101 lines
2.8 KiB
Markdown
101 lines
2.8 KiB
Markdown
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---
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id: 5900f3ac1000cf542c50febf
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title: 'Problem 64: Odd period square roots'
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challengeType: 5
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forumTopicId: 302176
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dashedName: problem-64-odd-period-square-roots
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---
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# --description--
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All square roots are periodic when written as continued fractions and can be written in the form:
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$\\displaystyle \\quad \\quad \\sqrt{N}=a_0+\\frac 1 {a_1+\\frac 1 {a_2+ \\frac 1 {a3+ \\dots}}}$
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For example, let us consider $\\sqrt{23}:$:
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$\\quad \\quad \\sqrt{23}=4+\\sqrt{23}-4=4+\\frac 1 {\\frac 1 {\\sqrt{23}-4}}=4+\\frac 1 {1+\\frac{\\sqrt{23}-3}7}$
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If we continue we would get the following expansion:
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$\\displaystyle \\quad \\quad \\sqrt{23}=4+\\frac 1 {1+\\frac 1 {3+ \\frac 1 {1+\\frac 1 {8+ \\dots}}}}$
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The process can be summarized as follows:
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$\\quad \\quad a_0=4, \\frac 1 {\\sqrt{23}-4}=\\frac {\\sqrt{23}+4} 7=1+\\frac {\\sqrt{23}-3} 7$
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$\\quad \\quad a_1=1, \\frac 7 {\\sqrt{23}-3}=\\frac {7(\\sqrt{23}+3)} {14}=3+\\frac {\\sqrt{23}-3} 2$
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$\\quad \\quad a_2=3, \\frac 2 {\\sqrt{23}-3}=\\frac {2(\\sqrt{23}+3)} {14}=1+\\frac {\\sqrt{23}-4} 7$
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$\\quad \\quad a_3=1, \\frac 7 {\\sqrt{23}-4}=\\frac {7(\\sqrt{23}+4)} 7=8+\\sqrt{23}-4$
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$\\quad \\quad a_4=8, \\frac 1 {\\sqrt{23}-4}=\\frac {\\sqrt{23}+4} 7=1+\\frac {\\sqrt{23}-3} 7$
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$\\quad \\quad a_5=1, \\frac 7 {\\sqrt{23}-3}=\\frac {7 (\\sqrt{23}+3)} {14}=3+\\frac {\\sqrt{23}-3} 2$
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$\\quad \\quad a_6=3, \\frac 2 {\\sqrt{23}-3}=\\frac {2(\\sqrt{23}+3)} {14}=1+\\frac {\\sqrt{23}-4} 7$
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$\\quad \\quad a_7=1, \\frac 7 {\\sqrt{23}-4}=\\frac {7(\\sqrt{23}+4)} {7}=8+\\sqrt{23}-4$
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It can be seen that the sequence is repeating. For conciseness, we use the notation $\\sqrt{23}=\[4;(1,3,1,8)]$, to indicate that the block (1,3,1,8) repeats indefinitely.
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The first ten continued fraction representations of (irrational) square roots are:
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$\\quad \\quad \\sqrt{2}=\[1;(2)]$, period = 1
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$\\quad \\quad \\sqrt{3}=\[1;(1,2)]$, period = 2
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$\\quad \\quad \\sqrt{5}=\[2;(4)]$, period = 1
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$\\quad \\quad \\sqrt{6}=\[2;(2,4)]$, period = 2
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$\\quad \\quad \\sqrt{7}=\[2;(1,1,1,4)]$, period = 4
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$\\quad \\quad \\sqrt{8}=\[2;(1,4)]$, period = 2
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$\\quad \\quad \\sqrt{10}=\[3;(6)]$, period = 1
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$\\quad \\quad \\sqrt{11}=\[3;(3,6)]$, period = 2
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$\\quad \\quad \\sqrt{12}=\[3;(2,6)]$, period = 2
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$\\quad \\quad \\sqrt{13}=\[3;(1,1,1,1,6)]$, period = 5
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Exactly four continued fractions, for $N \\le 13$, have an odd period.
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How many continued fractions for $N \\le 10\\,000$ have an odd period?
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# --hints--
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`oddPeriodSqrts()` should return a number.
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```js
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assert(typeof oddPeriodSqrts() === 'number');
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```
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`oddPeriodSqrts()` should return 1322.
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```js
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assert.strictEqual(oddPeriodSqrts(), 1322);
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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 oddPeriodSqrts() {
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return true;
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}
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oddPeriodSqrts();
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```
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# --solutions--
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```js
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// solution required
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```
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