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/**

* Problem 44 - Pentagon numbers

*

* @see {@link https://projecteuler.net/problem=44}

*

* Pentagonal numbers are generated by the formula, Pn=n(3n−1)/2. The first ten pentagonal numbers are:

* 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...

* It can be seen that P4 + P7 = 22 + 70 = 92 = P8. However, their difference, 70 − 22 = 48, is not pentagonal.

* Find the pair of pentagonal numbers, Pj and Pk, for which their sum and difference are pentagonal and D = |Pk − Pj| is minimised; what is the value of D?

*

* @author ddaniel27

*/

function problem44(k) {

if (k < 1) {

throw new Error('Invalid Input')

}

while (true) {

k++

const n = (k * (3 * k - 1)) / 2 // calculate Pk

for (let j = k - 1; j > 0; j--) {

const m = (j * (3 * j - 1)) / 2 // calculate all Pj < Pk

if (isPentagonal(n - m) && isPentagonal(n + m)) {

// Check sum and difference

return n - m // return D

}

}

}

}

/**

* Function to check if a number is pentagonal or not

* This function solves n

* applying the solution for a quadratic function

* @see {@link https://en.wikipedia.org/wiki/Quadratic_function}

*/

function isPentagonal(n) {

const pent = (Math.sqrt(24 * n + 1) + 1) / 6

return pent === Math.floor(pent)

}

export { problem44 }