(the sum where gcd...m) times (the sum where gcd...n)" ?
how did you step from "the sum where gcd...m and gcd...n" to "(the sum where gcd...m) times (the sum where gcd...n)" ?
- #3
JSuarez
- 402
- 1
The passage from the first line to the second is not correct; the set of a's such that gcd(a,m) = 1 may not be disjoint from the gcd(a,n) = 1.
One suggestion: instead of working with the sums, try using Euler's formula for \varphi:
\varphi\left(n\right) = n\left(\ 1 - \frac{1}{p_{1}}\right)\left(\ 1 - \frac{1}{p_{2}}\right)...\left(\ 1 - \frac{1}{p_{k}}\right)
Where p_{1},...,p_{n} are the prime factors of n.
One suggestion: instead of working with the sums, try using Euler's formula for \varphi:
\varphi\left(n\right) = n\left(\ 1 - \frac{1}{p_{1}}\right)\left(\ 1 - \frac{1}{p_{2}}\right)...\left(\ 1 - \frac{1}{p_{k}}\right)
Where p_{1},...,p_{n} are the prime factors of n.
- #4
- #5
icystrike
- 444
- 1
help please (=
- #6
JSuarez
- 402
- 1
Nobody questioned the validity of gcd(a,mn) = 1 iff gcd(a,m) = gcd(a,n) = 1; this is true. It's simply the way you split the sums that's a bit unclear, and could be improved.
- #7
icystrike
- 444
- 1
thanks ! but that split sum is valid?
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