The sieve of Eratosthenes handles primes up to an upper bound of 100,000,000. The number of primes is 5,761,455. Below are a few examples runs of the app. I have also created C source code for trial division and other factoring algorithms that use Professor Emeritus Arjen K. Lenstra’s Free Large Integer Package also known as lip.
Enter n (<= 18) below:
5
Enter k (<= 18) below:
0
1 1
Enter n (<= 18) below:
5
Enter k (<= 18) below:
1
5 5
Enter n (<= 18) below:
5
Enter k (<= 18) below:
2
10 10
Enter n (<= 18) below:
0
Enter n (<= 18) below:
0
Pascal's Triangle:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
1 11 55 165 330 462 462 330 165 55 11 1
1 12 66 220 495 792 924 792 495 220 66 12 1
1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1
1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1
1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1
1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1
1 17 136 680 2380 6188 12376 19448 24310 24310 19448 12376 6188 2380 680 136 17 1
1 18 153 816 3060 8568 18564 31824 43758 48620 43758 31824 18564 8568 3060 816 153 18 1
C:\Users\james\source\repos\BinomialCoefficeint\Debug\BinomialCoefficeint.exe (process 40028) exited with code 0.
Press any key to close this window . . .
// BinomialCoefficient.c (c) Monday, June 24, 2024
// by James Pate Williams, Jr. BA, BS, MSwE, PhD
#include <stdio.h>
#include <stdlib.h>
typedef long long ll;
ll** Binomial(ll n)
{
ll** C = (ll**)calloc(n + 1, sizeof(ll*));
if (C == NULL)
exit(-1);
for (int i = 0; i < n + 1; i++)
{
C[i] = (ll*)calloc(n + 1, sizeof(ll));
if (C[i] == NULL)
exit(-1);
}
if (n >= 0)
{
C[0][0] = 1;
}
if (n >= 1)
{
C[1][0] = 1;
C[1][1] = 1;
}
if (n >= 2)
{
for (int i = 2; i <= n; i++)
{
for (int j = 2; j <= n; j++)
{
C[i][j] = C[i - 1][j - 1] + C[i - 1][j];
}
}
}
return C;
}
ll Factorial(ll n)
{
ll fact = 1;
if (n > 1)
{
for (int i = 2; i <= n; i++)
fact = i * fact;
}
return fact;
}
ll BC(ll n, ll k)
{
return Factorial(n) / (Factorial(n - k) * Factorial(k));
}
int main()
{
int i = 0, j = 0;
ll** C = Binomial(20);
while (1)
{
char buffer[256] = { '\0' };
printf_s("Enter n (<= 18) below:\n");
scanf_s("%s", buffer, sizeof(buffer));
printf_s("\n");
ll n = atoll(buffer);
if (n == 0)
break;
printf_s("Enter k (<= 18) below:\n");
scanf_s("%s", buffer, sizeof(buffer));
printf_s("\n");
ll k = atoll(buffer);
printf_s("%lld\t%lld\n\n", C[n + 2][k + 2], BC(n, k));
}
printf_s("Pascal's Triangle:\n\n");
for (i = 2; i <= 20; i++)
{
for (j = 2; j <= 20; j++)
if (C[i][j] != 0)
printf_s("%5lld ", C[i][j]);
printf_s("\n");
}
for (i = 0; i <= 20; i++)
free(C[i]);
free(C);
}
The object of this C Win32 application is to find a multiple of 9 with its digits summing to a multiple of 9 also. The first column below is a multiple of 9 whose digits sum to 9 also. The second column is the sum of digits found in the column one number. The last column is the first column divided by 9.
Enter PRNG seed:
1
Enter number of bits (4 to 16):
4
9 9 1
Enter number of bits (4 to 16):
5
27 9 3
Enter number of bits (4 to 16):
6
45 9 5
Enter number of bits (4 to 16):
7
117 9 13
Enter number of bits (4 to 16):
8
252 9 28
Enter number of bits (4 to 16):
0
C:\Users\james\source\repos\CProductOf9Console\Debug\CProductOf9Console.exe (process 23280) exited with code 0.
Press any key to close this window . . .
Enter PRNG seed:
1
Enter number of bits (4 to 16):
9
369 18 41
Enter number of bits (4 to 16):
10
846 18 94
Enter number of bits (4 to 16):
11
1080 9 120
Enter number of bits (4 to 16):
12
3015 9 335
Enter number of bits (4 to 16):
13
5040 9 560
Enter number of bits (4 to 16):
14
10350 9 1150
Enter number of bits (4 to 16):
15
30870 18 3430
Enter number of bits (4 to 16):
16
57798 36 6422
Enter number of bits (4 to 16):
0
// CProductOf9Console.c (c) Sunday, June 23, 2024
// by James Pate Williams, Jr., BA, BS, MSwE, PhD
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
char nextStr[256], numbStr[256];
void ConvertToString(int number, int radix)
{
int i = 0;
while (number > 0)
{
nextStr[i++] = (char)(number % radix + '0');
number /= radix;
}
nextStr[i++] = '\0';
_strrev(nextStr);
}
int Sum(int next)
{
long sum = 0;
ConvertToString(next, 10);
for (int i = 0; i < (int)strlen(nextStr); i++)
sum += (long)nextStr[i] - '0';
if (sum % 9 == 0 && sum != 0)
return sum;
return -1;
}
long GetNext(int numBits, int* next)
{
long hi = 0, lo = 0, nine = 0;
nextStr[0] = '\0';
numbStr[0] = '\0';
if (numBits == 4)
{
while (1)
{
*next = 9 * (long)(rand() % 16);
if (*next != 0 && *next >= 8 && *next < 16)
{
nine = Sum(*next);
if (nine % 9 == 0)
return nine;
}
}
}
else if (numBits == 5)
{
while (1)
{
*next = 9 * (long)(rand() % 32);
if (*next >= 16 && *next < 32)
{
nine = Sum(*next);
if (nine % 9 == 0)
return nine;
}
}
}
else if (numBits == 6)
{
while (1)
{
*next = 9 * (long)(rand() % 64);
if (*next >= 32 && *next < 64)
{
nine = Sum(*next);
if (nine % 9 == 0)
return nine;
}
}
}
else if (numBits == 7)
{
while (1)
{
*next = 9 * (long)(rand() % 128);
if (*next >= 64 && *next < 128)
{
nine = Sum(*next);
if (nine % 9 == 0)
return nine;
}
}
}
else if (numBits == 8)
{
while (1)
{
*next = 9 * (long)(rand() % 256);
if (*next >= 128 && *next < 256)
{
nine = Sum(*next);
if (nine % 9 == 0)
return nine;
}
}
}
else if (numBits == 9)
{
while (1)
{
*next = 9 * (long)(rand() % 512);
if (*next >= 256 && *next < 512)
{
nine = Sum(*next);
if (nine % 9 == 0)
return nine;
}
}
}
else if (numBits == 10)
{
while (1)
{
*next = 9 * (long)(rand() % 1024);
if (*next >= 512 && *next < 1024)
{
nine = Sum(*next);
if (nine % 9 == 0)
return nine;
}
}
}
else if (numBits == 11)
{
while (1)
{
*next = 9 * (long)(rand() % 2048);
if (*next >= 1024 && *next < 2048)
{
nine = Sum(*next);
if (nine % 9 == 0)
return nine;
}
}
}
else if (numBits == 12)
{
while (1)
{
*next = 9 * (long)(rand() % 4096);
if (*next >= 2048 && *next < 4096)
{
nine = Sum(*next);
if (nine % 9 == 0)
return nine;
}
}
}
else if (numBits == 13)
{
while (1)
{
*next = 9 * (long)(rand() % 8192);
if (*next >= 4096 && *next < 8192)
{
nine = Sum(*next);
if (nine % 9 == 0)
return nine;
}
}
}
else if (numBits == 14)
{
while (1)
{
*next = 9 * (long)(rand() % 16384);
if (*next >= 8192 && *next < 16384)
{
nine = Sum(*next);
if (nine % 9 == 0)
return nine;
}
}
}
else if (numBits == 15)
{
while (1)
{
*next = 9 * (long)(rand() % 32768);
if (*next >= 16384 && *next < 32768)
{
nine = Sum(*next);
if (nine % 9 == 0)
return nine;
}
}
}
else if (numBits == 16)
{
while (1)
{
*next = 9 * (long)(rand() % 65536);
if (*next >= 32768 && *next < 65536)
{
nine = Sum(*next);
if (nine % 9 == 0)
return nine;
}
}
}
return -1;
}
int main()
{
char buffer[256] = { '\0' };
long seed = 0;
printf_s("Enter PRNG seed:\n");
scanf_s("%s", buffer, sizeof(buffer));
seed = atol(buffer);
srand((unsigned int)seed);
while (1)
{
int next = 0, nine = 0, numberBits = 0;
printf_s("Enter number of bits (4 to 16):\n");
scanf_s("%s", buffer, sizeof(buffer));
numberBits = atol(buffer);
if (numberBits == 0)
break;
if (numberBits < 4 || numberBits > 16)
{
printf_s("illegal number of bits must >= 4 and <= 16\n");
continue;
}
nine = GetNext(numberBits, &next);
if (nine == -1)
{
printf_s("illegal result, try again\n");
continue;
}
printf_s("%5ld\t%5ld\t%5ld\n", next, nine, next / 9);
}
return 0;
}
FreeLIP is a free large integer package solely created by Professor Emeritus Arjen K. Lenstra of the Number Field Sieve fame. He developed FreeLIP while he was an employee of AT&T – Lucent in the late 1980s. His copyright notice in the header file, lip.h, states copyright from 1989 to 1997. I have been using this excellent number theoretical package since the late 1990s. See the paper by Donald E. Knuth and Thomas J. Buckholtz for the formula for Tangent Numbers. I can’t remember where I got the Euler Numbers recurrence relation. I wrote a C# application in 2015 for computing Euler Numbers. The code below is in the vanilla C computer language. Excellent resources for the Euler and tangent numbers also known as zag numbers are:
For the last week or so I have been working my way through Chapter 3 The Solution of Nonlinear Equations found in the textbook “Numerical Analysis: An Algorithmic Approach” by S. D. Conte and Carl de Boor. I also used some C source code from “A Numerical Library in C for Scientists and Engineers” by H. T. Lau, PhD. I implemented twenty examples and exercises from the previously mentioned chapter.
Numerical Explorations 