What are self-powers?

Introduction

The series where every number is raised to itself is called self-powers. For example:

1¹+2²+3³+⋯+10¹⁰=10405071317

Here, we’ll use the in-built pow() method in Python.

The pow() method

The pow() method in python returns x^y if given two arguments. It takes an optional third argument that signifies the modulo operation. If the third parameter is present, it returns x^y mod z.

Syntax

pow(x, y, z)

Parameters

• x: It is the base number.
• y: It is the exponent number.
• z: It is the modulus number.

Return value

The method returns the x^y.

Problem statement

Find the last ten digits of the series: 1¹+2²+3³+⋯+10¹⁰

Solution

The most important thing to note about this issue is how quickly the terms’ sizes increase too quickly for today’s computers to be able to store them in memory. This can be avoided by making use of the fact that we only have to determine the sum’s 10 least significant digits.

We solve this problem using modular exponentiation, computing x^y mod m. Modular exponentiation is achieved with the square and multiply algorithm.

Hence, the following expression:

1¹+2²+3³+⋯+10¹⁰

can be converted to

1¹ mod 10¹⁰ + 2² mod 10¹⁰ + 3³ mod 10¹⁰ +⋯+10¹⁰ mod 10¹⁰

Now, the individual terms are restricted to 10 digits and can fit into a 64-bit word.

Code example

Let’s look at the code below:

Code explanation

• Line 1: We define the solve() method.

• Line 2: We define the target value.

• Line 3: We define the modulus value.

• Line 4: We calculate the individual terms of the series.

• Lines 5: We have the sum of the terms mod modulus.

• Line 9: We define the expected answer.

• Line 10: We compare the return value of the solve() method and the expected answer.