algorithm: Project Euler Problem 44 (#1188)

* [CREATE] Problem 044 from Project Euler

* [UPDATE] Code styling update

* [UPDATE] Change return condition, added an input for main function,
added tests for problem 44

* [UPDATE] minor styling fixes to standard javascript

* [UPDATE] Fix parentheses in main function return

Project-Euler/Problem044.js

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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 }

Project-Euler/test/Problem044.test.js

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+import { problem44 } from '../Problem044.js'
+
+describe('checking nth prime number', () => {
+  it('should be invalid input if number is negative', () => {
+    expect(() => problem44(-3)).toThrowError('Invalid Input')
+  })
+  it('should be invalid input if number is 0', () => {
+    expect(() => problem44(0)).toThrowError('Invalid Input')
+  })
+  // Project Euler Condition Check
+  test('if the number is greater or equal to 1', () => {
+    expect(problem44(1)).toBe(5482660)
+  })
+  // Project Euler Second Value for Condition Check
+  test('if the number is greater or equal to 2167', () => {
+    expect(problem44(2167)).toBe(8476206790)
+  })
+})