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)/2let p x = x(3x-1)/2let h x = x(2*x-1)
filter (\x -> 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 + 2×1^215 = 7 + 2×2^221 = 3 + 2×3^225 = 7 + 2×3^227 = 19 + 2×2^233 = 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 (\z -> x + 2^z) [1..y]
map (\x -> 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 (\x -> last(splitOn «.» x) == «0») [show(sqrt(fromIntegral(25-x)/2)) | x <- [1..25], isPrime x]
let validateOdd z = filter (\x -> last(splitOn «.» x) == «0») [show(sqrt(fromIntegral(z-x)/2)) | x <- [1..z], isPrime x]
filter (\x -> 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 × 715 = 3 × 5
The first three consecutive numbers to have three distinct prime factors are:
644 = 2² × 7 × 23645 = 3 × 5 × 43646 = 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 – chunksOfimport 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]