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perfect totient number

Synonym: 
totient perfect number
Type of Math Object: 
Definition
Major Section: 
Reference

Mathematics Subject Classification

11A25 no label found

Comments

Apologies in advance for the tenor of this question. In all likelihood it will appear to be an extremely gauche question of little importance distracting from topics considered more important, such as the curvature of space-time with an irrational number of dimensions. To me right now, it seems that the answer is obvious but I can't figure it out for some reason.

If an integer is of the form 4n + 1, doesn't that automatically rule out the possibility that it could be equiv 3 mod 4, that is, 4n - 1?

Yes, if n is also an integer, no otherwise. If n is 1/2 for example then 4n+1=3. But when n is an integer then 4n+1 = 1 modulo 4. Just follow the definition: (4n+1)-1=4n is divisible by 4. 1 is not 3 modulo 4. Proof: 3-1=2 is not divisible by 4 so 1 and 3 are not congruent modulo 4.

First, there is no need to apologize. There is nothing wrong
with your question. Had you been posting spam, that would
have been a different story. There is nothing inappropriate
in asking a mathematical question, no matter how elementary
or advanced it might be.

Second, the answer to your question is "yes". Assume we have an
integer x such that x = 4n + 1 for some n and also x = 4m - 1 for
some m. Then, we would have 4n + 1 = 4m - 1. Manipulating
algebra, this is equivlent to 2(n - m) = 1. This is impossible ---
twice an integer cannot equal 1. Hence the possibility is
ruled out.

I think I may have figured out the source of PrimeFan's confusion:

From my understanding of skimming the Ianucci paper Lisa mentions in her entry, I think what Mohan and Suryanarayana proved that there is some specific reason why if p is congruent to 3 mod 4 then 3p can't be a perfect totient number. There is some other reason why congruent to 0 or 2 it can't be a PTN. Of course I need to read the Mohan paper before saying this 100%.

You're exactly right, Anton, that's why I was confused. You're probably right about the 0 and 2 cases, too. Thanks for explaining that.

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