100 lines
2.0 KiB
Markdown
100 lines
2.0 KiB
Markdown
---
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id: 5900f39a1000cf542c50fead
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title: 'Problem 46: Goldbach''s other conjecture'
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challengeType: 5
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forumTopicId: 302134
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dashedName: problem-46-goldbachs-other-conjecture
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---
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# --description--
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It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.
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<div style='margin-left: 2em;'>
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9 = 7 + 2×1<sup>2</sup><br>
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15 = 7 + 2×2<sup>2</sup><br>
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21 = 3 + 2×3<sup>2</sup><br>
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25 = 7 + 2×3<sup>2</sup><br>
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27 = 19 + 2×2<sup>2</sup><br>
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33 = 31 + 2×1<sup>2</sup>
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</div>
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It turns out that the conjecture was false.
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What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?
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# --hints--
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`goldbachsOtherConjecture()` should return a number.
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```js
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assert(typeof goldbachsOtherConjecture() === 'number');
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```
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`goldbachsOtherConjecture()` should return 5777.
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```js
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assert.strictEqual(goldbachsOtherConjecture(), 5777);
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```
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# --seed--
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## --seed-contents--
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```js
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function goldbachsOtherConjecture() {
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return true;
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}
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goldbachsOtherConjecture();
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```
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# --solutions--
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```js
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function goldbachsOtherConjecture() { function isPrime(num) {
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if (num < 2) {
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return false;
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} else if (num === 2) {
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return true;
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}
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const sqrtOfNum = Math.floor(num ** 0.5);
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for (let i = 2; i <= sqrtOfNum + 1; i++) {
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if (num % i === 0) {
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return false;
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}
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}
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return true;
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}
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function isSquare(num) {
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return Math.sqrt(num) % 1 === 0;
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}
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// construct a list of prime numbers
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const primes = [];
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for (let i = 2; primes.length < 1000; i++) {
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if (isPrime(i)) primes.push(i);
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}
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let num = 3;
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let answer;
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while (!answer) {
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num += 2;
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if (!isPrime(num)) {
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let found = false;
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for (let primeI = 0; primeI < primes.length && !found; primeI++) {
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const square = (num - primes[primeI]) / 2;
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if (isSquare(square)) {
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found = true;
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break;
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}
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}
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if (!found) answer = num;
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}
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}
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return answer;
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}
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```
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