Are you curious to know the 2017th digit

Computer Science Level 4

$\Large 2017^{2016^{2015^{2014^{\cdot^{\cdot^{3^{2^{1}}}}}}}}$

What are the last 2017 digits of the decimal representation of the number above?

Submit your answer as the first 4 leftmost digits of this 2017-digit integer.

The answer is 9729.

1 solution

Jesse Nieminen
Feb 8, 2017

From the output of the following Java code, take the first $4$ digits and get $\boxed{9729}$ which is the answer.

(Could be optimized further by using Carmichael Lambda Function.)

import java.math.BigInteger;

public class Main3 {

    private static final BigInteger TWO = new BigInteger("2"), FIVE = new BigInteger("5");
    private static BigInteger[] mods = new BigInteger[2016];

    private static BigInteger eulerPhi(BigInteger big) {
        if(big.mod(TWO).signum() == 0) {
            big = big.divide(TWO);
        }
        if(big.mod(FIVE).signum() == 0) {
            big = big.divide(FIVE).multiply(TWO).multiply(TWO);
        }
        return big;
    }

    private static void computeMods() {
        mods[2015] = BigInteger.TEN.pow(2017);
        for (int i = mods.length-2; i >= 0; i--) {
            mods[i] = eulerPhi(mods[i+1]);
        }
    }


    public static void main(String[] args) {
        computeMods();

        BigInteger start = BigInteger.ONE;

        for (int i = 0; i < 2016; i++) {
            start = new BigInteger(""+(i+2)).modPow(start, mods[i]); 
        }

        System.out.println(start);

    }

}

Great When two different methods give the same result it is more interesting.

Mr X - 4 years, 4 months ago

OMG What a beautiful solution.

Rishabh Deep Singh - 4 years ago

Are you curious to know the 2017th digit

The answer is 9729.

1 solution

0 pending reports