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Merge pull request #25 from mdoumbouya/master
pythagoras, trigonometry, prime factorization, golden ratio
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‎en/golden-ratio/index.html‎

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<title>Golden Ratio</title>
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<meta name="viewport" content="width=device-width, initial-scale=1">
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<meta http-equiv="content-type" content="text/html; charset=UTF8">
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<!-- Bootstrap -->
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<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/4.0.0/css/bootstrap.min.css" integrity="sha384-Gn5384xqQ1aoWXA+058RXPxPg6fy4IWvTNh0E263XmFcJlSAwiGgFAW/dAiS6JXm" crossorigin="anonymous">
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<script src="https://maxcdn.bootstrapcdn.com/bootstrap/4.0.0/js/bootstrap.min.js" integrity="sha384-JZR6Spejh4U02d8jOt6vLEHfe/JQGiRRSQQxSfFWpi1MquVdAyjUar5+76PVCmYl" crossorigin="anonymous"></script>
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<link rel="stylesheet" href="https://use.fontawesome.com/releases/v5.7.0/css/all.css" integrity="sha384-lZN37f5QGtY3VHgisS14W3ExzMWZxybE1SJSEsQp9S+oqd12jhcu+A56Ebc1zFSJ" crossorigin="anonymous">
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<script src="https://cdn.jsdelivr.net/npm/sweetalert2@9"></script>
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<link href='https://fonts.googleapis.com/css?family=Source+Serif+Pro:400,600,700' rel='stylesheet' type='text/css'>
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<body>
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<div class="container container-course">
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<div class="row">
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<div class="col">
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<h1>Golden Ratio</h1>
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<p class="subtleHeading">Problem written by Moussa Doumbouya</p>
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<hr/>
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<p>
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When two numbers a and b are such that their ratio (a/b) is the same as the ratio
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of their sum to the largest of the two numbers (a+b)/a, they are said to be in the <a href="https://en.wikipedia.org/wiki/Golden_ratio">golden ratio.</a>
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<br /><br />
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The golden ratio is then: ϕ&nbsp;&nbsp;=&nbsp;&nbsp;(a+b) / a&nbsp;&nbsp;=&nbsp;&nbsp;a / b.
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<br /><br />
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The Fibonacci sequence is the sequence is such that the first two elements are 1,
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and any other element is the sum of the two elements preceeding it.
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<ul>
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<li>F(0) = 1</li>
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<li>F(1) = 1</li>
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<li>F(2) = F(1) + F(0) = 2</li>
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<li>F(3) = F(2) + F(1) = 3</li>
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<li>F(4) = F(3) + F(2) = 5</li>
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<li>F(5) = F(4) + F(3) = 8</li>
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<li>F(6) = F(5) + F(4) = 13</li>
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<li>F(7) = F(6) + F(5) = 21</li>
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<li>...</li>
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<li>F(n) = F(n-1) + F(n-2)</li>
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</ul>
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The ratio of consecutive Fibonaccy terms 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, etc. for larger and larger terms converges towards
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one number, ... the golden ratio!
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</p>
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<p>
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Write a program that estimates the golden ratio as the ratio of consecutive Fibonaccy terms.
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Here is a possible top-down decomposition.
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<ul>
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<li>A function that displays the golden ratio at position i (in Fibonacci sequence) for i going from 1 to 100</li>
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<li>A function that computes the golden ratio at position i, F(i) / F(i-1)</li>
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<li>A function that computes the Fibonaccy term i, F(i)</li>
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</ul>
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</p>
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<p>
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Here is a sample of the expected output of your program:
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</p>
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<pre class="console" style="width:400px">
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1.0
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2.0
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1.5
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1.6666666666666667
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1.6
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1.625
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1.6153846153846154
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1.619047619047619
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1.6176470588235294
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1.6181818181818182
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1.6179775280898876
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1.6180555555555556
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1.6180257510729614
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1.6180371352785146
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1.618032786885246
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1.618034447821682
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1.6180338134001253
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1.618034055727554
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1.6180339631667064
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1.6180339985218033
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1.618033985017358
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1.6180339901755971
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1.618033988205325
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1.618033988957902
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1.6180339886704431
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1.6180339887802426
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1.618033988738303
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1.6180339887543225
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1.6180339887482036
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1.6180339887505408
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1.6180339887496482
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1.618033988749989
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1.618033988749859
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1.6180339887499087
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1.6180339887498896
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1.618033988749897
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1.618033988749894
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1.6180339887498951
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1.6180339887498947
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1.618033988749895
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1.618033988749895
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1.618033988749895
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1.618033988749895
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1.618033988749895
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1.618033988749895
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1.618033988749895
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...
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</pre>
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<h2>Solution</h2>
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<pre class="console" id="editor" style="height:660.0px">
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def main():
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&quot;&quot;&quot;
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Show the ratio of consecutive Fibonaccy numbers
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&quot;&quot;&quot;
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for i in range(1, 100):
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print(golden(i))
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def golden(n):
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&quot;&quot;&quot;
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Return the ratio of fibonacci number (n) to fibonacci number (n-1)
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&quot;&quot;&quot;
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return fib(n)/fib(n-1)
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def fib(n):
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&quot;&quot;&quot;
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Compute fibonacci term n
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&quot;&quot;&quot;
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a = 1
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b = 1
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for i in range(1, n+1):
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next_element = a + b
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a = b
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b = next_element
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return a
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if __name__ == &#039;__main__&#039;:
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main()</pre>
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<script src="../../plugins/ace/ace.js" type="text/javascript" charset="utf-8"></script>
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<script>
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var editor = ace.edit("editor");
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editor.setTheme('ace/theme/eclipse');
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editor.getSession().setMode("ace/mode/python");
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editor.setReadOnly(true);
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editor.renderer.setShowGutter(false);
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editor.setFontSize("14px");
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/*editor.setTheme("ace/theme/eclipse");
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editor.getSession().setMode("ace/mode/java");*/
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</script>
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<hr/>
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</div>
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</div>
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</div>
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</body>

‎en/prime-factorization/index.html‎

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<title>Prime Factorization</title>
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<meta name="viewport" content="width=device-width, initial-scale=1">
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<meta http-equiv="content-type" content="text/html; charset=UTF8">
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<!-- Bootstrap -->
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<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/4.0.0/css/bootstrap.min.css" integrity="sha384-Gn5384xqQ1aoWXA+058RXPxPg6fy4IWvTNh0E263XmFcJlSAwiGgFAW/dAiS6JXm" crossorigin="anonymous">
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<link rel="stylesheet" type="text/css" href="../../style.css">
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<!-- Java Script -->
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<script src="https://ajax.googleapis.com/ajax/libs/jquery/3.4.1/jquery.min.js"></script>
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<script src="https://cdnjs.cloudflare.com/ajax/libs/popper.js/1.12.9/umd/popper.min.js" integrity="sha384-ApNbgh9B+Y1QKtv3Rn7W3mgPxhU9K/ScQsAP7hUibX39j7fakFPskvXusvfa0b4Q" crossorigin="anonymous"></script>
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<script src="https://maxcdn.bootstrapcdn.com/bootstrap/4.0.0/js/bootstrap.min.js" integrity="sha384-JZR6Spejh4U02d8jOt6vLEHfe/JQGiRRSQQxSfFWpi1MquVdAyjUar5+76PVCmYl" crossorigin="anonymous"></script>
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<!-- font awesome -->
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<link rel="stylesheet" href="https://use.fontawesome.com/releases/v5.7.0/css/all.css" integrity="sha384-lZN37f5QGtY3VHgisS14W3ExzMWZxybE1SJSEsQp9S+oqd12jhcu+A56Ebc1zFSJ" crossorigin="anonymous">
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<!-- SWAL -->
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<script src="https://cdn.jsdelivr.net/npm/sweetalert2@9"></script>
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<!-- Stanford -->
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<link href='https://fonts.googleapis.com/css?family=Source+Sans+Pro:300,400,600,700' rel='stylesheet' type='text/css'>
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<link href='https://fonts.googleapis.com/css?family=Source+Serif+Pro:400,600,700' rel='stylesheet' type='text/css'>
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<body>
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<div class="container container-course">
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<div class="row">
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<div class="col">
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<h1>Prime Factorization</h1>
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<p class="subtleHeading">Problem written by Moussa Doumbouya, discovered by Euclid</p>
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<hr/>
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<p>
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<a href="https://en.wikipedia.org/wiki/Integer_factorization#Prime_decomposition">Prime factorization</a> is the process of
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decomposing an integer into factors that are prime numbers.
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<a href="https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic">The fundamental theorem of arithmetic</a> states
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that every positive integer has a unique prime factorization. Here are examples of prime factorizations:
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<ul>
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<li>14 = 2 x 7</li>
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<li>25 = 5 x 5</li>
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<li>99 = 3 x 3 x 11</li>
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<li>255 = 3 x 5 x 17</li>
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<li>1234 = 2 x 617</li>
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</ul>
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</p>
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<p>
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Write a program that continually reads in an integer and outputs its prime factors. Here is a possible
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top-down decomposition of the problem.
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<ul>
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<li>
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A function that shows the prime factors of a number by repeatedly finding its smallest prime factor
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and repeating the same process on the remaining quotient until the remaining quotient is 1.
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</li>
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<li>
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A function that returns the smallest prime factor of its argument.
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That is, the smallest number in the interval [2, n] that is a prime number and that evenly divides the argument.
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</li>
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<li>
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A function that checks if its argument is prime.
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Note that a prime numer is a number that is not evenly divisible by any number other than 1 and itself.
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</li>
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</ul>
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</p>
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<p>
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Examples of sample inputs and outputs of your program:
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<pre>
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Enter an integer: <span class="blue">34</span>
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The prime factors of 34 are:
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2
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17
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Enter an integer: <span class="blue">28</span>
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The prime factors of 28 are:
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2
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2
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7
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Enter an integer: <span class="blue">147</span>
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The prime factors of 147 are:
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3
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7
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7
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</pre>
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</p>
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<h2>Solution</h2>
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<pre class="console" id="editor" style="height:1580.0px">&quot;&quot;&quot;
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This program performs prime factorization onto user-specified positive integers.
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&quot;&quot;&quot;
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def main():
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while True:
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n = read_positive_int(&quot;Enter an integer: &quot;)
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show_prime_factors(n)
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print(&quot;&quot;)
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def read_positive_int(prompt_msg):
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&quot;&quot;&quot;
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Reads a positive integer entered by the user.
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Args:
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prompt_msg (str): The prompt message to show the user when requesting the input.
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Returns:
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int: The positive integer entered by the user.
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&quot;&quot;&quot;
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n = int(input(prompt_msg))
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while n&lt;=0:
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print(&quot;Please enter a positive integer&quot;)
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n = int(input(prompt_msg))
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return n
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def show_prime_factors(n):
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&quot;&quot;&quot;
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Show the prime factors of n
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Args:
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n (int): The number to be factorized
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&quot;&quot;&quot;
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print(&quot;The prime factors of &quot; + str(n) + &quot; are: &quot;)
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while n&gt;1:
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p = smallest_prime_factor(n)
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print(p)
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n = n // p
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def smallest_prime_factor(n):
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&quot;&quot;&quot;
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Compute the smallest prime factor of n.
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Args:
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n (int): The number to be factorized
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Returns:
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int: The smallest number in the range [2, n] that is
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a divisor of n, and is a prime number
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&quot;&quot;&quot;
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for i in range(2, n+1):
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if n%i == 0 and is_prime(i):
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return i
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def is_prime(n):
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&quot;&quot;&quot;
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Test if the specified number is prime.
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Args:
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n (int): The number to be tested
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Returns:
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bool: True if n has no divisor in range [2, n[. False otherwise.
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&quot;&quot;&quot;
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for d in range(2, n):
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if n%d == 0:
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return False
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return True
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if __name__ == &#039;__main__&#039;:
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main()</pre>
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<script src="../../plugins/ace/ace.js" type="text/javascript" charset="utf-8"></script>
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<script>
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var editor = ace.edit("editor");
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editor.setTheme('ace/theme/eclipse');
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editor.getSession().setMode("ace/mode/python");
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editor.setReadOnly(true);
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editor.renderer.setShowGutter(false);
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editor.setFontSize("14px");
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/*editor.setTheme("ace/theme/eclipse");
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editor.getSession().setMode("ace/mode/java");*/
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</script>
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<hr/>
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</div>
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</div>
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</div>
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</body>

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