Problem

The sum of the primes below 10 is $2 + 3 + 5 + 7 = 17$.

Find the sum of all the primes below two million.

Problem

A Pythagorean triplet is a set of three natural numbers, $a \lt b \lt c$,
for which

$$a^2 + b^2 = c^2$$

For example, $3^2 + 4^2 = 9 + 16 = 25 = 5^2$.

There exists exactly one Pythagorean triplet for which $a + b + c = 1000$.
Find the product $a \times b \times c$.

This code is mainly based on Processing, a flexible software sketchbook for visualizing things and arts.

First, we derive a class named Perceptron with two useful method guess and train.

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int sign(float n) {
  if (n >= 0) {
    return 1;
  } else {
    return -1;
  }
}

class Perceptron {
  float[] weights = new float[2];
  float lr = 0.2;
  
  // constructor
  Perceptron() {
    // initialize the weight randomly
    for (int i = 0; i < weights.length; i++) {
      weights[i] = random(-1, 1);
    }
  }
  
  int guess(float[] inputs) {
    float sum = 0;
    for (int i = 0; i < weights.length; i++) {
      sum += inputs[i] * weights[i];
    }
    int output = sign(sum);
    return output;
  }
  
  void train(float[] inputs, int target) {
    int guess = guess(inputs);
    int error = target - guess;
    
    for (int i = 0; i < weights.length; i++) {
      weights[i] += error * inputs[i] * lr;
    }
  }
}

To specify the training procession using a simple perceptron, we activate a sketchbook named Training.

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class Point {
  float x;
  float y;
  int label;
  
  Point() {
    x = random(width);
    y = random(height);
    
    if (x > y) {
      label = 1;
    } else {
      label = -1;
    }
  }
  
  void show() {
    stroke(0);
    if (label == 1) {
      fill(255);
    } else {
      fill(0);
    }
    ellipse(x, y, 32, 32);
  }
}

Now, come to our basic purpose. Draw random samples and make trainings. Here is the main sketchbook file SimplePerceptron.

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Perceptron brain;

Point[] points = new Point[100];

int trainingIndex = 0;

void setup() {
  size(800, 800);
  brain = new Perceptron();
  
  for (int i = 0; i< points.length; i++) {
    points[i] = new Point();
  }
  float[] inputs = {-1, 0.5};
  int guess = brain.guess(inputs);
  println(guess);
}

void draw() {
  background(255);
  stroke(0);
  line(0, 0, width, height);
  for (Point pt : points) {
    pt.show();
  }
  
  for (Point pt : points) {
    float[] inputs = {pt.x, pt.y};
    
    int target = pt.label;
    int guess = brain.guess(inputs);
    
    if (guess == target) {
      fill(0, 255, 0);
    } else {
      fill(255, 0, 0);
    }
    
    noStroke();
    ellipse(pt.x, pt.y, 16, 16);
  }
  
  Point training = points[trainingIndex];
  float[] inputs = {training.x, training.y};
  int target = training.label;
  brain.train(inputs, target);
  trainingIndex++;
  if (trainingIndex == points.length) {
    trainingIndex = 0;
  }
}

Simply click the run button, Processing will help start an interative interface (similar to the canvas) and render the result below.

This code is mainly based on the p5js library, which provide us an accessible sketchbook to interactive with HTML objects.

Problem

The four adjacent digits in the 1000-digit number that have the greatest product are 9 × 9 × 8 × 9 = 5832.

73167176531330624919225119674426574742355349194934
96983520312774506326239578318016984801869478851843
85861560789112949495459501737958331952853208805511
12540698747158523863050715693290963295227443043557
66896648950445244523161731856403098711121722383113
62229893423380308135336276614282806444486645238749
30358907296290491560440772390713810515859307960866
70172427121883998797908792274921901699720888093776
65727333001053367881220235421809751254540594752243
52584907711670556013604839586446706324415722155397
53697817977846174064955149290862569321978468622482
83972241375657056057490261407972968652414535100474
82166370484403199890008895243450658541227588666881
16427171479924442928230863465674813919123162824586
17866458359124566529476545682848912883142607690042
24219022671055626321111109370544217506941658960408
07198403850962455444362981230987879927244284909188
84580156166097919133875499200524063689912560717606
05886116467109405077541002256983155200055935729725
71636269561882670428252483600823257530420752963450

Find the thirteen adjacent digits in the 1000-digit number that have the greatest product. What is the value of this product?

Problem

By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13,
we can see that the 6th prime is 13.

What is the 10001st prime number?