# Project Euler 48 Solution: Self powers

Problem 48

The series, 11 + 22 + 33 + ... + 1010 = 10405071317.

Find the last ten digits of the series, 11 + 22 + 33 + ... + 10001000.

## Solution

With the knowledge of how a sum and a product behaves under a modular congruence relation, the problem can be solved quickly:

$\sum\limits_{i=1}^n i^i \equiv \sum\limits_{i=1}^n (i^i \bmod m) \pmod{m}$

Now we have isolated the exponential term $$i^i \bmod m$$, which can be solved under a modular congruence like this:

$i^i \equiv \prod\limits_{j=1}^i i \equiv \prod\limits_{j=1}^i (i\bmod m) \pmod{m}$

That's enough already to solve this problem, but we can optimize the inner loop by using Exponentiation by squaring, by stating the products recursively so that

$b^e \pmod{m} \equiv \begin{cases}{(b\bmod m)} , &\text{if }e=1\\{(b^2\bmod m)}^{\frac {e-1}{2}}, &\text{if }e\text{ is odd}\\{(b^2\bmod m)}^{\frac{e}{2}} , &{\text{if }}e{\text{ is even}}.\end{cases}$

By checking the least significant bit of the exponent, we can decide which case must be used so that the algorithm can be implemented like this in Python:

def modpow(b, e, m):
r = 1
while e > 0:
if e & 1:
r = (r * b) % m
b = (b * b) % m
e = e >> 1
return r


Using the modpow function, the formula stated before can be implemented easily with a proper chosen $$m$$ to get the last 10 digits:

def powersum(n, m):
s = 0
for i in range(1, n + 1):
s+= modpow(i, i, m)
s%= m
return s

print powersum(1000, 10000000000)

The modpow implementation isn't necessary, since Python's pow function has the same capabilities when passing three parameters. However, calling library functions is only half of the fun.

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