# New types of congruences?

Well these congruences are pretty basic and trivial but I didn't find anything about them in books or on net.

It is interesting to note the following congruences.

$1) \phi(a)^n \equiv \phi(a) \mod a$ Here , $$a$$ is arbitary prime number and $$n$$ is any odd number.

$2) \phi(a)^{n'} \equiv 1\mod a$ Here $$a$$ is any arbitary prime and $$n'$$ is an arbitary even number.

2 years, 2 months ago

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Comment deleted Mar 06, 2016

Yes, they are true... it amounts to $$(-1)^n=-1$$ for odd $$n$$ and $$(-1)^n=1$$ for even $$n$$.

- 2 years, 2 months ago

Not actually , as $$\phi (a)$$ can't be less than one right ? @Otto Bretscher

- 2 years, 2 months ago

$$\phi(a)\equiv -1\pmod{a}$$

- 2 years, 2 months ago

Ah , how could I miss that !! anyway can you construct any problem which could be solved by these congruences ? I had one problem in my mind , simplifying the towers of integers becomes a lot easy by these congruences...@Otto Bretscher

- 2 years, 2 months ago

You know, I'm a guy with a short attention span. I soon get tired of a certain "type" of problem (like those congruences) and move on to something else. I'm sure you can come up with great examples.

- 2 years, 2 months ago

Comment deleted Mar 06, 2016

who knows ;)

- 2 years, 2 months ago

Experienced Mathematician like you ;) @Otto Bretscher

- 2 years, 2 months ago