Comparing two numbers written in index form like 2¹¹ and 3⁷ is not difficult, as any calculator would confirm that 2¹¹ = 2048 < 3⁷ = 2187.

However, confirming that 632382⁵¹⁸⁰⁶¹ > 519432⁵²⁵⁸⁰⁶ would be much more difficult, as both numbers contain over three million digits.

Using base_exp.txt (right click and ‘Save Link/Target As…’), a 22K text file containing one thousand lines with a base/exponent pair on each line, determine which line number has the greatest numerical value.

NOTE: The first two lines in the file represent the numbers in the example given above.

Analysis

Very simple to compare the product of the exponent by the log of the base and keep the maximum value and respective line number.

Solution

Runs < 1 second in Python.

from math import log
 
mv, ml, ln = 0, 0, 0 
for line in file('base_exp.txt'): 
  ln += 1
  b, e = line.split(',') 
  v = int(e) * log(int(b)) 
  if v > mv: 
    mv, ml = v, ln 
print "Answer to PE99 = ", ml

Comments

The first known prime found to exceed one million digits was discovered in 1999, and is a Mersenne prime of the form 2^6972593-1; it contains exactly 2,098,960 digits. Subsequently other Mersenne primes, of the form 2^p-1, have been found which contain more digits.

However, in 2004 there was found a massive non-Mersenne prime which contains 2,357,207 digits: 28433×2^7830457+1.
Find the last ten digits of this prime number.

Analysis

We recognized we could reuse the understanding from Problem 48.

Solution

Runs < 1 seconds in Python.

def PE97(a, b, c, m):
    return (c * pow(a, b, 10**m) + 1) % 10**m
print "Answer to PE97 = ",PE97(2, 7830457, 28433, 10)

Comments

• See also problem 48 & problem 188.
• Perl allows underscores to be used in numeric literals for readability.
• Don’t forget that leading zeros are not printed.

You can reduce the number of iterations significantly by breaking out the multiplication as follows:

my $max_n = 7_830_457;
my $m = 1e10;
my $nx = 1;
 
$nx = ($nx * 2**100) % $m for (1..$max_n/100);
$nx = ($nx * 2) % $m for (1..$max_n%100);
 
print "Answer to PE97 = ",($nx * 28_433 + 1) % $m,"n";

This version runs in a fraction of a second in Perl.