Problem 45

Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:

Triangle Tn=n(n+1)/2 1, 3, 6, 10, 15, … Pentagonal Pn=n(3n−1)/2 1, 5, 12, 22, 35, … Hexagonal Hn=n(2n−1) 1, 6, 15, 28, 45, …

It can be verified that T285 = P165 = H143 = 40755.

Find the next triangle number that is also pentagonal and hexagonal.

Solution

import Data.List

let t x = x(x+1)/2 let p x = x(3x-1)/2 let h x = x(2*x-1)

filter (-> p(x) elemIndex [t x | x <- [1..500]] /= Nothing) [1..500]

Problem 46

It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.

9 = 7 + 2×1^2 15 = 7 + 2×2^2 21 = 3 + 2×3^2 25 = 7 + 2×3^2 27 = 19 + 2×2^2 33 = 31 + 2×1^2

It turns out that the conjecture was false.

What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?

Solution

import Data.List.Split – splitOn

let square x y = map (-> x + 2^z) [1..y]

map (-> square x) $ filter isPrime [1..100]

map (div 2) $ map (21-) $ filter isPrime [1..21]

[sqrt(fromIntegral(25-x)/2) | x <- [1..25], isPrime x]

filter (-> last(splitOn “.” x) == “0”) [show(sqrt(fromIntegral(25-x)/2)) | x <- [1..25], isPrime x]

let validateOdd z = filter (-> last(splitOn “.” x) == “0”) [show(sqrt(fromIntegral(z-x)/2)) | x <- [1..z], isPrime x]

filter (-> length(validateOdd(x)) == 0) [x | x <- [1..10000], (not . isPrime) x, odd x]

Problem 47

The first two consecutive numbers to have two distinct prime factors are:

14 = 2 × 7 15 = 3 × 5

The first three consecutive numbers to have three distinct prime factors are:

644 = 2² × 7 × 23 645 = 3 × 5 × 43 646 = 2 × 17 × 19.

Find the first four consecutive integers to have four distinct prime factors. What is the first of these numbers?

Solution

import Data.List.Split – chunksOf import Data.List.Grouping – splitEvery

let divisors x = 1:[ y | y <- [2..(x div 2)], x mod y == 0] ++ [x]

let isPrime x = divisors x == [1,x]

filter ( x -> (sum(x) - head(x) * 4) == 4) $ chunksOf 4 $ take 10000 [n | n <- [1..], (length $ filter isPrime $ divisors n) == 5]

filter ( x -> (sum(x) - head(x) * 4) == 4) $ chunksOf 4 [n | n <- [10000..100000], (length $ filter isPrime $ divisors n) == 5]