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Add more solutions and minor improvements

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Added solutions for problem 66, 67, 68, 69 and 70 in C and python.
Also, minor improvements to the code for a few older problems.
This commit is contained in:
daniele 2019-09-29 13:26:34 +02:00
parent 8e8ac9931a
commit 3a09f3d4f5
Signed by: fuxino
GPG Key ID: 6FE25B4A3EE16FDA
26 changed files with 1484 additions and 76 deletions
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9
C/p018.c
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@ -74,9 +74,16 @@ int main(int argc, char **argv)
fclose(fp);
/* Use function implemented in projecteuler.c to find the maximum path.*/
/* Use the function implemented in projecteuler.c to find the maximum path.*/
max = find_max_path(triang, 15);
for(i = 0; i < 100; i++)
{
free(triang[i]);
}
free(triang);
clock_gettime(CLOCK_MONOTONIC, &end);
elapsed = (end.tv_sec - start.tv_sec) + (double)(end.tv_nsec - start.tv_nsec) / 1000000000;

9
C/p049.c
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@ -14,7 +14,7 @@
#define N 10000
int check_digits(int a, int b);
int is_permutation(int a, int b);
int *primes;
@ -45,7 +45,7 @@ int main(int argc, char **argv)
/* If i, i+j and i+2*j are all primes and they have
* all the same digits, the result has been found.*/
if(i + 2 * j < N && primes[i+j] && primes[i+2*j] &&
check_digits(i, i+j) && check_digits(i, i+2*j))
is_permutation(i, i+j) && is_permutation(i, i+2*j))
{
found = 1;
break;
@ -73,23 +73,26 @@ int main(int argc, char **argv)
return 0;
}
int check_digits(int a, int b)
int is_permutation(int a, int b)
{
int i;
int digits1[10] = {0}, digits2[10] = {0};
/* Get digits of a.*/
while(a > 0)
{
digits1[a%10]++;
a /= 10;
}
/* Get digits of b.*/
while(b > 0)
{
digits2[b%10]++;
b /= 10;
}
/* If they're not the same, return 0.*/
for(i = 0; i < 10; i++)
{
if(digits1[i] != digits2[i])

8
C/p052.c
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@ -6,7 +6,7 @@
#include <stdlib.h>
#include <time.h>
int have_same_digits(int a, int b);
int is_permutation(int a, int b);
int main(int argc, char **argv)
{
@ -21,8 +21,8 @@ int main(int argc, char **argv)
/* Brute force approach, try every integer number until the desired one is found.*/
while(1)
{
if(have_same_digits(i, 2*i) && have_same_digits(i, 3*i) && have_same_digits(i, 4*i) &&
have_same_digits(i, 5*i) && have_same_digits(i, 6*i))
if(is_permutation(i, 2*i) && is_permutation(i, 3*i) && is_permutation(i, 4*i) &&
is_permutation(i, 5*i) && is_permutation(i, 6*i))
{
break;
}
@ -42,7 +42,7 @@ int main(int argc, char **argv)
return 0;
}
int have_same_digits(int a, int b)
int is_permutation(int a, int b)
{
int i;
int digits1[10] = {0}, digits2[10] = {0};

9
C/p062.c
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@ -9,7 +9,7 @@
#define N 10000
int check_digits(long int a, long int b);
int is_permutation(long int a, long int b);
int count_digits(long int a);
int main(int argc, char **argv)
@ -38,7 +38,7 @@ int main(int argc, char **argv)
* they can't be permutations).*/
for(j = i + 1; j < N && count_digits(cubes[j]) == count_digits(cubes[i]); j++)
{
if(check_digits(cubes[i], cubes[j]))
if(is_permutation(cubes[i], cubes[j]))
{
count++;
}
@ -85,23 +85,26 @@ int count_digits(long int a)
return count;
}
int check_digits(long int a, long int b)
int is_permutation(int a, int b)
{
int i;
int digits1[10] = {0}, digits2[10] = {0};
/* Get digits of a.*/
while(a > 0)
{
digits1[a%10]++;
a /= 10;
}
/* Get digits of b.*/
while(b > 0)
{
digits2[b%10]++;
b /= 10;
}
/* If they're not the same, return 0.*/
for(i = 0; i < 10; i++)
{
if(digits1[i] != digits2[i])

18
C/p064.c
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@ -50,7 +50,8 @@ int is_square(int n);
int main(int argc, char **argv)
{
int i, count = 0;
int i, count = 0, period;
int *fraction;
double elapsed;
struct timespec start, end;
@ -60,12 +61,23 @@ int main(int argc, char **argv)
{
/* Perfect squares are obviously not represented as continued fractions.
* For all other numbers, calculate their period and check if it's odd.*/
if(!is_square(i) && period_cf(i) % 2 != 0)
if(!is_square(i))
{
count++;
if((fraction = build_sqrt_cont_fraction(i, &period, 300)) == NULL)
{
fprintf(stderr, "Error! Build_cont_fraction function returned NULL\n");
return 1;
}
if(period % 2 != 0)
{
count++;
}
}
}
free(fraction);
clock_gettime(CLOCK_MONOTONIC, &end);
elapsed = (end.tv_sec - start.tv_sec) + (double)(end.tv_nsec - start.tv_nsec) / 1000000000;

2
C/p065.c
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@ -77,7 +77,7 @@ int main(int argc, char **argv)
clock_gettime(CLOCK_MONOTONIC, &end);
elapsed=(end.tv_sec-start.tv_sec)+(double)(end.tv_nsec-start.tv_nsec)/1000000000;
elapsed = (end.tv_sec - start.tv_sec) + (double)(end.tv_nsec - start.tv_nsec) / 1000000000;
printf("Project Euler, Problem 65\n");
printf("Answer: %d\n", sum);

92
C/p066.c Normal file
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@ -0,0 +1,92 @@
/* Consider quadratic Diophantine equations of the form:
*
* x^2 – Dy^2 = 1
*
* For example, when D=13, the minimal solution in x is 649^2 – 13×180^2 = 1.
*
* It can be assumed that there are no solutions in positive integers when D is square.
*
* By finding minimal solutions in x for D = {2, 3, 5, 6, 7}, we obtain the following:
*
* 3^2 – 2×2^2 = 1
* 2^2 – 3×1^2 = 1
* 9^2 – 5×4^2 = 1
* 5^2 – 6×2^2 = 1
* 8^2 – 7×3^2 = 1
*
* Hence, by considering minimal solutions in x for D ≤ 7, the largest x is obtained when D=5.
*
* Find the value of D ≤ 1000 in minimal solutions of x for which the largest value of x is obtained.*/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <time.h>
#include <gmp.h>
#include "projecteuler.h"
int is_square(int n);
int main(int argc, char **argv)
{
int i, max_d = -1;
mpz_t x, max;
double elapsed;
struct timespec start, end;
clock_gettime(CLOCK_MONOTONIC, &start);
// mpz_init(x);
mpz_init_set_ui(max, 0);
for(i = 2; i <= 1000; i++)
{
if(!is_square(i))
{
/* Solve the Diophantine equation x^2-D*y^2=1 (Pell equation)*/
if(pell_eq(i, x) == -1)
{
fprintf(stderr, "Error! Pell_eq function failed\n");
return 1;
}
if(mpz_cmp(x, max) > 0)
{
mpz_set(max, x);
max_d = i;
}
}
}
mpz_clears(x, max, NULL);
clock_gettime(CLOCK_MONOTONIC, &end);
elapsed = (end.tv_sec - start.tv_sec) + (double)(end.tv_nsec - start.tv_nsec) / 1000000000;
printf("Project Euler, Problem 66\n");
printf("Answer: %d\n", max_d);
printf("Elapsed time: %.9lf seconds\n", elapsed);
return 0;
}
int is_square(int n)
{
int m;
double p;
p = sqrt(n);
m = p;
if(p == m)
{
return 1;
}
else
{
return 0;
}
}

82
C/p067.c Normal file
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@ -0,0 +1,82 @@
/* By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.
*
* 3
* 7 4
* 2 4 6
* 8 5 9 3
*
* That is, 3 + 7 + 4 + 9 = 23.
* Find the maximum total from top to bottom in triangle.txt (right click and 'Save Link/Target As...'), a 15K text file containing a triangle
* with one-hundred rows.
*
* NOTE: This is a much more difficult version of Problem 18. It is not possible to try every route to solve this problem, as there are 299 altogether!
* If you could check one trillion (1012) routes every second it would take over twenty billion years to check them all.
* There is an efficient algorithm to solve it. ;o)*/
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
#include "projecteuler.h"
int main(int argc, char **argv)
{
int i, j, max;
int **triang;
double elapsed;
FILE *fp;
struct timespec start, end;
clock_gettime(CLOCK_MONOTONIC, &start);
if((triang = (int **)malloc(100*sizeof(int *))) == NULL)
{
fprintf(stderr, "Error while allocating memory\n");
return 1;
}
for(i = 1; i <= 100; i++)
{
if((triang[i-1] = (int *)malloc(i*sizeof(int))) == NULL)
{
fprintf(stderr, "Error while allocating memory\n");
return 1;
}
}
if((fp = fopen("triangle.txt", "r")) == NULL)
{
fprintf(stderr, "Error while opening file %s\n", "triangle.txt");
return 1;
}
for(i = 1; i <= 100; i++)
{
for(j = 0; j < i; j++)
{
fscanf(fp, "%d", &triang[i-1][j]);
}
}
fclose(fp);
/* Use the function implemented in projecteuler.c to find the maximum path.*/
max = find_max_path(triang, 100);
for(i = 0; i < 100; i++)
{
free(triang[i]);
}
free(triang);
clock_gettime(CLOCK_MONOTONIC, &end);
elapsed = (end.tv_sec - start.tv_sec) + (double)(end.tv_nsec - start.tv_nsec) / 1000000000;
printf("Project Euler, Problem 67\n");
printf("Answer: %d\n", max);
printf("Elapsed time: %.9lf seconds\n", elapsed);
return 0;
}

217
C/p068.c Normal file
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@ -0,0 +1,217 @@
/* Consider the following "magic" 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine.
*
* 4
* \
* \
* 3
* / \
* / \
* 1------2------6
* /
* /
* 5
*
* Working clockwise, and starting from the group of three with the numerically lowest external node (4,3,2 in this example),
* each solution can be described uniquely. For example, the above solution can be described by the set: 4,3,2; 6,2,1; 5,1,3.
*
* It is possible to complete the ring with four different totals: 9, 10, 11, and 12. There are eight solutions in total.
*
* Total Solution Set
* 9 4,2,3; 5,3,1; 6,1,2
* 9 4,3,2; 6,2,1; 5,1,3
* 10 2,3,5; 4,5,1; 6,1,3
* 10 2,5,3; 6,3,1; 4,1,5
* 11 1,4,6; 3,6,2; 5,2,4
* 11 1,6,4; 5,4,2; 3,2,6
* 12 1,5,6; 2,6,4; 3,4,5
* 12 1,6,5; 3,5,4; 2,4,6
*
* By concatenating each group it is possible to form 9-digit strings; the maximum string for a 3-gon ring is 432621513.
*
* Using the numbers 1 to 10, and depending on arrangements, it is possible to form 16- and 17-digit strings. What is the maximum 16-digit string
* for a "magic" 5-gon ring?
*
* O
* \
* O
* / \ O
* / \ /
* O O
* / \ /
* O O--O---O
* \
* O
*/
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
#include <time.h>
#include "projecteuler.h"
int *eval_ring(int **ring, int n);
long int array_to_long(int *n, int len);
int compare(void *a, void *b);
int main(int argc, char **argv)
{
int i;
int *eval, **ring;
long int n, max = 0;
double elapsed;
struct timespec start, end;
clock_gettime(CLOCK_MONOTONIC, &start);
if((ring = (int **)malloc(10*sizeof(int *))) == NULL)
{
fprintf(stderr, "Error while allocating memory\n");
return 1;
}
for(i = 0; i < 10; i++)
{
if((ring[i] = (int *)malloc(sizeof(int))) == NULL)
{
fprintf(stderr, "Error while allocating memory\n");
return 1;
}
*ring[i] = i + 1;
}
/* Check if the first permutation of values is a possible solution.*/
eval = eval_ring(ring, 5);
/* Convert the vector into an integer number.*/
n = array_to_long(eval, 15);
if(n > max)
{
max = n;
}
free(eval);
/* Generate all possible permutations, for each one check if
* it's a possible solution for the ring and save the maximum.*/
while(next_permutation((void **)ring, 10, compare) != -1)
{
eval = eval_ring(ring, 5);
n = array_to_long(eval, 15);
if(n > max)
{
max = n;
}
free(eval);
}
clock_gettime(CLOCK_MONOTONIC, &end);
elapsed = (end.tv_sec - start.tv_sec) + (double)(end.tv_nsec - start.tv_nsec) / 1000000000;
printf("Project Euler, Problem 68\n");
printf("Answer: %ld\n", max);
printf("Elapsed time: %.9lf seconds\n", elapsed);
return 0;
}
int compare(void *a, void *b)
{
int *n1, *n2;
n1 = (int *)a;
n2 = (int *)b;
return *n1 - *n2;
}
/* Function to evaluate the ring. The ring is represented as a vector of 2*n elements,
* the first n elements represent the external nodes of the ring, the next n elements
* represent the internal ring.*/
int *eval_ring(int **ring, int n)
{
int i, j, magic_val, val;
int *res;
for(i = 1; i < n; i++)
{
/* We need to start from the lowest external node, so if
* the first element in the vector is not the lowest of
* the first n elements (the external elements), the configuration
* is not a valid one.*/
if(*ring[i] < *ring[0])
{
return NULL;
}
}
if((res = (int *)malloc(3*n*sizeof(int))) == NULL)
{
fprintf(stderr, "Error while allocating memory\n");
return NULL;
}
/* Each group of three must have the same value.*/
magic_val = *ring[0] + *ring[n] + *ring[n+1];
for(i = 0, j = 0; i < n; i++, j += 3)
{
/* We need to find the maximum 16-digit string, this is
* possible only if the element "10" is used only once,
* i.e. if it's one of the external nodes.*/
if(*ring[n+i] == 10 || *ring[n+(i+1)%n] == 10)
{
return NULL;
}
/* Check that the value of the current three-element group
* is the "magic" value.*/
val = *ring[i] + *ring[n+i] + *ring[n+(i+1)%n];
if(val != magic_val)
{
return NULL;
}
/* Save the current element group in the result string.*/
res[j] = *ring[i];
res[j+1] = *ring[n+i];
res[j+2] = *ring[n+(i+1)%n];
}
return res;
}
/* Function to convert the vector into a long int.*/
long int array_to_long(int *n, int len)
{
int i, k = 0;
long int res = 0;
if(n == NULL)
{
return 0;
}
for(i = len - 1; i >= 0; i--)
{
res += n[i] * pow(10, k);
if(n[i] >= 10)
{
k += 2;
}
else
{
k++;
}
}
return res;
}

66
C/p069.c Normal file
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@ -0,0 +1,66 @@
/* Euler's Totient function, φ(n) [sometimes called the phi function], is used to determine the number of numbers less than n which are
* relatively prime to n. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, φ(9)=6.
*
* n Relatively Prime φ(n) n/φ(n)
* 2 1 1 2
* 3 1,2 2 1.5
* 4 1,3 2 2
* 5 1,2,3,4 4 1.25
* 6 1,5 2 3
* 7 1,2,3,4,5,6 6 1.1666...
* 8 1,3,5,7 4 2
* 9 1,2,4,5,7,8 6 1.5
* 10 1,3,7,9 4 2.5
*
* It can be seen that n=6 produces a maximum n/φ(n) for n ≤ 10.
*
* Find the value of n ≤ 1,000,000 for which n/φ(n) is a maximum.*/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <time.h>
#include "projecteuler.h"
#define N 1000000
int main(int argc, char **argv)
{
int i, res = 1;
double elapsed;
struct timespec start, end;
clock_gettime(CLOCK_MONOTONIC, &start);
i = 1;
/* Using Euler's formula, phi(n)=n*prod(1-1/p), where p are the distinct
* primes that divide n. So n/phi(n)=1/prod(1-1/p). To find the maximum
* value of this function, the denominator must be minimized. This happens
* when n has the most distinct small prime factor, i.e. to find the solution
* we need to multiply the smallest consecutive primes until the result is
* larger than 1000000.*/
while(res < N)
{
i++;
if(is_prime(i))
{
res *= i;
}
}
/* We need the previous value, because we want i<1000000.*/
res /= i;
clock_gettime(CLOCK_MONOTONIC, &end);
elapsed = (end.tv_sec-start.tv_sec) + (double)(end.tv_nsec - start.tv_nsec) / 1000000000;
printf("Project Euler, Problem 69\n");
printf("Answer: %d\n", res);
printf("Elapsed time: %.9lf seconds\n", elapsed);
return 0;
}

93
C/p070.c Normal file
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@ -0,0 +1,93 @@
/* Euler's Totient function, φ(n) [sometimes called the phi function], is used to determine the number of positive numbers
* less than or equal to n which are relatively prime to n. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and
* relatively prime to nine, φ(9)=6.
* The number 1 is considered to be relatively prime to every positive number, so φ(1)=1.
*
* Interestingly, φ(87109)=79180, and it can be seen that 87109 is a permutation of 79180.
*
* Find the value of n, 1 < n < 10^7, for which φ(n) is a permutation of n and the ratio n/φ(n) produces a minimum.*/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <time.h>
#include <float.h>
#include "projecteuler.h"
#define N 10000000
int is_permutation(int a, int b);
int main(int argc, char **argv)
{
int i, a, b, p, n = -1;
int *primes;
double elapsed, min = DBL_MAX;
struct timespec start, end;
clock_gettime(CLOCK_MONOTONIC, &start);
primes = sieve(N);
for(i = 2; i < N; i++)
{
/* When n is prime, phi(n)=(n-1), so to minimize n/phi(n) we should
* use n prime. But n-1 can't be a permutation of n. The second best
* bet is to use semiprimes. For a semiprime n=p*q, phi(n)=(p-1)(q-1).
* So we check if a number is semiprime, if yes calculate phi, finally
* check if phi(n) is a permutation of n and update the minimum if it's
* smaller.*/
if(is_semiprime(i, &a, &b, primes))
{
p = phi_semiprime(i, a, b);
if(is_permutation(p, i) && (double)i / p < min)
{
n = i;
min = (double)i / p;
}
}
}
clock_gettime(CLOCK_MONOTONIC, &end);
elapsed = (end.tv_sec - start.tv_sec) + (double)(end.tv_nsec - start.tv_nsec) / 1000000000;
printf("Project Euler, Problem 70\n");
printf("Answer: %d\n", n);
printf("Elapsed time: %.9lf seconds\n", elapsed);
return 0;
}
int is_permutation(int a, int b)
{
int i;
int digits1[10] = {0}, digits2[10] = {0};
/* Get digits of a.*/
while(a > 0)
{
digits1[a%10]++;
a /= 10;
}
/* Get digits of b.*/
while(b > 0)
{
digits2[b%10]++;
b /= 10;
}
/* If they're not the same, return 0.*/
for(i = 0; i < 10; i++)
{
if(digits1[i] != digits2[i])
{
return 0;
}
}
return 1;
}

204
C/projecteuler.c
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@ -478,23 +478,215 @@ int is_pentagonal(long int n)
* d_(n+1)=(S-m_(n+1)^2)/d_n
* a_(n+1)=floor((sqrt(S)+m_(n+1))/d_(n+1))=floor((a_0+m_(n+1))/d_(n+1))
* if a_i=2*a_0, the algorithm ends.*/
int period_cf(int n)
int *build_sqrt_cont_fraction(int i, int *period, int l)
{
int mn = 0, mn1, dn = 1, dn1, a0, an, an1, count = 0;
int mn = 0, mn1, dn = 1, dn1, a0, an, an1, count = 0, j;
int *fraction;
a0 = floor(sqrt(n));
if((fraction = (int *)malloc(l*sizeof(int))) == NULL)
{
return NULL;
}
j = 0;
a0 = floor(sqrt(i));
an = a0;
fraction[j] = an;
j++;
do
{
mn1 = dn * an - mn;
dn1 = (n - mn1 * mn1) / dn;
dn1 = (i - mn1 * mn1)/ dn;
an1 = floor((a0+mn1)/dn1);
mn = mn1;
dn = dn1;
an = an1;
count++;
fraction[j] = an;
j++;
}while(an != 2 * a0);
return count;
fraction[j] = -1;
*period = count;
return fraction;
}
/* Function to solve the Diophantine equation in the form x^2-Dy^2=1
* (Pell equation) using continued fractions.*/
int pell_eq(int d, mpz_t x)
{
int found = 0, j, period;
mpz_t n1, n2, n3, d1, d2, d3, sol, tmp;
int *fraction;
/* Find the continued fraction for sqrt(d).*/
if((fraction = build_sqrt_cont_fraction(d, &period, 100)) == NULL)
{
return -1;
}
/* Calculate the first convergent of the continued fraction.*/
mpz_init_set_ui(n1, 0);
mpz_init_set_ui(n2, 1);
mpz_init_set_ui(d1, 1);
mpz_init_set_ui(d2, 0);
mpz_inits(n3, d3, sol, tmp, NULL);
j = 0;
mpz_mul_ui(n3, n2, fraction[j]);
mpz_add(n3, n3, n1);
mpz_mul_ui(d3, d2, fraction[j]);
mpz_add(d3, d3, d1);
j++;
/* Check if x=n, y=d solve the equation x^2-Dy^2=1.*/
mpz_mul(sol, n3, n3);
mpz_mul(tmp, d3, d3);
mpz_mul_ui(tmp, tmp, d);
mpz_sub(sol, sol, tmp);
if(mpz_cmp_ui(sol, 1) == 0)
{
mpz_set(x, n3);
found = 1;
free(fraction);
mpz_clears(n1, n2, n3, d1, d2, d3, sol, tmp, NULL);
}
/* Until a solution is found, calculate the next convergent
* and check if x=n and y=d solve the equation.*/
while(!found)
{
mpz_set(n1, n2);
mpz_set(n2, n3);
mpz_set(d1, d2);
mpz_set(d2, d3);
mpz_mul_ui(n3, n2, fraction[j]);
mpz_add(n3, n3, n1);
mpz_mul_ui(d3, d2, fraction[j]);
mpz_add(d3, d3, d1);
mpz_mul(sol, n3, n3);
mpz_mul(tmp, d3, d3);
mpz_mul_ui(tmp, tmp, d);
mpz_sub(sol, sol, tmp);
if(mpz_cmp_ui(sol, 1) == 0)
{
mpz_set(x, n3);
found = 1;
free(fraction);
mpz_clears(n1, n2, n3, d1, d2, d3, sol, tmp, NULL);
}
j++;
if(fraction[j] == -1)
{
j = 1;
}
}
return 0;
}
/* Function to check if a number is semiprime. Parameters include
* pointers to p and q to return the factors values and a list of
* primes.*/
int is_semiprime(int n, int *p, int *q, int *primes)
{
int i, limit;
/* If n is prime, it's not semiprime.*/
if(primes[n])
{
return 0;
}
/* Check if n is semiprime and one of the factors is 2.*/
if(n % 2 == 0)
{
if(primes[n/2])
{
*p = 2;
*q = n / 2;
return 1;
}
else
{
return 0;
}
}
/* Check if n is semiprime and one of the factors is 3.*/
else if(n % 3 == 0)
{
if(primes[n/3])
{
*p = 3;
*q = n / 3;
return 1;
}
else
{
return 0;
}
}
/*Any number can have only one prime factor greater than its
square root, so we can stop checking at this point.*/
limit = floor(sqrt(n));
/* Every prime other than 2 and 3 is in the form 6k+1 or 6k-1.
* If I check all those value no prime factors of the number
* will be missed. For each of these possible primes, check if
* they are prime, then if the number is semiprime with using
* that factor.*/
for(i = 5; i <= limit; i += 6)
{
if(primes[i] && n % i == 0)
{
if(primes[n/i])
{
*p = i;
*q = n / i;
return 1;
}
else
{
return 0;
}
}
else if(primes[i+2] && n % (i + 2) == 0)
{
if(primes[n/(i+2)])
{
*p = i + 2;
*q = n / (i + 2);
return 1;
}
else
{
return 0;
}
}
}
return 0;
}
/* If n=pq is semiprime, phi(n)=(p-1)(q-1)=pq-p-q+1=n-(p+4)+1
* if p!=q. If p=q (n is a square), phi(n)=n-p.*/
int phi_semiprime(int n, int p, int q)
{
if(p == q)
{
return n - p;
}
else
{
return n - (p + q) + 1;
}
}

5
C/projecteuler.h
View File

@ -19,6 +19,9 @@ void quick_sort(void **array, int l, int r, int (*cmp)(void *lv, void *rv));
int next_permutation(void **perm, int n, int (*cmp)(void *a, void *b));
int is_pandigital(int value, int n);
int is_pentagonal(long int n);
int period_cf(int n);
int *build_sqrt_cont_fraction(int i, int *period, int l);
int pell_eq(int i, mpz_t x);
int is_semiprime(int n, int *p, int *q, int *primes);
int phi_semiprime(int n, int p, int q);
#endif

100
C/triangle.txt Normal file
View File

@ -0,0 +1,100 @@
59
73 41
52 40 09
26 53 06 34
10 51 87 86 81
61 95 66 57 25 68
90 81 80 38 92 67 73
30 28 51 76 81 18 75 44
84 14 95 87 62 81 17 78 58
21 46 71 58 02 79 62 39 31 09
56 34 35 53 78 31 81 18 90 93 15
78 53 04 21 84 93 32 13 97 11 37 51
45 03 81 79 05 18 78 86 13 30 63 99 95
39 87 96 28 03 38 42 17 82 87 58 07 22 57
06 17 51 17 07 93 09 07 75 97 95 78 87 08 53
67 66 59 60 88 99 94 65 55 77 55 34 27 53 78 28
76 40 41 04 87 16 09 42 75 69 23 97 30 60 10 79 87
12 10 44 26 21 36 32 84 98 60 13 12 36 16 63 31 91 35
70 39 06 05 55 27 38 48 28 22 34 35 62 62 15 14 94 89 86
66 56 68 84 96 21 34 34 34 81 62 40 65 54 62 05 98 03 02 60
38 89 46 37 99 54 34 53 36 14 70 26 02 90 45 13 31 61 83 73 47
36 10 63 96 60 49 41 05 37 42 14 58 84 93 96 17 09 43 05 43 06 59
66 57 87 57 61 28 37 51 84 73 79 15 39 95 88 87 43 39 11 86 77 74 18
54 42 05 79 30 49 99 73 46 37 50 02 45 09 54 52 27 95 27 65 19 45 26 45
71 39 17 78 76 29 52 90 18 99 78 19 35 62 71 19 23 65 93 85 49 33 75 09 02
33 24 47 61 60 55 32 88 57 55 91 54 46 57 07 77 98 52 80 99 24 25 46 78 79 05
92 09 13 55 10 67 26 78 76 82 63 49 51 31 24 68 05 57 07 54 69 21 67 43 17 63 12
24 59 06 08 98 74 66 26 61 60 13 03 09 09 24 30 71 08 88 70 72 70 29 90 11 82 41 34
66 82 67 04 36 60 92 77 91 85 62 49 59 61 30 90 29 94 26 41 89 04 53 22 83 41 09 74 90
48 28 26 37 28 52 77 26 51 32 18 98 79 36 62 13 17 08 19 54 89 29 73 68 42 14 08 16 70 37
37 60 69 70 72 71 09 59 13 60 38 13 57 36 09 30 43 89 30 39 15 02 44 73 05 73 26 63 56 86 12
55 55 85 50 62 99 84 77 28 85 03 21 27 22 19 26 82 69 54 04 13 07 85 14 01 15 70 59 89 95 10 19
04 09 31 92 91 38 92 86 98 75 21 05 64 42 62 84 36 20 73 42 21 23 22 51 51 79 25 45 85 53 03 43 22
75 63 02 49 14 12 89 14 60 78 92 16 44 82 38 30 72 11 46 52 90 27 08 65 78 03 85 41 57 79 39 52 33 48
78 27 56 56 39 13 19 43 86 72 58 95 39 07 04 34 21 98 39 15 39 84 89 69 84 46 37 57 59 35 59 50 26 15 93
42 89 36 27 78 91 24 11 17 41 05 94 07 69 51 96 03 96 47 90 90 45 91 20 50 56 10 32 36 49 04 53 85 92 25 65
52 09 61 30 61 97 66 21 96 92 98 90 06 34 96 60 32 69 68 33 75 84 18 31 71 50 84 63 03 03 19 11 28 42 75 45 45
61 31 61 68 96 34 49 39 05 71 76 59 62 67 06 47 96 99 34 21 32 47 52 07 71 60 42 72 94 56 82 83 84 40 94 87 82 46
01 20 60 14 17 38 26 78 66 81 45 95 18 51 98 81 48 16 53 88 37 52 69 95 72 93 22 34 98 20 54 27 73 61 56 63 60 34 63
93 42 94 83 47 61 27 51 79 79 45 01 44 73 31 70 83 42 88 25 53 51 30 15 65 94 80 44 61 84 12 77 02 62 02 65 94 42 14 94
32 73 09 67 68 29 74 98 10 19 85 48 38 31 85 67 53 93 93 77 47 67 39 72 94 53 18 43 77 40 78 32 29 59 24 06 02 83 50 60 66
32 01 44 30 16 51 15 81 98 15 10 62 86 79 50 62 45 60 70 38 31 85 65 61 64 06 69 84 14 22 56 43 09 48 66 69 83 91 60 40 36 61
92 48 22 99 15 95 64 43 01 16 94 02 99 19 17 69 11 58 97 56 89 31 77 45 67 96 12 73 08 20 36 47 81 44 50 64 68 85 40 81 85 52 09
91 35 92 45 32 84 62 15 19 64 21 66 06 01 52 80 62 59 12 25 88 28 91 50 40 16 22 99 92 79 87 51 21 77 74 77 07 42 38 42 74 83 02 05
46 19 77 66 24 18 05 32 02 84 31 99 92 58 96 72 91 36 62 99 55 29 53 42 12 37 26 58 89 50 66 19 82 75 12 48 24 87 91 85 02 07 03 76 86
99 98 84 93 07 17 33 61 92 20 66 60 24 66 40 30 67 05 37 29 24 96 03 27 70 62 13 04 45 47 59 88 43 20 66 15 46 92 30 04 71 66 78 70 53 99
67 60 38 06 88 04 17 72 10 99 71 07 42 25 54 05 26 64 91 50 45 71 06 30 67 48 69 82 08 56 80 67 18 46 66 63 01 20 08 80 47 07 91 16 03 79 87
18 54 78 49 80 48 77 40 68 23 60 88 58 80 33 57 11 69 55 53 64 02 94 49 60 92 16 35 81 21 82 96 25 24 96 18 02 05 49 03 50 77 06 32 84 27 18 38
68 01 50 04 03 21 42 94 53 24 89 05 92 26 52 36 68 11 85 01 04 42 02 45 15 06 50 04 53 73 25 74 81 88 98 21 67 84 79 97 99 20 95 04 40 46 02 58 87
94 10 02 78 88 52 21 03 88 60 06 53 49 71 20 91 12 65 07 49 21 22 11 41 58 99 36 16 09 48 17 24 52 36 23 15 72 16 84 56 02 99 43 76 81 71 29 39 49 17
64 39 59 84 86 16 17 66 03 09 43 06 64 18 63 29 68 06 23 07 87 14 26 35 17 12 98 41 53 64 78 18 98 27 28 84 80 67 75 62 10 11 76 90 54 10 05 54 41 39 66
43 83 18 37 32 31 52 29 95 47 08 76 35 11 04 53 35 43 34 10 52 57 12 36 20 39 40 55 78 44 07 31 38 26 08 15 56 88 86 01 52 62 10 24 32 05 60 65 53 28 57 99
03 50 03 52 07 73 49 92 66 80 01 46 08 67 25 36 73 93 07 42 25 53 13 96 76 83 87 90 54 89 78 22 78 91 73 51 69 09 79 94 83 53 09 40 69 62 10 79 49 47 03 81 30
71 54 73 33 51 76 59 54 79 37 56 45 84 17 62 21 98 69 41 95 65 24 39 37 62 03 24 48 54 64 46 82 71 78 33 67 09 16 96 68 52 74 79 68 32 21 13 78 96 60 09 69 20 36
73 26 21 44 46 38 17 83 65 98 07 23 52 46 61 97 33 13 60 31 70 15 36 77 31 58 56 93 75 68 21 36 69 53 90 75 25 82 39 50 65 94 29 30 11 33 11 13 96 02 56 47 07 49 02
76 46 73 30 10 20 60 70 14 56 34 26 37 39 48 24 55 76 84 91 39 86 95 61 50 14 53 93 64 67 37 31 10 84 42 70 48 20 10 72 60 61 84 79 69 65 99 73 89 25 85 48 92 56 97 16
03 14 80 27 22 30 44 27 67 75 79 32 51 54 81 29 65 14 19 04 13 82 04 91 43 40 12 52 29 99 07 76 60 25 01 07 61 71 37 92 40 47 99 66 57 01 43 44 22 40 53 53 09 69 26 81 07
49 80 56 90 93 87 47 13 75 28 87 23 72 79 32 18 27 20 28 10 37 59 21 18 70 04 79 96 03 31 45 71 81 06 14 18 17 05 31 50 92 79 23 47 09 39 47 91 43 54 69 47 42 95 62 46 32 85
37 18 62 85 87 28 64 05 77 51 47 26 30 65 05 70 65 75 59 80 42 52 25 20 44 10 92 17 71 95 52 14 77 13 24 55 11 65 26 91 01 30 63 15 49 48 41 17 67 47 03 68 20 90 98 32 04 40 68
90 51 58 60 06 55 23 68 05 19 76 94 82 36 96 43 38 90 87 28 33 83 05 17 70 83 96 93 06 04 78 47 80 06 23 84 75 23 87 72 99 14 50 98 92 38 90 64 61 58 76 94 36 66 87 80 51 35 61 38
57 95 64 06 53 36 82 51 40 33 47 14 07 98 78 65 39 58 53 06 50 53 04 69 40 68 36 69 75 78 75 60 03 32 39 24 74 47 26 90 13 40 44 71 90 76 51 24 36 50 25 45 70 80 61 80 61 43 90 64 11
18 29 86 56 68 42 79 10 42 44 30 12 96 18 23 18 52 59 02 99 67 46 60 86 43 38 55 17 44 93 42 21 55 14 47 34 55 16 49 24 23 29 96 51 55 10 46 53 27 92 27 46 63 57 30 65 43 27 21 20 24 83
81 72 93 19 69 52 48 01 13 83 92 69 20 48 69 59 20 62 05 42 28 89 90 99 32 72 84 17 08 87 36 03 60 31 36 36 81 26 97 36 48 54 56 56 27 16 91 08 23 11 87 99 33 47 02 14 44 73 70 99 43 35 33
90 56 61 86 56 12 70 59 63 32 01 15 81 47 71 76 95 32 65 80 54 70 34 51 40 45 33 04 64 55 78 68 88 47 31 47 68 87 03 84 23 44 89 72 35 08 31 76 63 26 90 85 96 67 65 91 19 14 17 86 04 71 32 95
37 13 04 22 64 37 37 28 56 62 86 33 07 37 10 44 52 82 52 06 19 52 57 75 90 26 91 24 06 21 14 67 76 30 46 14 35 89 89 41 03 64 56 97 87 63 22 34 03 79 17 45 11 53 25 56 96 61 23 18 63 31 37 37 47
77 23 26 70 72 76 77 04 28 64 71 69 14 85 96 54 95 48 06 62 99 83 86 77 97 75 71 66 30 19 57 90 33 01 60 61 14 12 90 99 32 77 56 41 18 14 87 49 10 14 90 64 18 50 21 74 14 16 88 05 45 73 82 47 74 44
22 97 41 13 34 31 54 61 56 94 03 24 59 27 98 77 04 09 37 40 12 26 87 09 71 70 07 18 64 57 80 21 12 71 83 94 60 39 73 79 73 19 97 32 64 29 41 07 48 84 85 67 12 74 95 20 24 52 41 67 56 61 29 93 35 72 69
72 23 63 66 01 11 07 30 52 56 95 16 65 26 83 90 50 74 60 18 16 48 43 77 37 11 99 98 30 94 91 26 62 73 45 12 87 73 47 27 01 88 66 99 21 41 95 80 02 53 23 32 61 48 32 43 43 83 14 66 95 91 19 81 80 67 25 88
08 62 32 18 92 14 83 71 37 96 11 83 39 99 05 16 23 27 10 67 02 25 44 11 55 31 46 64 41 56 44 74 26 81 51 31 45 85 87 09 81 95 22 28 76 69 46 48 64 87 67 76 27 89 31 11 74 16 62 03 60 94 42 47 09 34 94 93 72
56 18 90 18 42 17 42 32 14 86 06 53 33 95 99 35 29 15 44 20 49 59 25 54 34 59 84 21 23 54 35 90 78 16 93 13 37 88 54 19 86 67 68 55 66 84 65 42 98 37 87 56 33 28 58 38 28 38 66 27 52 21 81 15 08 22 97 32 85 27
91 53 40 28 13 34 91 25 01 63 50 37 22 49 71 58 32 28 30 18 68 94 23 83 63 62 94 76 80 41 90 22 82 52 29 12 18 56 10 08 35 14 37 57 23 65 67 40 72 39 93 39 70 89 40 34 07 46 94 22 20 05 53 64 56 30 05 56 61 88 27
23 95 11 12 37 69 68 24 66 10 87 70 43 50 75 07 62 41 83 58 95 93 89 79 45 39 02 22 05 22 95 43 62 11 68 29 17 40 26 44 25 71 87 16 70 85 19 25 59 94 90 41 41 80 61 70 55 60 84 33 95 76 42 63 15 09 03 40 38 12 03 32
09 84 56 80 61 55 85 97 16 94 82 94 98 57 84 30 84 48 93 90 71 05 95 90 73 17 30 98 40 64 65 89 07 79 09 19 56 36 42 30 23 69 73 72 07 05 27 61 24 31 43 48 71 84 21 28 26 65 65 59 65 74 77 20 10 81 61 84 95 08 52 23 70
47 81 28 09 98 51 67 64 35 51 59 36 92 82 77 65 80 24 72 53 22 07 27 10 21 28 30 22 48 82 80 48 56 20 14 43 18 25 50 95 90 31 77 08 09 48 44 80 90 22 93 45 82 17 13 96 25 26 08 73 34 99 06 49 24 06 83 51 40 14 15 10 25 01
54 25 10 81 30 64 24 74 75 80 36 75 82 60 22 69 72 91 45 67 03 62 79 54 89 74 44 83 64 96 66 73 44 30 74 50 37 05 09 97 70 01 60 46 37 91 39 75 75 18 58 52 72 78 51 81 86 52 08 97 01 46 43 66 98 62 81 18 70 93 73 08 32 46 34
96 80 82 07 59 71 92 53 19 20 88 66 03 26 26 10 24 27 50 82 94 73 63 08 51 33 22 45 19 13 58 33 90 15 22 50 36 13 55 06 35 47 82 52 33 61 36 27 28 46 98 14 73 20 73 32 16 26 80 53 47 66 76 38 94 45 02 01 22 52 47 96 64 58 52 39
88 46 23 39 74 63 81 64 20 90 33 33 76 55 58 26 10 46 42 26 74 74 12 83 32 43 09 02 73 55 86 54 85 34 28 23 29 79 91 62 47 41 82 87 99 22 48 90 20 05 96 75 95 04 43 28 81 39 81 01 28 42 78 25 39 77 90 57 58 98 17 36 73 22 63 74 51
29 39 74 94 95 78 64 24 38 86 63 87 93 06 70 92 22 16 80 64 29 52 20 27 23 50 14 13 87 15 72 96 81 22 08 49 72 30 70 24 79 31 16 64 59 21 89 34 96 91 48 76 43 53 88 01 57 80 23 81 90 79 58 01 80 87 17 99 86 90 72 63 32 69 14 28 88 69
37 17 71 95 56 93 71 35 43 45 04 98 92 94 84 96 11 30 31 27 31 60 92 03 48 05 98 91 86 94 35 90 90 08 48 19 33 28 68 37 59 26 65 96 50 68 22 07 09 49 34 31 77 49 43 06 75 17 81 87 61 79 52 26 27 72 29 50 07 98 86 01 17 10 46 64 24 18 56
51 30 25 94 88 85 79 91 40 33 63 84 49 67 98 92 15 26 75 19 82 05 18 78 65 93 61 48 91 43 59 41 70 51 22 15 92 81 67 91 46 98 11 11 65 31 66 10 98 65 83 21 05 56 05 98 73 67 46 74 69 34 08 30 05 52 07 98 32 95 30 94 65 50 24 63 28 81 99 57
19 23 61 36 09 89 71 98 65 17 30 29 89 26 79 74 94 11 44 48 97 54 81 55 39 66 69 45 28 47 13 86 15 76 74 70 84 32 36 33 79 20 78 14 41 47 89 28 81 05 99 66 81 86 38 26 06 25 13 60 54 55 23 53 27 05 89 25 23 11 13 54 59 54 56 34 16 24 53 44 06
13 40 57 72 21 15 60 08 04 19 11 98 34 45 09 97 86 71 03 15 56 19 15 44 97 31 90 04 87 87 76 08 12 30 24 62 84 28 12 85 82 53 99 52 13 94 06 65 97 86 09 50 94 68 69 74 30 67 87 94 63 07 78 27 80 36 69 41 06 92 32 78 37 82 30 05 18 87 99 72 19 99
44 20 55 77 69 91 27 31 28 81 80 27 02 07 97 23 95 98 12 25 75 29 47 71 07 47 78 39 41 59 27 76 13 15 66 61 68 35 69 86 16 53 67 63 99 85 41 56 08 28 33 40 94 76 90 85 31 70 24 65 84 65 99 82 19 25 54 37 21 46 33 02 52 99 51 33 26 04 87 02 08 18 96
54 42 61 45 91 06 64 79 80 82 32 16 83 63 42 49 19 78 65 97 40 42 14 61 49 34 04 18 25 98 59 30 82 72 26 88 54 36 21 75 03 88 99 53 46 51 55 78 22 94 34 40 68 87 84 25 30 76 25 08 92 84 42 61 40 38 09 99 40 23 29 39 46 55 10 90 35 84 56 70 63 23 91 39
52 92 03 71 89 07 09 37 68 66 58 20 44 92 51 56 13 71 79 99 26 37 02 06 16 67 36 52 58 16 79 73 56 60 59 27 44 77 94 82 20 50 98 33 09 87 94 37 40 83 64 83 58 85 17 76 53 02 83 52 22 27 39 20 48 92 45 21 09 42 24 23 12 37 52 28 50 78 79 20 86 62 73 20 59
54 96 80 15 91 90 99 70 10 09 58 90 93 50 81 99 54 38 36 10 30 11 35 84 16 45 82 18 11 97 36 43 96 79 97 65 40 48 23 19 17 31 64 52 65 65 37 32 65 76 99 79 34 65 79 27 55 33 03 01 33 27 61 28 66 08 04 70 49 46 48 83 01 45 19 96 13 81 14 21 31 79 93 85 50 05
92 92 48 84 59 98 31 53 23 27 15 22 79 95 24 76 05 79 16 93 97 89 38 89 42 83 02 88 94 95 82 21 01 97 48 39 31 78 09 65 50 56 97 61 01 07 65 27 21 23 14 15 80 97 44 78 49 35 33 45 81 74 34 05 31 57 09 38 94 07 69 54 69 32 65 68 46 68 78 90 24 28 49 51 45 86 35
41 63 89 76 87 31 86 09 46 14 87 82 22 29 47 16 13 10 70 72 82 95 48 64 58 43 13 75 42 69 21 12 67 13 64 85 58 23 98 09 37 76 05 22 31 12 66 50 29 99 86 72 45 25 10 28 19 06 90 43 29 31 67 79 46 25 74 14 97 35 76 37 65 46 23 82 06 22 30 76 93 66 94 17 96 13 20 72
63 40 78 08 52 09 90 41 70 28 36 14 46 44 85 96 24 52 58 15 87 37 05 98 99 39 13 61 76 38 44 99 83 74 90 22 53 80 56 98 30 51 63 39 44 30 91 91 04 22 27 73 17 35 53 18 35 45 54 56 27 78 48 13 69 36 44 38 71 25 30 56 15 22 73 43 32 69 59 25 93 83 45 11 34 94 44 39 92
12 36 56 88 13 96 16 12 55 54 11 47 19 78 17 17 68 81 77 51 42 55 99 85 66 27 81 79 93 42 65 61 69 74 14 01 18 56 12 01 58 37 91 22 42 66 83 25 19 04 96 41 25 45 18 69 96 88 36 93 10 12 98 32 44 83 83 04 72 91 04 27 73 07 34 37 71 60 59 31 01 54 54 44 96 93 83 36 04 45
30 18 22 20 42 96 65 79 17 41 55 69 94 81 29 80 91 31 85 25 47 26 43 49 02 99 34 67 99 76 16 14 15 93 08 32 99 44 61 77 67 50 43 55 87 55 53 72 17 46 62 25 50 99 73 05 93 48 17 31 70 80 59 09 44 59 45 13 74 66 58 94 87 73 16 14 85 38 74 99 64 23 79 28 71 42 20 37 82 31 23
51 96 39 65 46 71 56 13 29 68 53 86 45 33 51 49 12 91 21 21 76 85 02 17 98 15 46 12 60 21 88 30 92 83 44 59 42 50 27 88 46 86 94 73 45 54 23 24 14 10 94 21 20 34 23 51 04 83 99 75 90 63 60 16 22 33 83 70 11 32 10 50 29 30 83 46 11 05 31 17 86 42 49 01 44 63 28 60 07 78 95 40
44 61 89 59 04 49 51 27 69 71 46 76 44 04 09 34 56 39 15 06 94 91 75 90 65 27 56 23 74 06 23 33 36 69 14 39 05 34 35 57 33 22 76 46 56 10 61 65 98 09 16 69 04 62 65 18 99 76 49 18 72 66 73 83 82 40 76 31 89 91 27 88 17 35 41 35 32 51 32 67 52 68 74 85 80 57 07 11 62 66 47 22 67
65 37 19 97 26 17 16 24 24 17 50 37 64 82 24 36 32 11 68 34 69 31 32 89 79 93 96 68 49 90 14 23 04 04 67 99 81 74 70 74 36 96 68 09 64 39 88 35 54 89 96 58 66 27 88 97 32 14 06 35 78 20 71 06 85 66 57 02 58 91 72 05 29 56 73 48 86 52 09 93 22 57 79 42 12 01 31 68 17 59 63 76 07 77
73 81 14 13 17 20 11 09 01 83 08 85 91 70 84 63 62 77 37 07 47 01 59 95 39 69 39 21 99 09 87 02 97 16 92 36 74 71 90 66 33 73 73 75 52 91 11 12 26 53 05 26 26 48 61 50 90 65 01 87 42 47 74 35 22 73 24 26 56 70 52 05 48 41 31 18 83 27 21 39 80 85 26 08 44 02 71 07 63 22 05 52 19 08 20
17 25 21 11 72 93 33 49 64 23 53 82 03 13 91 65 85 02 40 05 42 31 77 42 05 36 06 54 04 58 07 76 87 83 25 57 66 12 74 33 85 37 74 32 20 69 03 97 91 68 82 44 19 14 89 28 85 85 80 53 34 87 58 98 88 78 48 65 98 40 11 57 10 67 70 81 60 79 74 72 97 59 79 47 30 20 54 80 89 91 14 05 33 36 79 39
60 85 59 39 60 07 57 76 77 92 06 35 15 72 23 41 45 52 95 18 64 79 86 53 56 31 69 11 91 31 84 50 44 82 22 81 41 40 30 42 30 91 48 94 74 76 64 58 74 25 96 57 14 19 03 99 28 83 15 75 99 01 89 85 79 50 03 95 32 67 44 08 07 41 62 64 29 20 14 76 26 55 48 71 69 66 19 72 44 25 14 01 48 74 12 98 07
64 66 84 24 18 16 27 48 20 14 47 69 30 86 48 40 23 16 61 21 51 50 26 47 35 33 91 28 78 64 43 68 04 79 51 08 19 60 52 95 06 68 46 86 35 97 27 58 04 65 30 58 99 12 12 75 91 39 50 31 42 64 70 04 46 07 98 73 98 93 37 89 77 91 64 71 64 65 66 21 78 62 81 74 42 20 83 70 73 95 78 45 92 27 34 53 71 15
30 11 85 31 34 71 13 48 05 14 44 03 19 67 23 73 19 57 06 90 94 72 57 69 81 62 59 68 88 57 55 69 49 13 07 87 97 80 89 05 71 05 05 26 38 40 16 62 45 99 18 38 98 24 21 26 62 74 69 04 85 57 77 35 58 67 91 79 79 57 86 28 66 34 72 51 76 78 36 95 63 90 08 78 47 63 45 31 22 70 52 48 79 94 15 77 61 67 68
23 33 44 81 80 92 93 75 94 88 23 61 39 76 22 03 28 94 32 06 49 65 41 34 18 23 08 47 62 60 03 63 33 13 80 52 31 54 73 43 70 26 16 69 57 87 83 31 03 93 70 81 47 95 77 44 29 68 39 51 56 59 63 07 25 70 07 77 43 53 64 03 94 42 95 39 18 01 66 21 16 97 20 50 90 16 70 10 95 69 29 06 25 61 41 26 15 59 63 35

2
Python/p018.py
View File

@ -54,7 +54,7 @@ def main():
for i in range(l):
triang[i] = list(map(int, triang[i]))
# Use function implemented in projecteuler.c to find the maximum path.
# Use the function implemented in projecteuler.c to find the maximum path.
max_ = find_max_path(triang, 15)
end = default_timer()

21
Python/p049.py
View File

@ -13,24 +13,6 @@ from numpy import zeros
from timeit import default_timer
from projecteuler import sieve
def check_digits(a, b):
digits1 = zeros(10, int)
digits2 = zeros(10, int)
while a > 0:
digits1[a%10] = digits1[a%10] + 1
a = a // 10
while b > 0:
digits2[b%10] = digits2[b%10] + 1
b = b // 10
for i in range(10):
if digits1[i] != digits2[i]:
return False
return True
def main():
start = default_timer()
@ -51,7 +33,8 @@ def main():
# If i, i+j and i+2*j are all primes and they have
# all the same digits, the result has been found.
if i + 2 * j < N and primes[i+j] == 1 and primes[i+2*j] == 1 and\
check_digits(i, i+j) and check_digits(i, i+2*j):
''.join(sorted(str(i))) == ''.join(sorted(str(i+j))) and\
''.join(sorted(str(i))) == ''.join(sorted(str(i+2*j))):
found = 1
break
if(found):

28
Python/p052.py
View File

@ -8,27 +8,6 @@ from numpy import zeros
from timeit import default_timer
def have_same_digits(a, b):
digits1 = zeros(10, int)
digits2 = zeros(10, int)
# Get digits of a.
while a > 0:
digits1[a%10] = digits1[a%10] + 1
a = a // 10
# Get digits of b.
while b > 0:
digits2[b%10] = digits2[b%10] + 1
b = b // 10
# If they're not the same, return 0.
for i in range(10):
if digits1[i] != digits2[i]:
return 0
return 1
def main():
start = default_timer()
@ -36,8 +15,11 @@ def main():
# Brute force approach, try every integer number until the desired one is found.
while True:
if have_same_digits(i, 2*i) and have_same_digits(i, 3*i) and have_same_digits(i, 4*i) and\
have_same_digits(i, 5*i) and have_same_digits(i, 6*i):
if ''.join(sorted(str(i))) == ''.join(sorted(str(2*i))) and\
''.join(sorted(str(i))) == ''.join(sorted(str(3*i))) and\
''.join(sorted(str(i))) == ''.join(sorted(str(4*i))) and\
''.join(sorted(str(i))) == ''.join(sorted(str(5*i))) and\
''.join(sorted(str(i))) == ''.join(sorted(str(6*i))):
break
i = i + 1

4
Python/p055.py
View File

@ -44,11 +44,11 @@ def is_lychrel(n):
# If the sum is palindrome, the number is not a Lychrel number.
if is_palindrome(tmp, 10):
return 0
return False
n = tmp
return 1
return True
def main():
start = default_timer()

14
Python/p064.py
View File

@ -45,16 +45,16 @@
from math import floor, sqrt
from timeit import default_timer
from projecteuler import period_cf
from projecteuler import build_sqrt_cont_fraction
def is_square(n):
p = sqrt(n)
m = int(p)
if p == m:
return 1
return True
else:
return 0
return False
def main():
start = default_timer()
@ -64,8 +64,12 @@ def main():
for i in range(2, 10000):
# Perfect squares are obviously not represented as continued fractions.
# For all other numbers, calculate their period and check if it's odd.
if not is_square(i) and period_cf(i) % 2 != 0:
count = count + 1
if not is_square(i):
# period_cf(i) % 2 != 0:
fraction, period = build_sqrt_cont_fraction(i, 300)
if period % 2 != 0:
count = count + 1
end = default_timer()

60
Python/p066.py Normal file
View File

@ -0,0 +1,60 @@
#!/usr/bin/python
# Consider quadratic Diophantine equations of the form:
#
# x^2 – Dy^2 = 1
#
# For example, when D=13, the minimal solution in x is 649^2 – 13×180^2 = 1.
#
# It can be assumed that there are no solutions in positive integers when D is square.
#
# By finding minimal solutions in x for D = {2, 3, 5, 6, 7}, we obtain the following:
#
# 3^2 – 2×2^2 = 1
# 2^2 – 3×1^2 = 1
# 9^2 – 5×4^2 = 1
# 5^2 – 6×2^2 = 1
# 8^2 – 7×3^2 = 1
#
# Hence, by considering minimal solutions in x for D ≤ 7, the largest x is obtained when D=5.
#
# Find the value of D ≤ 1000 in minimal solutions of x for which the largest value of x is obtained.
from math import sqrt
from timeit import default_timer
from projecteuler import pell_eq
def is_square(n):
p = sqrt(n)
m = int(p)
if p == m:
return True
else:
return False
def main():
start = default_timer()
max_ = 0
max_d = -1
for i in range(2, 1001):
if not is_square(i):
# Solve the Diophantine equation x^2-D*y^2=1 (Pell equation)
x = pell_eq(i)
if x > max_:
max_ = x
max_d = i
end = default_timer()
print('Project Euler, Problem 66')
print('Answer: {}'.format(max_d))
print('Elapsed time: {:.9f} seconds'.format(end - start))
if __name__ == '__main__':
main()

53
Python/p067.py Normal file
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#!/usr/bin/python3
# By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.
#
# 3
# 7 4
# 2 4 6
# 8 5 9 3
#
# That is, 3 + 7 + 4 + 9 = 23.
# Find the maximum total from top to bottom in triangle.txt (right click and 'Save Link/Target As...'), a 15K text file containing a triangle
# with one-hundred rows.
#
# NOTE: This is a much more difficult version of Problem 18. It is not possible to try every route to solve this problem, as there are 299 altogether!
# If you could check one trillion (1012) routes every second it would take over twenty billion years to check them all.
# There is an efficient algorithm to solve it. ;o)
from timeit import default_timer
from projecteuler import find_max_path
def main():
start = default_timer()
try:
fp = open('triangle.txt', 'r')
except:
print('Error while opening file {}'.format('triangle.txt'))
exit(1)
triang = list()
for line in fp:
triang.append(line.strip('\n').split())
fp.close()
l = len(triang)
for i in range(l):
triang[i] = list(map(int, triang[i]))
# Use the function implemented in projecteuler.c to find the maximum path.
max_ = find_max_path(triang, 100)
end = default_timer()
print('Project Euler, Problem 67')
print('Answer: {}'.format(max_))
print('Elapsed time: {:.9f} seconds'.format(end - start))
if __name__ == '__main__':
main()

137
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#!/usr/bin/python3
# Consider the following "magic" 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine.
#
# 4
# \
# \
# 3
# / \
# / \
# 1------2------6
# /
# /
# 5
#
# Working clockwise, and starting from the group of three with the numerically lowest external node (4,3,2 in this example),
# each solution can be described uniquely. For example, the above solution can be described by the set: 4,3,2; 6,2,1; 5,1,3.
#
# It is possible to complete the ring with four different totals: 9, 10, 11, and 12. There are eight solutions in total.
#
# Total Solution Set
# 9 4,2,3; 5,3,1; 6,1,2
# 9 4,3,2; 6,2,1; 5,1,3
# 10 2,3,5; 4,5,1; 6,1,3
# 10 2,5,3; 6,3,1; 4,1,5
# 11 1,4,6; 3,6,2; 5,2,4
# 11 1,6,4; 5,4,2; 3,2,6
# 12 1,5,6; 2,6,4; 3,4,5
# 12 1,6,5; 3,5,4; 2,4,6
#
# By concatenating each group it is possible to form 9-digit strings; the maximum string for a 3-gon ring is 432621513.
#
# Using the numbers 1 to 10, and depending on arrangements, it is possible to form 16- and 17-digit strings. What is the maximum 16-digit string
# for a "magic" 5-gon ring?
#
# O
# \
# O
# / \ O
# / \ /
# O O
# / \ /
# O O--O---O
# \
# O
#
from itertools import permutations
from timeit import default_timer
# Function to evaluate the ring. The ring is represented as a vector of 2*n elements,
# the first n elements represent the external nodes of the ring, the next n elements
# represent the internal ring.
def eval_ring(ring, n):
for i in range(1, n):
# We need to start from the lowest external node, so if
# the first element in the vector is not the lowest of
# the first n elements (the external elements), the configuration
# is not a valid one.
if ring[i] < ring[0]:
return None
res = [0] * 3 * n
# Each group of three must have the same value.
magic_val = ring[0] + ring[n] + ring[n+1]
j = 0
for i in range(n):
# We need to find the maximum 16-digit string, this is
# possible only if the element "10" is used only once,
# i.e. if it's one of the external nodes.
if ring[n+i] == 10 or ring[n+(i+1)%n] == 10:
return None
# Check that the value of the current three-element group
# is the "magic" value.
val = ring[i] + ring[n+i] + ring[n+(i+1)%n]
if val != magic_val :
return None
# Save the current element group in the result string.
res[j] = ring[i]
res[j+1] = ring[n+i]
res[j+2] = ring[n+(i+1)%n]
j = j + 3
return res
def list_to_int(l):
if l == None:
return 0
res = 0
k = 0
for i in reversed(l):
res = res + i * 10 ** k
if i >= 10:
k = k + 2
else:
k = k + 1
return res
def main():
start = default_timer()
# Generate all possible permutations, for each one check if
# it's a possible solution for the ring and save the maximum
rings = list(permutations(list(range(1, 11))))
max_ = 0
n = None
for ring in rings:
eval_ = eval_ring(ring, 5);
# Convert the list into an integer number.
n = list_to_int(eval_)
if n > max_:
max_ = n
end = default_timer()
print('Project Euler, Problem 68')
print('Answer: {}'.format(max_))
print('Elapsed time: {:.9f} seconds'.format(end - start))
if __name__ == '__main__':
main()

55
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#!/usr/bin/python3
# Euler's Totient function, φ(n) [sometimes called the phi function], is used to determine the number of numbers less than n which are
# relatively prime to n. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, φ(9)=6.
#
# n Relatively Prime φ(n) n/φ(n)
# 2 1 1 2
# 3 1,2 2 1.5
# 4 1,3 2 2
# 5 1,2,3,4 4 1.25
# 6 1,5 2 3
# 7 1,2,3,4,5,6 6 1.1666...
# 8 1,3,5,7 4 2
# 9 1,2,4,5,7,8 6 1.5
# 10 1,3,7,9 4 2.5
#
# It can be seen that n=6 produces a maximum n/φ(n) for n ≤ 10.
#
# Find the value of n ≤ 1,000,000 for which n/φ(n) is a maximum.
from timeit import default_timer
from projecteuler import is_prime
def main():
start = default_timer()
N = 1000000
i = 1
res = 1
# Using Euler's formula, phi(n)=n*prod(1-1/p), where p are the distinct
# primes that divide n. So n/phi(n)=1/prod(1-1/p). To find the maximum
# value of this function, the denominator must be minimized. This happens
# when n has the most distinct small prime factor, i.e. to find the solution
# we need to multiply the smallest consecutive primes until the result is
# larger than 1000000.
while res < N:
i = i + 1
if is_prime(i):
res = res * i
# We need the previous value, because we want i<1000000
res = res // i
end = default_timer()
print('Project Euler, Problem 69')
print('Answer: {}'.format(res))
print('Elapsed time: {:.9f} seconds'.format(end - start))
if __name__ == '__main__':
main()

51
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#!/usr/bin/python3
# Euler's Totient function, φ(n) [sometimes called the phi function], is used to determine the number of positive numbers
# less than or equal to n which are relatively prime to n. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and
# relatively prime to nine, φ(9)=6.
# The number 1 is considered to be relatively prime to every positive number, so φ(1)=1.
#
# Interestingly, φ(87109)=79180, and it can be seen that 87109 is a permutation of 79180.
#
# Find the value of n, 1 < n < 10^7, for which φ(n) is a permutation of n and the ratio n/φ(n) produces a minimum.
from numpy import zeros
from timeit import default_timer
from projecteuler import sieve, is_semiprime, phi_semiprime
def main():
start = default_timer()
N = 10000000
n = -1
min_ = float('inf')
semi_p = False
primes = sieve(N)
for i in range(2, N):
# When n is prime, phi(n)=(n-1), so to minimize n/phi(n) we should
# use n prime. But n-1 can't be a permutation of n. The second best
# bet is to use semiprimes. For a semiprime n=p*q, phi(n)=(p-1)(q-1).
# So we check if a number is semiprime, if yes calculate phi, finally
# check if phi(n) is a permutation of n and update the minimum if it's
# smaller.
semi_p, a, b = is_semiprime(i, primes)
if semi_p == True:
p = phi_semiprime(i, a, b)
if ''.join(sorted(str(p))) == ''.join(sorted(str(i))) and i / p < min_:
n = i
min_ = i / p
end = default_timer()
print('Project Euler, Problem 70')
print('Answer: {}'.format(n))
print('Elapsed time: {:.9f} seconds'.format(end - start))
if __name__ == '__main__':
main()

121
Python/projecteuler.py
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@ -190,24 +190,137 @@ def is_pentagonal(n):
# d_(n+1)=(S-m_(n+1)^2)/d_n
# a_(n+1)=floor((sqrt(S)+m_(n+1))/d_(n+1))=floor((a_0+m_(n+1))/d_(n+1))
# if a_i=2*a_0, the algorithm ends.
def period_cf(n):
def build_sqrt_cont_fraction(i, l):
mn = 0
dn = 1
count = 0
a0 = floor(sqrt(n))
fraction = [0] * l
j = 0
a0 = floor(sqrt(i))
an = a0
fraction[j] = an
j = j + 1
while True:
mn1 = dn * an - mn
dn1 = (n - mn1 * mn1) // dn
dn1 = (i - mn1 * mn1)/ dn
an1 = floor((a0+mn1)/dn1)
mn = mn1
dn = dn1
an = an1
count = count + 1
fraction[j] = an
j = j + 1
if an == 2 * a0:
break
return count
fraction[j] = -1
return fraction, count
# Function to solve the Diophantine equation in the form x^2-Dy^2=1
# (Pell equation) using continued fractions.
def pell_eq(d):
# Find the continued fraction for sqrt(d).
fraction, period = build_sqrt_cont_fraction(d, 100)
# Calculate the first convergent of the continued fraction.
n1 = 0
n2 = 1
d1 = 1
d2 = 0
j = 0
n3 = fraction[j] * n2 + n1
d3 = fraction[j] * d2 + d1
j = j + 1
# Check if x=n, y=d solve the equation x^2-Dy^2=1.
sol = n3 * n3 - d * d3 * d3
if sol == 1:
return n3
# Until a solution is found, calculate the next convergent
# and check if x=n and y=d solve the equation.
while True:
n1 = n2
n2 = n3
d1 = d2
d2 = d3
n3 = fraction[j] * n2 + n1
d3 = fraction[j] * d2 + d1
sol = n3 * n3 - d * d3 * d3
if sol == 1:
return n3
j = j + 1
if fraction[j] == -1:
j = 1
# Function to check if a number is semiprime. Parameters include
# pointers to p and q to return the factors values and a list of
# primes.
def is_semiprime(n, primes):
# If n is prime, it's not semiprime.
if primes[n] == 1:
return False, -1, -1
# Check if n is semiprime and one of the factors is 2.
if n % 2 == 0:
if primes[n//2] == 1:
p = 2
q = n // 2
return True, p, q
else:
return False, -1, -1
# Check if n is semiprime and one of the factors is 3.
elif n % 3 == 0:
if primes[n//3] == 1:
p = 3
q = n // 3
return True, p, q
else:
return False, -1, -1
# Any number can have only one prime factor greater than its
# square root, so we can stop checking at this point.
limit = floor(sqrt(n)) + 1
# Every prime other than 2 and 3 is in the form 6k+1 or 6k-1.
# If I check all those value no prime factors of the number
# will be missed. For each of these possible primes, check if
# they are prime, then if the number is semiprime with using
# that factor.
for i in range(5, limit, 6):
if primes[i] == 1 and n % i == 0:
if primes[n//i] == 1:
p = i
q = n // i
return True, p, q
else:
return False, -1, -1
elif primes[i+2] == 1 and n % (i + 2) == 0:
if primes[n//(i+2)] == 1:
p = i + 2
q = n // (i + 2)
return True, p, q
else:
return False, -1, -1
return False, -1, -1
# If n=pq is semiprime, phi(n)=(p-1)(q-1)=pq-p-q+1=n-(p+4)+1
# if p!=q. If p=q (n is a square), phi(n)=n-p.
def phi_semiprime(n, p, q):
if p == q:
return n - p
else:
return n - (p + q) + 1

100
Python/triangle.txt Normal file
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@ -0,0 +1,100 @@
59
73 41
52 40 09
26 53 06 34
10 51 87 86 81
61 95 66 57 25 68
90 81 80 38 92 67 73
30 28 51 76 81 18 75 44
84 14 95 87 62 81 17 78 58
21 46 71 58 02 79 62 39 31 09
56 34 35 53 78 31 81 18 90 93 15
78 53 04 21 84 93 32 13 97 11 37 51
45 03 81 79 05 18 78 86 13 30 63 99 95
39 87 96 28 03 38 42 17 82 87 58 07 22 57
06 17 51 17 07 93 09 07 75 97 95 78 87 08 53
67 66 59 60 88 99 94 65 55 77 55 34 27 53 78 28
76 40 41 04 87 16 09 42 75 69 23 97 30 60 10 79 87
12 10 44 26 21 36 32 84 98 60 13 12 36 16 63 31 91 35
70 39 06 05 55 27 38 48 28 22 34 35 62 62 15 14 94 89 86
66 56 68 84 96 21 34 34 34 81 62 40 65 54 62 05 98 03 02 60
38 89 46 37 99 54 34 53 36 14 70 26 02 90 45 13 31 61 83 73 47
36 10 63 96 60 49 41 05 37 42 14 58 84 93 96 17 09 43 05 43 06 59
66 57 87 57 61 28 37 51 84 73 79 15 39 95 88 87 43 39 11 86 77 74 18
54 42 05 79 30 49 99 73 46 37 50 02 45 09 54 52 27 95 27 65 19 45 26 45
71 39 17 78 76 29 52 90 18 99 78 19 35 62 71 19 23 65 93 85 49 33 75 09 02
33 24 47 61 60 55 32 88 57 55 91 54 46 57 07 77 98 52 80 99 24 25 46 78 79 05
92 09 13 55 10 67 26 78 76 82 63 49 51 31 24 68 05 57 07 54 69 21 67 43 17 63 12
24 59 06 08 98 74 66 26 61 60 13 03 09 09 24 30 71 08 88 70 72 70 29 90 11 82 41 34
66 82 67 04 36 60 92 77 91 85 62 49 59 61 30 90 29 94 26 41 89 04 53 22 83 41 09 74 90
48 28 26 37 28 52 77 26 51 32 18 98 79 36 62 13 17 08 19 54 89 29 73 68 42 14 08 16 70 37
37 60 69 70 72 71 09 59 13 60 38 13 57 36 09 30 43 89 30 39 15 02 44 73 05 73 26 63 56 86 12
55 55 85 50 62 99 84 77 28 85 03 21 27 22 19 26 82 69 54 04 13 07 85 14 01 15 70 59 89 95 10 19
04 09 31 92 91 38 92 86 98 75 21 05 64 42 62 84 36 20 73 42 21 23 22 51 51 79 25 45 85 53 03 43 22
75 63 02 49 14 12 89 14 60 78 92 16 44 82 38 30 72 11 46 52 90 27 08 65 78 03 85 41 57 79 39 52 33 48
78 27 56 56 39 13 19 43 86 72 58 95 39 07 04 34 21 98 39 15 39 84 89 69 84 46 37 57 59 35 59 50 26 15 93
42 89 36 27 78 91 24 11 17 41 05 94 07 69 51 96 03 96 47 90 90 45 91 20 50 56 10 32 36 49 04 53 85 92 25 65
52 09 61 30 61 97 66 21 96 92 98 90 06 34 96 60 32 69 68 33 75 84 18 31 71 50 84 63 03 03 19 11 28 42 75 45 45
61 31 61 68 96 34 49 39 05 71 76 59 62 67 06 47 96 99 34 21 32 47 52 07 71 60 42 72 94 56 82 83 84 40 94 87 82 46
01 20 60 14 17 38 26 78 66 81 45 95 18 51 98 81 48 16 53 88 37 52 69 95 72 93 22 34 98 20 54 27 73 61 56 63 60 34 63
93 42 94 83 47 61 27 51 79 79 45 01 44 73 31 70 83 42 88 25 53 51 30 15 65 94 80 44 61 84 12 77 02 62 02 65 94 42 14 94
32 73 09 67 68 29 74 98 10 19 85 48 38 31 85 67 53 93 93 77 47 67 39 72 94 53 18 43 77 40 78 32 29 59 24 06 02 83 50 60 66
32 01 44 30 16 51 15 81 98 15 10 62 86 79 50 62 45 60 70 38 31 85 65 61 64 06 69 84 14 22 56 43 09 48 66 69 83 91 60 40 36 61
92 48 22 99 15 95 64 43 01 16 94 02 99 19 17 69 11 58 97 56 89 31 77 45 67 96 12 73 08 20 36 47 81 44 50 64 68 85 40 81 85 52 09
91 35 92 45 32 84 62 15 19 64 21 66 06 01 52 80 62 59 12 25 88 28 91 50 40 16 22 99 92 79 87 51 21 77 74 77 07 42 38 42 74 83 02 05
46 19 77 66 24 18 05 32 02 84 31 99 92 58 96 72 91 36 62 99 55 29 53 42 12 37 26 58 89 50 66 19 82 75 12 48 24 87 91 85 02 07 03 76 86
99 98 84 93 07 17 33 61 92 20 66 60 24 66 40 30 67 05 37 29 24 96 03 27 70 62 13 04 45 47 59 88 43 20 66 15 46 92 30 04 71 66 78 70 53 99
67 60 38 06 88 04 17 72 10 99 71 07 42 25 54 05 26 64 91 50 45 71 06 30 67 48 69 82 08 56 80 67 18 46 66 63 01 20 08 80 47 07 91 16 03 79 87
18 54 78 49 80 48 77 40 68 23 60 88 58 80 33 57 11 69 55 53 64 02 94 49 60 92 16 35 81 21 82 96 25 24 96 18 02 05 49 03 50 77 06 32 84 27 18 38
68 01 50 04 03 21 42 94 53 24 89 05 92 26 52 36 68 11 85 01 04 42 02 45 15 06 50 04 53 73 25 74 81 88 98 21 67 84 79 97 99 20 95 04 40 46 02 58 87
94 10 02 78 88 52 21 03 88 60 06 53 49 71 20 91 12 65 07 49 21 22 11 41 58 99 36 16 09 48 17 24 52 36 23 15 72 16 84 56 02 99 43 76 81 71 29 39 49 17
64 39 59 84 86 16 17 66 03 09 43 06 64 18 63 29 68 06 23 07 87 14 26 35 17 12 98 41 53 64 78 18 98 27 28 84 80 67 75 62 10 11 76 90 54 10 05 54 41 39 66
43 83 18 37 32 31 52 29 95 47 08 76 35 11 04 53 35 43 34 10 52 57 12 36 20 39 40 55 78 44 07 31 38 26 08 15 56 88 86 01 52 62 10 24 32 05 60 65 53 28 57 99
03 50 03 52 07 73 49 92 66 80 01 46 08 67 25 36 73 93 07 42 25 53 13 96 76 83 87 90 54 89 78 22 78 91 73 51 69 09 79 94 83 53 09 40 69 62 10 79 49 47 03 81 30
71 54 73 33 51 76 59 54 79 37 56 45 84 17 62 21 98 69 41 95 65 24 39 37 62 03 24 48 54 64 46 82 71 78 33 67 09 16 96 68 52 74 79 68 32 21 13 78 96 60 09 69 20 36
73 26 21 44 46 38 17 83 65 98 07 23 52 46 61 97 33 13 60 31 70 15 36 77 31 58 56 93 75 68 21 36 69 53 90 75 25 82 39 50 65 94 29 30 11 33 11 13 96 02 56 47 07 49 02
76 46 73 30 10 20 60 70 14 56 34 26 37 39 48 24 55 76 84 91 39 86 95 61 50 14 53 93 64 67 37 31 10 84 42 70 48 20 10 72 60 61 84 79 69 65 99 73 89 25 85 48 92 56 97 16
03 14 80 27 22 30 44 27 67 75 79 32 51 54 81 29 65 14 19 04 13 82 04 91 43 40 12 52 29 99 07 76 60 25 01 07 61 71 37 92 40 47 99 66 57 01 43 44 22 40 53 53 09 69 26 81 07
49 80 56 90 93 87 47 13 75 28 87 23 72 79 32 18 27 20 28 10 37 59 21 18 70 04 79 96 03 31 45 71 81 06 14 18 17 05 31 50 92 79 23 47 09 39 47 91 43 54 69 47 42 95 62 46 32 85
37 18 62 85 87 28 64 05 77 51 47 26 30 65 05 70 65 75 59 80 42 52 25 20 44 10 92 17 71 95 52 14 77 13 24 55 11 65 26 91 01 30 63 15 49 48 41 17 67 47 03 68 20 90 98 32 04 40 68
90 51 58 60 06 55 23 68 05 19 76 94 82 36 96 43 38 90 87 28 33 83 05 17 70 83 96 93 06 04 78 47 80 06 23 84 75 23 87 72 99 14 50 98 92 38 90 64 61 58 76 94 36 66 87 80 51 35 61 38
57 95 64 06 53 36 82 51 40 33 47 14 07 98 78 65 39 58 53 06 50 53 04 69 40 68 36 69 75 78 75 60 03 32 39 24 74 47 26 90 13 40 44 71 90 76 51 24 36 50 25 45 70 80 61 80 61 43 90 64 11
18 29 86 56 68 42 79 10 42 44 30 12 96 18 23 18 52 59 02 99 67 46 60 86 43 38 55 17 44 93 42 21 55 14 47 34 55 16 49 24 23 29 96 51 55 10 46 53 27 92 27 46 63 57 30 65 43 27 21 20 24 83
81 72 93 19 69 52 48 01 13 83 92 69 20 48 69 59 20 62 05 42 28 89 90 99 32 72 84 17 08 87 36 03 60 31 36 36 81 26 97 36 48 54 56 56 27 16 91 08 23 11 87 99 33 47 02 14 44 73 70 99 43 35 33
90 56 61 86 56 12 70 59 63 32 01 15 81 47 71 76 95 32 65 80 54 70 34 51 40 45 33 04 64 55 78 68 88 47 31 47 68 87 03 84 23 44 89 72 35 08 31 76 63 26 90 85 96 67 65 91 19 14 17 86 04 71 32 95
37 13 04 22 64 37 37 28 56 62 86 33 07 37 10 44 52 82 52 06 19 52 57 75 90 26 91 24 06 21 14 67 76 30 46 14 35 89 89 41 03 64 56 97 87 63 22 34 03 79 17 45 11 53 25 56 96 61 23 18 63 31 37 37 47
77 23 26 70 72 76 77 04 28 64 71 69 14 85 96 54 95 48 06 62 99 83 86 77 97 75 71 66 30 19 57 90 33 01 60 61 14 12 90 99 32 77 56 41 18 14 87 49 10 14 90 64 18 50 21 74 14 16 88 05 45 73 82 47 74 44
22 97 41 13 34 31 54 61 56 94 03 24 59 27 98 77 04 09 37 40 12 26 87 09 71 70 07 18 64 57 80 21 12 71 83 94 60 39 73 79 73 19 97 32 64 29 41 07 48 84 85 67 12 74 95 20 24 52 41 67 56 61 29 93 35 72 69
72 23 63 66 01 11 07 30 52 56 95 16 65 26 83 90 50 74 60 18 16 48 43 77 37 11 99 98 30 94 91 26 62 73 45 12 87 73 47 27 01 88 66 99 21 41 95 80 02 53 23 32 61 48 32 43 43 83 14 66 95 91 19 81 80 67 25 88
08 62 32 18 92 14 83 71 37 96 11 83 39 99 05 16 23 27 10 67 02 25 44 11 55 31 46 64 41 56 44 74 26 81 51 31 45 85 87 09 81 95 22 28 76 69 46 48 64 87 67 76 27 89 31 11 74 16 62 03 60 94 42 47 09 34 94 93 72
56 18 90 18 42 17 42 32 14 86 06 53 33 95 99 35 29 15 44 20 49 59 25 54 34 59 84 21 23 54 35 90 78 16 93 13 37 88 54 19 86 67 68 55 66 84 65 42 98 37 87 56 33 28 58 38 28 38 66 27 52 21 81 15 08 22 97 32 85 27
91 53 40 28 13 34 91 25 01 63 50 37 22 49 71 58 32 28 30 18 68 94 23 83 63 62 94 76 80 41 90 22 82 52 29 12 18 56 10 08 35 14 37 57 23 65 67 40 72 39 93 39 70 89 40 34 07 46 94 22 20 05 53 64 56 30 05 56 61 88 27
23 95 11 12 37 69 68 24 66 10 87 70 43 50 75 07 62 41 83 58 95 93 89 79 45 39 02 22 05 22 95 43 62 11 68 29 17 40 26 44 25 71 87 16 70 85 19 25 59 94 90 41 41 80 61 70 55 60 84 33 95 76 42 63 15 09 03 40 38 12 03 32
09 84 56 80 61 55 85 97 16 94 82 94 98 57 84 30 84 48 93 90 71 05 95 90 73 17 30 98 40 64 65 89 07 79 09 19 56 36 42 30 23 69 73 72 07 05 27 61 24 31 43 48 71 84 21 28 26 65 65 59 65 74 77 20 10 81 61 84 95 08 52 23 70
47 81 28 09 98 51 67 64 35 51 59 36 92 82 77 65 80 24 72 53 22 07 27 10 21 28 30 22 48 82 80 48 56 20 14 43 18 25 50 95 90 31 77 08 09 48 44 80 90 22 93 45 82 17 13 96 25 26 08 73 34 99 06 49 24 06 83 51 40 14 15 10 25 01
54 25 10 81 30 64 24 74 75 80 36 75 82 60 22 69 72 91 45 67 03 62 79 54 89 74 44 83 64 96 66 73 44 30 74 50 37 05 09 97 70 01 60 46 37 91 39 75 75 18 58 52 72 78 51 81 86 52 08 97 01 46 43 66 98 62 81 18 70 93 73 08 32 46 34
96 80 82 07 59 71 92 53 19 20 88 66 03 26 26 10 24 27 50 82 94 73 63 08 51 33 22 45 19 13 58 33 90 15 22 50 36 13 55 06 35 47 82 52 33 61 36 27 28 46 98 14 73 20 73 32 16 26 80 53 47 66 76 38 94 45 02 01 22 52 47 96 64 58 52 39
88 46 23 39 74 63 81 64 20 90 33 33 76 55 58 26 10 46 42 26 74 74 12 83 32 43 09 02 73 55 86 54 85 34 28 23 29 79 91 62 47 41 82 87 99 22 48 90 20 05 96 75 95 04 43 28 81 39 81 01 28 42 78 25 39 77 90 57 58 98 17 36 73 22 63 74 51
29 39 74 94 95 78 64 24 38 86 63 87 93 06 70 92 22 16 80 64 29 52 20 27 23 50 14 13 87 15 72 96 81 22 08 49 72 30 70 24 79 31 16 64 59 21 89 34 96 91 48 76 43 53 88 01 57 80 23 81 90 79 58 01 80 87 17 99 86 90 72 63 32 69 14 28 88 69
37 17 71 95 56 93 71 35 43 45 04 98 92 94 84 96 11 30 31 27 31 60 92 03 48 05 98 91 86 94 35 90 90 08 48 19 33 28 68 37 59 26 65 96 50 68 22 07 09 49 34 31 77 49 43 06 75 17 81 87 61 79 52 26 27 72 29 50 07 98 86 01 17 10 46 64 24 18 56
51 30 25 94 88 85 79 91 40 33 63 84 49 67 98 92 15 26 75 19 82 05 18 78 65 93 61 48 91 43 59 41 70 51 22 15 92 81 67 91 46 98 11 11 65 31 66 10 98 65 83 21 05 56 05 98 73 67 46 74 69 34 08 30 05 52 07 98 32 95 30 94 65 50 24 63 28 81 99 57
19 23 61 36 09 89 71 98 65 17 30 29 89 26 79 74 94 11 44 48 97 54 81 55 39 66 69 45 28 47 13 86 15 76 74 70 84 32 36 33 79 20 78 14 41 47 89 28 81 05 99 66 81 86 38 26 06 25 13 60 54 55 23 53 27 05 89 25 23 11 13 54 59 54 56 34 16 24 53 44 06
13 40 57 72 21 15 60 08 04 19 11 98 34 45 09 97 86 71 03 15 56 19 15 44 97 31 90 04 87 87 76 08 12 30 24 62 84 28 12 85 82 53 99 52 13 94 06 65 97 86 09 50 94 68 69 74 30 67 87 94 63 07 78 27 80 36 69 41 06 92 32 78 37 82 30 05 18 87 99 72 19 99
44 20 55 77 69 91 27 31 28 81 80 27 02 07 97 23 95 98 12 25 75 29 47 71 07 47 78 39 41 59 27 76 13 15 66 61 68 35 69 86 16 53 67 63 99 85 41 56 08 28 33 40 94 76 90 85 31 70 24 65 84 65 99 82 19 25 54 37 21 46 33 02 52 99 51 33 26 04 87 02 08 18 96
54 42 61 45 91 06 64 79 80 82 32 16 83 63 42 49 19 78 65 97 40 42 14 61 49 34 04 18 25 98 59 30 82 72 26 88 54 36 21 75 03 88 99 53 46 51 55 78 22 94 34 40 68 87 84 25 30 76 25 08 92 84 42 61 40 38 09 99 40 23 29 39 46 55 10 90 35 84 56 70 63 23 91 39
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92 92 48 84 59 98 31 53 23 27 15 22 79 95 24 76 05 79 16 93 97 89 38 89 42 83 02 88 94 95 82 21 01 97 48 39 31 78 09 65 50 56 97 61 01 07 65 27 21 23 14 15 80 97 44 78 49 35 33 45 81 74 34 05 31 57 09 38 94 07 69 54 69 32 65 68 46 68 78 90 24 28 49 51 45 86 35
41 63 89 76 87 31 86 09 46 14 87 82 22 29 47 16 13 10 70 72 82 95 48 64 58 43 13 75 42 69 21 12 67 13 64 85 58 23 98 09 37 76 05 22 31 12 66 50 29 99 86 72 45 25 10 28 19 06 90 43 29 31 67 79 46 25 74 14 97 35 76 37 65 46 23 82 06 22 30 76 93 66 94 17 96 13 20 72
63 40 78 08 52 09 90 41 70 28 36 14 46 44 85 96 24 52 58 15 87 37 05 98 99 39 13 61 76 38 44 99 83 74 90 22 53 80 56 98 30 51 63 39 44 30 91 91 04 22 27 73 17 35 53 18 35 45 54 56 27 78 48 13 69 36 44 38 71 25 30 56 15 22 73 43 32 69 59 25 93 83 45 11 34 94 44 39 92
12 36 56 88 13 96 16 12 55 54 11 47 19 78 17 17 68 81 77 51 42 55 99 85 66 27 81 79 93 42 65 61 69 74 14 01 18 56 12 01 58 37 91 22 42 66 83 25 19 04 96 41 25 45 18 69 96 88 36 93 10 12 98 32 44 83 83 04 72 91 04 27 73 07 34 37 71 60 59 31 01 54 54 44 96 93 83 36 04 45
30 18 22 20 42 96 65 79 17 41 55 69 94 81 29 80 91 31 85 25 47 26 43 49 02 99 34 67 99 76 16 14 15 93 08 32 99 44 61 77 67 50 43 55 87 55 53 72 17 46 62 25 50 99 73 05 93 48 17 31 70 80 59 09 44 59 45 13 74 66 58 94 87 73 16 14 85 38 74 99 64 23 79 28 71 42 20 37 82 31 23
51 96 39 65 46 71 56 13 29 68 53 86 45 33 51 49 12 91 21 21 76 85 02 17 98 15 46 12 60 21 88 30 92 83 44 59 42 50 27 88 46 86 94 73 45 54 23 24 14 10 94 21 20 34 23 51 04 83 99 75 90 63 60 16 22 33 83 70 11 32 10 50 29 30 83 46 11 05 31 17 86 42 49 01 44 63 28 60 07 78 95 40
44 61 89 59 04 49 51 27 69 71 46 76 44 04 09 34 56 39 15 06 94 91 75 90 65 27 56 23 74 06 23 33 36 69 14 39 05 34 35 57 33 22 76 46 56 10 61 65 98 09 16 69 04 62 65 18 99 76 49 18 72 66 73 83 82 40 76 31 89 91 27 88 17 35 41 35 32 51 32 67 52 68 74 85 80 57 07 11 62 66 47 22 67
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73 81 14 13 17 20 11 09 01 83 08 85 91 70 84 63 62 77 37 07 47 01 59 95 39 69 39 21 99 09 87 02 97 16 92 36 74 71 90 66 33 73 73 75 52 91 11 12 26 53 05 26 26 48 61 50 90 65 01 87 42 47 74 35 22 73 24 26 56 70 52 05 48 41 31 18 83 27 21 39 80 85 26 08 44 02 71 07 63 22 05 52 19 08 20
17 25 21 11 72 93 33 49 64 23 53 82 03 13 91 65 85 02 40 05 42 31 77 42 05 36 06 54 04 58 07 76 87 83 25 57 66 12 74 33 85 37 74 32 20 69 03 97 91 68 82 44 19 14 89 28 85 85 80 53 34 87 58 98 88 78 48 65 98 40 11 57 10 67 70 81 60 79 74 72 97 59 79 47 30 20 54 80 89 91 14 05 33 36 79 39
60 85 59 39 60 07 57 76 77 92 06 35 15 72 23 41 45 52 95 18 64 79 86 53 56 31 69 11 91 31 84 50 44 82 22 81 41 40 30 42 30 91 48 94 74 76 64 58 74 25 96 57 14 19 03 99 28 83 15 75 99 01 89 85 79 50 03 95 32 67 44 08 07 41 62 64 29 20 14 76 26 55 48 71 69 66 19 72 44 25 14 01 48 74 12 98 07
64 66 84 24 18 16 27 48 20 14 47 69 30 86 48 40 23 16 61 21 51 50 26 47 35 33 91 28 78 64 43 68 04 79 51 08 19 60 52 95 06 68 46 86 35 97 27 58 04 65 30 58 99 12 12 75 91 39 50 31 42 64 70 04 46 07 98 73 98 93 37 89 77 91 64 71 64 65 66 21 78 62 81 74 42 20 83 70 73 95 78 45 92 27 34 53 71 15
30 11 85 31 34 71 13 48 05 14 44 03 19 67 23 73 19 57 06 90 94 72 57 69 81 62 59 68 88 57 55 69 49 13 07 87 97 80 89 05 71 05 05 26 38 40 16 62 45 99 18 38 98 24 21 26 62 74 69 04 85 57 77 35 58 67 91 79 79 57 86 28 66 34 72 51 76 78 36 95 63 90 08 78 47 63 45 31 22 70 52 48 79 94 15 77 61 67 68
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