http://projecteuler.net/problem=99

Largest exponential

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 , a  text file containing one thousand lines with a base/exponent pair on each line, determine which line number has the greatest numerical value.

http://projecteuler.net/problem=97

Large non-Mersenne prime

The first known prime found to exceed one million digits was discovered in 1999, and is a Mersenne prime of the form 2^69725931; it contains exactly 2,098,960 digits. Subsequently other Mersenne primes, of the form 2^p1, have been found which contain more digits.
However, in 2004 there was found a massive non-Mersenne prime which contains 2,357,207 digits: 284332^7830457+1.
Find the last ten digits of this prime number.

http://projecteuler.net/problem=58

Spiral primes

Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed.

37 36 35 34 33 32 31
38 17 16 15 14 13 30
39 18  5  4  3 12 29
40 19  6  1  2 11 28
41 20  7  8  9 10 27
42 21 22 23 24 25 26
43 44 45 46 47 48 49

It is interesting to note that the odd squares lie along the bottom right diagonal, but what is more interesting is that 8 out of the 13 numbers lying along both diagonals are prime; that is, a ratio of 8/13  62%.
If one complete new layer is wrapped around the spiral above, a square spiral with side length 9 will be formed. If this process is continued, what is the side length of the square spiral for which the ratio of primes along both diagonals first falls below 10%?

http://projecteuler.net/problem=63

Powerful digit counts

The 5-digit number, 16807=7⁵, is also a fifth power. Similarly, the 9-digit number, 134217728=8⁹, is a ninth power.
How many n-digit positive integers exist which are also an nth power?

http://projecteuler.net/problem=53

Combinatoric selections

There are exactly ten ways of selecting three from five, 12345:
123, 124, 125, 134, 135, 145, 234, 235, 245, and 345
In combinatorics, we use the notation, ⁵C₃ = 10.

where r n, n! = n(n1)…321, and 0! = 1.

How many, not necessarily distinct, values of  ⁿC_r, for 1  n  100, are greater than one-million?

http://projecteuler.net/problem=56

Powerful digit sum

A googol (10¹⁰⁰) is a massive number: one followed by one-hundred zeros; 100¹⁰⁰ is almost unimaginably large: one followed by two-hundred zeros. Despite their size, the sum of the digits in each number is only 1.

Considering natural numbers of the form, a^b, where a, b  100, what is the maximum digital sum?

http://projecteuler.net/problem=28

Number spiral diagonals

Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows:

21 22 23 24 25
20  7  8  9 10
19  6  1  2 11
18  5  4  3 12
17 16 15 14 13

It can be verified that the sum of the numbers on the diagonals is 101.

What is the sum of the numbers on the diagonals in a 1001 by 1001 spiral formed in the same way?

http://projecteuler.net/problem=40

Champernowne’s constant

An irrational decimal fraction is created by concatenating the positive integers:

0.123456789101112131415161718192021…

If d_n represents the n^th digit of the fractional part, find the value of the following expression.

d₁  d₁₀  d₁₀₀  d₁₀₀₀  d₁₀₀₀₀  d₁₀₀₀₀₀  d_1000000

http://projecteuler.net/problem=48

Self Powers

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

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

Recently, I’ve been typing more Ruby then usual. In Ruby you can have a fairly big number.
http://projecteuler.net/problem=25

1000-digit Fibonacci number

The Fibonacci sequence is defined by the recurrence relation:

F_n = F_n1 + F_n2, where F₁ = 1 and F₂ = 1.

The 12th term, F12, is the first term to contain three digits.

What is the first term in the Fibonacci sequence to contain 1000 digits?