Lagrange Theorem etc.

**particlejohn** · August 31st 2008, 09:18 PM

Define a sequence $\{a_{i} \}$ by $a_1 = 4$ and $a_{i+1} = 4^{a_{i}}$ for $i \geq 1$ . Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_{i}$ .

It would take a long time to do this by brute force. What exactly does this mean (this was a hint that was given)?: If $4$ does not divide $n$ , then $4^{a} \mod n$ is determined by $a \mod \phi(n)$ . So for example, $4^{a}\mod15$ is determined by $a \mod \phi(15)$ .

**badgerigar** · September 1st 2008, 12:50 AM

I think you are expected to know that if $gcd(a,m) =1$ then $a^{\phi(m)}\equiv 1 \quad (mod m)$ , where phi is the Euler phi function

I suspect you are capable of sussing things out from there.

Thread: Lagrange Theorem etc.

LinkBack

Thread Tools

Display

Lagrange Theorem etc.

Similar Math Help Forum Discussions

using the Lagrange Theorem

Prove Wilson's theorem by Lagrange's theorem

Lagrange's Theorem(?)

Lagrange's Mean Value Theorem

Lagrange Theorem.

Search Tags