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calculating $a^b \!\mod c$

I have a number of form: $p^n + p$, where $p$ is a prime number and $n$ can be any large number, for example, say $10^{12}$.

What is the generic algorithm to compute $(p^n + p) \pmod k$, where $k$ is a huge number say $k=1000000007$.

Thanks!

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marked as duplicate by hardmath, Ross Millikan, Noah Snyder, Chris Eagle, no identity Oct 8 '12 at 9:29

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Since adding $p$ modulo $k$ is about as easy as any operation can be, perhaps the generic question is about exponentiating $p^n$ modulo $k$, a topic dealt with in previous questions. –  hardmath Oct 2 '12 at 13:32
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Possible duplicate: calculating $a^b$ mod $c$. Also two more times since then, Modular exponentiation? and how to calculate $f^x$ using fast binary exponentiation? –  hardmath Oct 2 '12 at 15:11

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up vote 1 down vote accepted

As you already know (a+b)mod n = ((a mod n) + (b mod n)) mod n . So I guess addition here is not a problem.

The real question seems to be on $p^n$ mod k where n is large. For that, have a look at Modular Exponentiation on wikipedia.

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thank you , i will check it out. –  JCH Oct 2 '12 at 16:56

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