Special Pythagorean triplet: Project Euler Problem 9

Problem Statement:
A Pythagorean triplet is a set of three natural numbers, $a, for which,
$a^2+b^2=c^2$.

For example, $3^2+4^2=9+16=25=5^2$.

There exists exactly one Pythagorean triplet for which a + b + c = 1000.
Find the product abc.

Summation of primes: Project Euler Problem 10

Problem Statement:
The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.
Find the sum of all the primes below two million.

Largest product in a series: Project Euler Problem 8

Problem Statement:
Find the greatest product of five consecutive digits in the 1000-digit number.

73167176531330624919225119674426574742355349194934
96983520312774506326239578318016984801869478851843
85861560789112949495459501737958331952853208805511
12540698747158523863050715693290963295227443043557
66896648950445244523161731856403098711121722383113
62229893423380308135336276614282806444486645238749
30358907296290491560440772390713810515859307960866
70172427121883998797908792274921901699720888093776
65727333001053367881220235421809751254540594752243
52584907711670556013604839586446706324415722155397
53697817977846174064955149290862569321978468622482
83972241375657056057490261407972968652414535100474
82166370484403199890008895243450658541227588666881
16427171479924442928230863465674813919123162824586
17866458359124566529476545682848912883142607690042
24219022671055626321111109370544217506941658960408
07198403850962455444362981230987879927244284909188
84580156166097919133875499200524063689912560717606
05886116467109405077541002256983155200055935729725
71636269561882670428252483600823257530420752963450

Sum square difference: Project Euler Problem 6

Problem Statement:

The sum of the squares of the first ten natural numbers is,
$1^2 + 2^2 + ... + 10^2 = 385$

The square of the sum of the first ten natural numbers is,
$(1 + 2 + ... + 10)^2 = 55^2 = 3025$

Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is $3025-385=2640$.

Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.

Factorial digit sum: Project Euler Problem 20

Problem Statement:
$n!$ means $n\times(n-1)\times...\times2\times1$

For example, $10! = 10 \times 9 \times ... \times 3 \times 2 \times 1 = 3628800$,
and the sum of the digits in the number 10! is $3 + 6 + 2 + 8 + 8 + 0 + 0 = 27$.

Find the sum of the digits in the number 100!

10001st prime: Project Euler Problem 7

Problem Statement:
By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.

What is the 10 001st prime number?