Let f(n) be the largest positive integer x less than 10^{9} such that the last 9 digits of n^{x} form the number x (including leading zeros), or zero if no such integer exists.

For example:

f(4) = 411728896 (4^{411728896} = …490411728896)

f(10) = 0

f(157) = 743757 (157^{743757} = …567000743757)

Σf(n), 2 ≤ n ≤ 10^{3} = 442530011399

Find Σf(n), 2 ≤ n ≤ 10^{6}.

Analysis

Well, just follow the instructions and employ Pypy to get it to run under 1 minute. This is not much different from other problems that deal with large numbers where the pow() function and the third modulus parameter are used.

I should work on improving the algorithm so I wouldn’t need Pypy to bail me out of a time constraint. It takes about 67 seconds in Python.

Project Euler 455 Solution

Runs < 32 seconds in Pypy.

L, s = 1000000, 0
for n in xrange(2, L):
x0, x1 = 0, n
while x0 != x1 and x1:
x0, x1 = x1, pow(n, x0, 10**9)
s += x1
print "Project Euler 455 Solution =", s

Slowly swipe from either end beginning with the white vertical bar to get an idea of the starting or ending digits. For less drama, just double click the answer area. The distance between the two bars will give you an idea of the magnitude. Touch devices can tap and hold the center of the box between the two bars and choose define to reveal the answer.