Category Archives: Euler

Ruby: Project Euler Problem # 97

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http://projecteuler.net/problem=97

Large non-Mersenne prime

The first known prime found to exceed one million digits was discovered in 1999, and is a Mersenne prime of the form 26972593−1; it contains exactly 2,098,960 digits. Subsequently other Mersenne primes, of the form 2p−1, have been found which contain more digits.
However, in 2004 there was found a massive non-Mersenne prime which contains 2,357,207 digits: 28433×27830457+1.
Find the last ten digits of this prime number.

Ruby: Project Euler Problem # 58

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http://projecteuler.net/problem=58

Spiral primes

Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed.

37 36 35 34 33 32 31
38 17 16 15 14 13 30
39 18  5  4  3 12 29
40 19  6  1  2 11 28
41 20  7  8  9 10 27
42 21 22 23 24 25 26
43 44 45 46 47 48 49

It is interesting to note that the odd squares lie along the bottom right diagonal, but what is more interesting is that 8 out of the 13 numbers lying along both diagonals are prime; that is, a ratio of 8/13 ≈ 62%.
If one complete new layer is wrapped around the spiral above, a square spiral with side length 9 will be formed. If this process is continued, what is the side length of the square spiral for which the ratio of primes along both diagonals first falls below 10%?

Ruby: Solution to Project Euler Problem # 28

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http://projecteuler.net/problem=28

Number spiral diagonals

Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows:

21 22 23 24 25
20  7  8  9 10
19  6  1  2 11
18  5  4  3 12
17 16 15 14 13

It can be verified that the sum of the numbers on the diagonals is 101.

What is the sum of the numbers on the diagonals in a 1001 by 1001 spiral formed in the same way?

Ruby: Solution to Project Euler Problem # 25

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Recently, I’ve been typing more Ruby then usual. In Ruby you can have a fairly big number.
http://projecteuler.net/problem=25

1000-digit Fibonacci number

The Fibonacci sequence is defined by the recurrence relation:

Fn = Fn−1 + Fn−2, where F1 = 1 and F2 = 1.

The 12th term, F12, is the first term to contain three digits.

What is the first term in the Fibonacci sequence to contain 1000 digits?

C#: Project Euler Solution to Problem 27

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http://projecteuler.net/problem=27

Considering quadratics of the form:

n² + an + b, where |a| < 1000 and |b| < 1000 where |n| is the modulus/absolute value of n e.g. |11| = 11 and |-4| = 4 Find the product of the coefficients, a and b, for the quadratic expression that produces the maximum number of primes for consecutive values of n, starting with n = 0. 

static void Main(string[] args)
{
    //n² + an + b
    int maxprimes = 0;
    int maxproduct = 0;
    for (int a = 0; a < 1000; a++)
    {
        for (int b = 0; b < 1000; b++)
        {
            int prime = countPrime(a, b);
            if (maxprimes < prime)
            {
                maxprimes = prime;
                maxproduct= (-a*b );
            }
        }
    }
    Console.WriteLine(maxproduct);
}
 
public static int countPrime(int a, int b)
{
    int count = 0;
    int n = 0;
    while (true)
    {
        double result = Math.Pow(n, 2.00) - (a * n) + b;
        if (isPrime((int)result) && result >= 0)
            count++;
        else
            return count;
        n++;
    }
}
 
public static bool isPrime(int n)
{
    if (n == 1)
        return false;
    if (n == 2)
        return true;
    for (int i = 2; i < n; ++i)
    {
        if ((n % i) == 0)
            return false;
    }
    return true;
}