Category: C Applications

Blog Entry © Sunday, July 20, 2025, by James Pate Williams, Jr. Example and Two Exercises from a Numerical Analysis Textbook

Happy 59th Anniversary of the historic moon landing by Neil Armstrong, Buzz Aldrin, and Michael Collins in Apolla 11.

Blog Entry July 20 2025Download

Newtonian Gravity and Cosmology © Friday, July 18, 2025, by James Pate Williams, Jr. in Conjunction with the Microsoft Copilot AI

Newtonian Gravity and CosmologyDownload

Some General Relativity Mathematics by James Pate Williams, Jr. (c) July 13, 2025

Some Relativity MathDownload
// SomeRelativityMath.cpp : Defines the entry point for the application.
// https://arxiv.org/pdf/2506.09946

#include "pch.h"
#include "framework.h"
#include "SomeRelativityMath.h"
#include <stdio.h>
#include <vector>

#define MAX_LOADSTRING 100

typedef struct tagPoint2d
{
    double x, y;
} POINT2D, * PPOINT2D;

// Global Variables:
HINSTANCE hInst;                                // current instance
WCHAR szTitle[MAX_LOADSTRING];                  // The title bar text
WCHAR szWindowClass[MAX_LOADSTRING];            // the main window class name
char text[8192];
std::vector<POINT2D> points1, points2;
std::vector<POINT2D> points3, points4;

// Forward declarations of functions included in this code module:
ATOM                MyRegisterClass(HINSTANCE hInstance);
BOOL                InitInstance(HINSTANCE, int);
LRESULT CALLBACK    WndProc(HWND, UINT, WPARAM, LPARAM);
INT_PTR CALLBACK    About(HWND, UINT, WPARAM, LPARAM);
INT_PTR CALLBACK    DrawGraph(HWND, UINT, WPARAM, LPARAM);

int APIENTRY wWinMain(_In_ HINSTANCE hInstance,
                     _In_opt_ HINSTANCE hPrevInstance,
                     _In_ LPWSTR    lpCmdLine,
                     _In_ int       nCmdShow)
{
    UNREFERENCED_PARAMETER(hPrevInstance);
    UNREFERENCED_PARAMETER(lpCmdLine);

    // TODO: Place code here.

    // Initialize global strings
    LoadStringW(hInstance, IDS_APP_TITLE, szTitle, MAX_LOADSTRING);
    LoadStringW(hInstance, IDC_SOMERELATIVITYMATH, szWindowClass, MAX_LOADSTRING);
    MyRegisterClass(hInstance);

    // Perform application initialization:
    if (!InitInstance (hInstance, nCmdShow))
    {
        return FALSE;
    }

    HACCEL hAccelTable = LoadAccelerators(hInstance, MAKEINTRESOURCE(IDC_SOMERELATIVITYMATH));

    MSG msg;

    // Main message loop:
    while (GetMessage(&msg, nullptr, 0, 0))
    {
        if (!TranslateAccelerator(msg.hwnd, hAccelTable, &msg))
        {
            TranslateMessage(&msg);
            DispatchMessage(&msg);
        }
    }

    return (int) msg.wParam;
}

//
//  FUNCTION: MyRegisterClass()
//
//  PURPOSE: Registers the window class.
//
ATOM MyRegisterClass(HINSTANCE hInstance)
{
    WNDCLASSEXW wcex = { 0 };

    wcex.cbSize = sizeof(WNDCLASSEX);

    wcex.style          = CS_HREDRAW | CS_VREDRAW;
    wcex.lpfnWndProc    = WndProc;
    wcex.cbClsExtra     = 0;
    wcex.cbWndExtra     = 0;
    wcex.hInstance      = hInstance;
    wcex.hIcon          = LoadIcon(hInstance, MAKEINTRESOURCE(IDI_SOMERELATIVITYMATH));
    wcex.hCursor        = LoadCursor(nullptr, IDC_ARROW);
    wcex.hbrBackground  = (HBRUSH)(COLOR_WINDOW+1);
    wcex.lpszMenuName   = MAKEINTRESOURCEW(IDC_SOMERELATIVITYMATH);
    wcex.lpszClassName  = szWindowClass;
    wcex.hIconSm        = LoadIcon(wcex.hInstance, MAKEINTRESOURCE(IDI_SMALL));

    return RegisterClassExW(&wcex);
}

//
//   FUNCTION: InitInstance(HINSTANCE, int)
//
//   PURPOSE: Saves instance handle and creates main window
//
//   COMMENTS:
//
//        In this function, we save the instance handle in a global variable and
//        create and display the main program window.
//
BOOL InitInstance(HINSTANCE hInstance, int nCmdShow)
{
   hInst = hInstance; // Store instance handle in our global variable

   HWND hWnd = CreateWindowW(szWindowClass, szTitle, WS_OVERLAPPEDWINDOW,
      CW_USEDEFAULT, 0, CW_USEDEFAULT, 0, nullptr, nullptr, hInstance, nullptr);

   if (!hWnd)
   {
      return FALSE;
   }

   ShowWindow(hWnd, nCmdShow);
   UpdateWindow(hWnd);

   return TRUE;
}

static void FindMinMax(
    double& xMin, double& xMax,
    double& yMin, double& yMax)
{
    // uses global 2D double points structure

    xMin = yMin = DBL_MAX;
    xMax = yMax = DBL_MIN;

    for (size_t i = 0; i < points1.size(); i++)
    {
        POINT2D pt = points1[i];
        double x = pt.x;
        double y = pt.y;

        if (x < xMin)
            xMin = x;
        if (x > xMax)
            xMax = x;
        if (y < yMin)
            yMin = y;
        if (y > yMax)
            yMax = y;
    }

    for (size_t i = 0; i < points2.size(); i++)
    {
        POINT2D pt = points2[i];
        double x = pt.x;
        double y = pt.y;

        if (x < xMin)
            xMin = x;
        if (x > xMax)
            xMax = x;
        if (y < yMin)
            yMin = y;
        if (y > yMax)
            yMax = y;
    }

    for (size_t i = 0; i < points3.size(); i++)
    {
        POINT2D pt = points3[i];
        double x = pt.x;
        double y = pt.y;

        if (x < xMin)
            xMin = x;
        if (x > xMax)
            xMax = x;
        if (y < yMin)
            yMin = y;
        if (y > yMax)
            yMax = y;
    }

    for (size_t i = 0; i < points4.size(); i++)
    {
        POINT2D pt = points4[i];
        double x = pt.x;
        double y = pt.y;

        if (x < xMin)
            xMin = x;
        if (x > xMax)
            xMax = x;
        if (y < yMin)
            yMin = y;
        if (y > yMax)
            yMax = y;
    }
}

static void DrawFormattedText(HDC hdc, char text[], RECT rect)
{
    // Draw the text with formatting options
    DrawTextA(hdc, text, -1, &rect, DT_SINGLELINE | DT_NOCLIP);
}

//
//  FUNCTION: WndProc(HWND, UINT, WPARAM, LPARAM)
//
//  PURPOSE: Processes messages for the main window.
//
//  WM_COMMAND  - process the application menu
//  WM_PAINT    - Paint the main window
//  WM_DESTROY  - post a quit message and return
//
//
LRESULT CALLBACK WndProc(HWND hWnd, UINT message, WPARAM wParam, LPARAM lParam)
{
    switch (message)
    {
    case WM_CREATE:
    {
        double P1 = 0, P2 = 0, P3 = 0, P4 = 0;
        int numberPts = 1024;
        double x0 = 1, h = 0.5 / numberPts, x = x0;

        points1.resize(numberPts);
        points2.resize(numberPts);
        points3.resize(numberPts);
        points4.resize(numberPts);

        for (int i = 0; i < numberPts; i++)
        {
            P1 = x + 1.0;
            P2 = 3.0 * x * x + 4.0 * x + 2.0;
            P3 = 15.0 * x * x * x + 16.0 * x * x + 6.0 * x;
            P4 = 105.0 * x * x * x * x + 156.0 * x * x * x +
                94.0 * x * x + 24.0 * x;
            POINT2D pt = { 0 };
            pt.x = x;
            pt.y = log(P1);
            points1[i] = pt;
            pt.y = log(P2);
            points2[i] = pt;
            pt.y = log(P3);
            points3[i] = pt;
            pt.y = log(P4);
            points4[i] = pt;
            x += h;
        }
        break;
    }
    case WM_COMMAND:
        {
            int wmId = LOWORD(wParam);
            // Parse the menu selections:
            switch (wmId)
            {
            case IDM_ABOUT:
                DialogBox(hInst, MAKEINTRESOURCE(IDD_ABOUTBOX), hWnd, About);
                break;
            case IDM_EXIT:
                DestroyWindow(hWnd);
                break;
            default:
                return DefWindowProc(hWnd, message, wParam, lParam);
            }
        }
        break;
    case WM_PAINT:
    {
        double h = 0, pi = 0, plm = 0, theta = 0;
        double xMax = 0, xMin = 0, yMax = 0, yMin = 0;
        FindMinMax(xMin, xMax, yMin, yMax);
        float xSpan = (float)(xMax - xMin);
        float ySpan = (float)(yMax - yMin);
        RECT rect = { };
        GetClientRect(hWnd, &rect);
        float width = (float)(rect.right - rect.left + 1);
        float height = (float)(rect.bottom - rect.top - 32 + 1);
        float sx0 = 2.0f * width / 16.0f;
        float sx1 = 14.0f * width / 16.0f;
        float sy0 = 2.0f * height / 16.0f;
        float sy1 = 14.0f * height / 16.0f;
        float deltaX = xSpan / 8.0f;
        float deltaY = ySpan / 8.0f;
        float xSlope = (sx1 - sx0) / xSpan;
        float xInter = (float)(sx0 - xSlope * xMin);
        float ySlope = (sy0 - sy1) / ySpan;
        float yInter = (float)(sy0 - ySlope * yMax);
        float px = 0, py = 0, sx = 0, sy = 0;
        PAINTSTRUCT ps;
        POINT wPt = { };
        HDC hdc = BeginPaint(hWnd, &ps);
        int i = 0;
        float x = (float)xMin;
        float y = (float)yMax;
        px = x;
        py = y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        MoveToEx(hdc, (int)sx, (int)sy0, &wPt);
        char buffer[128] = { };

        while (i <= 8)
        {
            sx = xSlope * x + xInter;
            wPt.x = wPt.y = 0;
            MoveToEx(hdc, (int)sx, (int)sy0, &wPt);
            LineTo(hdc, (int)sx, (int)sy1);

            sprintf_s(buffer, "%3.2f", x);
            SIZE size = { };
            GetTextExtentPoint32A(
                hdc,
                buffer,
                strlen(buffer),
                &size);
            RECT textRect = { };
            textRect.left = (long)(sx - size.cx / 2.0f);
            textRect.right = (long)(sx + size.cx / 2.0f);
            textRect.top = (long)sy1;
            textRect.bottom = (long)(sy1 + size.cy / 2.0f);
            DrawFormattedText(hdc, buffer, textRect);
            x += deltaX;
            i++;
        }

        i = 0;
        y = (float)yMin;

        while (i <= 8)
        {
            sy = ySlope * y + yInter;
            wPt.x = wPt.y = 0;
            MoveToEx(hdc, (int)sx0, (int)sy, &wPt);
            LineTo(hdc, (int)sx, (int)sy);

            if (i != 0)
            {
                sprintf_s(buffer, "%3.1f", y);
                SIZE size = { };
                GetTextExtentPoint32A(
                    hdc,
                    buffer,
                    strlen(buffer),
                    &size);
                RECT textRect = { };
                textRect.left = (long)(sx0 - size.cx - size.cx / 5.0f);
                textRect.right = (long)(sx0 - size.cx / 2.0f);
                textRect.top = (long)(sy - size.cy / 2.0f);
                textRect.bottom = (long)(sy + size.cy / 2.0f);
                DrawFormattedText(hdc, buffer, textRect);
            }

            y += deltaY;
            i++;
        }

        HGDIOBJ rPenNew = NULL;
        HGDIOBJ hPenOld = NULL;
        rPenNew = CreatePen(PS_SOLID, 2, RGB(255, 0, 0));
        hPenOld = SelectObject(hdc, rPenNew);

        px = (float)points1[0].x;
        py = (float)points1[0].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        for (size_t j = 1; j < points1.size(); j++)
        {
            px = (float)points1[j].x;
            py = (float)points1[j].y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            LineTo(hdc, (int)sx, (int)sy);
        }

        SelectObject(hdc, hPenOld);
        DeleteObject(rPenNew);

        HGDIOBJ gPenNew = CreatePen(PS_SOLID, 2, RGB(0, 255, 0));
        hPenOld = SelectObject(hdc, gPenNew);

        px = (float)points2[0].x;
        py = (float)points2[0].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        for (size_t j = 1; j < points2.size(); j++)
        {
            px = (float)points2[j].x;
            py = (float)points2[j].y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            LineTo(hdc, (int)sx, (int)sy);
        }

        SelectObject(hdc, hPenOld);
        DeleteObject(gPenNew);

        HGDIOBJ bPenNew = CreatePen(PS_SOLID, 2, RGB(0, 0, 255));
        hPenOld = SelectObject(hdc, bPenNew);

        px = (float)points3[0].x;
        py = (float)points3[0].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        for (size_t j = 1; j < points3.size(); j++)
        {
            px = (float)points3[j].x;
            py = (float)points3[j].y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            LineTo(hdc, (int)sx, (int)sy);
        }

        SelectObject(hdc, hPenOld);
        DeleteObject(bPenNew);

        HGDIOBJ pPenNew = CreatePen(PS_SOLID, 2, RGB(128, 0, 128));
        hPenOld = SelectObject(hdc, pPenNew);

        px = (float)points4[0].x;
        py = (float)points4[0].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        for (size_t j = 1; j < points4.size(); j++)
        {
            px = (float)points4[j].x;
            py = (float)points4[j].y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            LineTo(hdc, (int)sx, (int)sy);
        }

        SelectObject(hdc, hPenOld);
        DeleteObject(pPenNew);

        EndPaint(hWnd, &ps);
    }
    break;
    case WM_DESTROY:
        PostQuitMessage(0);
        break;
    default:
        return DefWindowProc(hWnd, message, wParam, lParam);
    }
    return 0;
}

// Message handler for about box.
INT_PTR CALLBACK About(HWND hDlg, UINT message, WPARAM wParam, LPARAM lParam)
{
    UNREFERENCED_PARAMETER(lParam);
    switch (message)
    {
    case WM_INITDIALOG:
        return (INT_PTR)TRUE;

    case WM_COMMAND:
        if (LOWORD(wParam) == IDOK || LOWORD(wParam) == IDCANCEL)
        {
            EndDialog(hDlg, LOWORD(wParam));
            return (INT_PTR)TRUE;
        }
        break;
    }
    return (INT_PTR)FALSE;
}

static INT_PTR CALLBACK DrawGraph(
    HWND hDlg, UINT message, WPARAM wParam, LPARAM lParam)
{
    UNREFERENCED_PARAMETER(lParam);
    char line[256] = { };
    switch (message)
    {
    case WM_INITDIALOG:
        GetWindowTextA(hDlg, line, 256);
        strcat_s(text, 8192, " ");
        strcat_s(text, 8192, line);
        SetWindowTextA(hDlg, text);
        return (INT_PTR)TRUE;

    case WM_COMMAND:
        if (LOWORD(wParam) == IDOK || LOWORD(wParam) == IDCANCEL)
        {
            EndDialog(hDlg, LOWORD(wParam));
            return (INT_PTR)TRUE;
        }
        break;
    case WM_PAINT:
        double h = 0, pi = 0, plm = 0, theta = 0;
        double xMax = 0, xMin = 0, yMax = 0, yMin = 0;
        FindMinMax(xMin, xMax, yMin, yMax);
        float xSpan = (float)(xMax - xMin);
        float ySpan = (float)(yMax - yMin);
        RECT rect = { };
        GetClientRect(hDlg, &rect);
        float width = (float)(rect.right - rect.left + 1);
        float height = (float)(rect.bottom - rect.top - 32 + 1);
        float sx0 = 2.0f * width / 16.0f;
        float sx1 = 14.0f * width / 16.0f;
        float sy0 = 2.0f * height / 16.0f;
        float sy1 = 14.0f * height / 16.0f;
        float deltaX = xSpan / 8.0f;
        float deltaY = ySpan / 8.0f;
        float xSlope = (sx1 - sx0) / xSpan;
        float xInter = (float)(sx0 - xSlope * xMin);
        float ySlope = (sy0 - sy1) / ySpan;
        float yInter = (float)(sy0 - ySlope * yMax);
        float px = 0, py = 0, sx = 0, sy = 0;
        PAINTSTRUCT ps;
        POINT wPt = { };
        HDC hdc = BeginPaint(hDlg, &ps);
        int i = 0;
        float x = (float)xMin;
        float y = (float)yMax;
        px = x;
        py = y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        MoveToEx(hdc, (int)sx, (int)sy0, &wPt);
        char buffer[128] = { };

        while (i <= 8)
        {
            sx = xSlope * x + xInter;
            wPt.x = wPt.y = 0;
            MoveToEx(hdc, (int)sx, (int)sy0, &wPt);
            LineTo(hdc, (int)sx, (int)sy1);

            sprintf_s(buffer, "%3.0f", x);
            SIZE size = { };
            GetTextExtentPoint32A(
                hdc,
                buffer,
                strlen(buffer),
                &size);
            RECT textRect = { };
            textRect.left = (long)(sx - size.cx / 2.0f);
            textRect.right = (long)(sx + size.cx / 2.0f);
            textRect.top = (long)sy1;
            textRect.bottom = (long)(sy1 + size.cy / 2.0f);
            DrawFormattedText(hdc, buffer, textRect);
            x += deltaX;
            i++;
        }

        i = 0;
        y = (float)yMin;

        while (i <= 8)
        {
            sy = ySlope * y + yInter;
            wPt.x = wPt.y = 0;
            MoveToEx(hdc, (int)sx0, (int)sy, &wPt);
            LineTo(hdc, (int)sx, (int)sy);

            if (i != 0)
            {
                sprintf_s(buffer, "%3.0f", y);
                SIZE size = { };
                GetTextExtentPoint32A(
                    hdc,
                    buffer,
                    strlen(buffer),
                    &size);
                RECT textRect = { };
                textRect.left = (long)(sx0 - size.cx - size.cx / 5.0f);
                textRect.right = (long)(sx0 - size.cx / 2.0f);
                textRect.top = (long)(sy - size.cy / 2.0f);
                textRect.bottom = (long)(sy + size.cy / 2.0f);
                DrawFormattedText(hdc, buffer, textRect);
            }

            y += deltaY;
            i++;
        }

        HGDIOBJ rPenNew = NULL;
        HGDIOBJ hPenOld = NULL;
        rPenNew = CreatePen(PS_SOLID, 2, RGB(255, 0, 0));
        hPenOld = SelectObject(hdc, rPenNew);

        px = (float)points1[1].x;
        py = (float)points1[1].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        for (size_t j = 1; j < points1.size(); j++)
        {
            px = (float)points1[j].x;
            py = (float)points1[j].y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            LineTo(hdc, (int)sx, (int)sy);
        }

        SelectObject(hdc, hPenOld);
        DeleteObject(rPenNew);

        px = (float)points2[1].x;
        py = (float)points2[1].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        HGDIOBJ gPenNew = CreatePen(PS_SOLID, 2, RGB(0, 255, 0));
        hPenOld = SelectObject(hdc, gPenNew);
        px = (float)points2[1].x;
        py = (float)points2[1].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        for (size_t j = 1; j < points2.size(); j++)
        {
            px = (float)points2[j].x;
            py = (float)points2[j].y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            LineTo(hdc, (int)sx, (int)sy);
        }

        SelectObject(hdc, hPenOld);
        DeleteObject(gPenNew);

        px = (float)points3[1].x;
        py = (float)points3[1].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        HGDIOBJ bPenNew = CreatePen(PS_SOLID, 2, RGB(0, 0, 255));
        hPenOld = SelectObject(hdc, bPenNew);
        px = (float)points3[1].x;
        py = (float)points3[1].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        for (size_t j = 1; j < points3.size(); j++)
        {
            px = (float)points3[j].x;
            py = (float)points3[j].y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            LineTo(hdc, (int)sx, (int)sy);
        }

        SelectObject(hdc, hPenOld);
        DeleteObject(bPenNew);

        px = (float)points4[1].x;
        py = (float)points4[1].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        HGDIOBJ tPenNew = CreatePen(PS_SOLID, 2, RGB(128, 0, 128));
        hPenOld = SelectObject(hdc, tPenNew);
        px = (float)points4[1].x;
        py = (float)points4[1].y;
        sx = xSlope * px + xInter;
        sy = ySlope * py + yInter;
        wPt.x = wPt.y = 0;
        MoveToEx(hdc, (int)sx, (int)sy, &wPt);

        for (size_t j = 1; j < points4.size(); j++)
        {
            px = (float)points4[j].x;
            py = (float)points4[j].y;
            sx = xSlope * px + xInter;
            sy = ySlope * py + yInter;
            LineTo(hdc, (int)sx, (int)sy);
        }

        SelectObject(hdc, hPenOld);
        DeleteObject(tPenNew);

        EndPaint(hDlg, &ps);
        return (INT_PTR)FALSE;
        break;
    }

    return (INT_PTR)FALSE;
}

Historical Attendance Data for the First United Methodist Church (FUMC) of LaGrange, Georgia for the Year 2006 and Modeling the Data Using Various Polynomials and Least Squares Fitting by James Pate Williams, Jr., BA, BS, Master of Software Engineering and Doctor of Philosophy in Computer Science

Church Data Polynomial Fitting 2006Download

Historical Attendance Data for the Four Services Offered by the First United Methodist Church of LaGrange, Georgia, Compiled and Analyzed by James Pate Williams, Jr.

Church Attendance BPNN 2006 to 2008Download

Blog Entry (c) Thursday, June 19, 2025, Factorizations of Some Composite Integers Using Two Different Computer Languages and Methods

Factorizations of the Seventh Fermat Number
and Other Composite Integers Using the
Pollard-Shor-Williams C# App and the LIP
Lenstra Elliptic Curve Method

It took 20.1 hours for J. M. Pollard to 
factor the Seventh Fermat Number in
December 1988. He was using an 8-bit
Philips P2012 personal computer with
64k RAM and two 640k floppy drives.
His seven programs were written in the
Pascal computer language. The current
author was using a 64-bit Core i5 Dell
Latitude 3410 notebook computer with
8 GB RAM and a 235 GB solid state
hard drive. The computer language was
Windows 32 vanilla C in the Release x64
Configuration. The operating system was
Windows 11 Pro with the Visual Studio
2022 Community Version Integrated
Development Environment and C#.
 
2^128+1
340282366920938463463374607431768211457 39
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 1
59649589127497217 p 17
5704689200685129054721 p 22
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:20:31.359
Function Evaluations = 661379586
2^128+1
340282366920938463463374607431768211457 39
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 2
59649589127497217 p 17
5704689200685129054721 p 22
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:12:31.146
Function Evaluations = 283327140
2^128+1
340282366920938463463374607431768211457 39
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 3
59649589127497217 p 17
5704689200685129054721 p 22
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:21:53.637
Function Evaluations = 371472150
2^128+1
340282366920938463463374607431768211457 39
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 4
59649589127497217 p 17
5704689200685129054721 p 22
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:15:25.712
Function Evaluations = 197866620
2^128+1
340282366920938463463374607431768211457 39
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 5
59649589127497217 p 17
5704689200685129054721 p 22
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:03:06.030
Function Evaluations = 19501012
2^128+1
340282366920938463463374607431768211457 39
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 6
59649589127497217 p 17
5704689200685129054721 p 22
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:12:25.076
Function Evaluations = 198404541
2^128+1
340282366920938463463374607431768211457 39
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 7
59649589127497217 p 17
5704689200685129054721 p 22
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:20:59.172
Function Evaluations = 168943987
2^128+1
340282366920938463463374607431768211457 39
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 8
59649589127497217 p 17
5704689200685129054721 p 22
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:04:58.588
Function Evaluations = 13754504

Using Lenstra's Elliptic Curve Method

enter the number to be factored below:
2^128+1
340282366920938463463374607431768211457
number has 39 digits
== Menu ==
1 Cohen's Brent-Pollard Method
2 Cohen's Trial Division
3 Lenstra's Elliptic Curve Method
4 Pollard p-1 Method First Stage
5 Pollard p-1 Both Stages
6 Pollard-Shor-Williams Method
7 Exit Application
Enter an option '1' to '7':
3
factorization is complete
Runtime in seconds:
 1.00000 sec.
original number has 39-decimal digits
'c' means composite and 'p' means prime
59649589127497217 17-decimal digits p
5704689200685129054721 22-decimal digits p
enter the number to be factored below:

Miscellaneous numbers using the C# app:

2^32+1
4294967297 10
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 1
641 p 3
6700417 p 7
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:00:00.034
Function Evaluations = 57
2^41-1
2199023255551 13
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 1
13367 p 5
164511353 p 9
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:00:00.003
Function Evaluations = 1128
2^67-1
147573952589676412927 21
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 1
193707721 p 9
761838257287 p 12
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:00:00.065
Function Evaluations = 30519
2^144-3
22300745198530623141535718272648361505980413 44
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 1
492729991333 p 12
45259565260477899162010980272761 p 32
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:00:04.804
Function Evaluations = 2458746
2^32+1
4294967297 10
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 5
641 p 3
6700417 p 7
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:00:00.001
Function Evaluations = 81
2^41-1
2199023255551 13
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 5
13367 p 5
164511353 p 9
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:00:00.002
Function Evaluations = 174
2^67-1
147573952589676412927 21
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 5
193707721 p 9
761838257287 p 12
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:00:00.050
Function Evaluations = 35949
2^144-3
22300745198530623141535718272648361505980413 44
Pseudo-Random Number Generator Seed = 1
Number of Tasks = 5
492729991333 p 12
45259565260477899162010980272761 p 32
Total Elapsed Runtime Hrs:Min:Sec.MS = 00:00:01.613
Function Evaluations = 632928

Now Lenstra's ECM:

enter the number to be factored below:
2^32+1
4294967297
number has 10 digits
== Menu ==
1 Cohen's Brent-Pollard Method
2 Cohen's Trial Division
3 Lenstra's Elliptic Curve Method
4 Pollard p-1 Method First Stage
5 Pollard p-1 Both Stages
6 Pollard-Shor-Williams Method
7 Exit Application
Enter an option '1' to '7':
3
factorization is complete
Runtime in seconds:
 0.00000 sec.
original number has 10-decimal digits
'c' means composite and 'p' means prime
641 3-decimal digits p
6700417 7-decimal digits p
enter the number to be factored below:
2^41-1
2199023255551
number has 13 digits
== Menu ==
1 Cohen's Brent-Pollard Method
2 Cohen's Trial Division
3 Lenstra's Elliptic Curve Method
4 Pollard p-1 Method First Stage
5 Pollard p-1 Both Stages
6 Pollard-Shor-Williams Method
7 Exit Application
Enter an option '1' to '7':
3
factorization is complete
Runtime in seconds:
 0.00000 sec.
original number has 13-decimal digits
'c' means composite and 'p' means prime
13367 5-decimal digits p
164511353 9-decimal digits p
enter the number to be factored below:
2^67-1
147573952589676412927
number has 21 digits
== Menu ==
1 Cohen's Brent-Pollard Method
2 Cohen's Trial Division
3 Lenstra's Elliptic Curve Method
4 Pollard p-1 Method First Stage
5 Pollard p-1 Both Stages
6 Pollard-Shor-Williams Method
7 Exit Application
Enter an option '1' to '7':
3
factorization is complete
Runtime in seconds:
 0.00000 sec.
original number has 21-decimal digits
'c' means composite and 'p' means prime
193707721 9-decimal digits p
761838257287 12-decimal digits p
enter the number to be factored below:
2^144-3
22300745198530623141535718272648361505980413
number has 44 digits
== Menu ==
1 Cohen's Brent-Pollard Method
2 Cohen's Trial Division
3 Lenstra's Elliptic Curve Method
4 Pollard p-1 Method First Stage
5 Pollard p-1 Both Stages
6 Pollard-Shor-Williams Method
7 Exit Application
Enter an option '1' to '7':
3
factorization is complete
Runtime in seconds:
 0.00000 sec.
original number has 44-decimal digits
'c' means composite and 'p' means prime
492729991333 12-decimal digits p
45259565260477899162010980272761 32-decimal digits p
enter the number to be factored below:

For the last number 2^144-3 it took J. M. Pollard's
factoring with cubic integers 47 hours in 1988.

Blog Entry © Thursday, June 5, 2025, by James Pate Williams, Jr., Analytic, Numeric, and Siacci’s Method for Solving Ballistic Trajectory Problems (Point-Mass Projectile Motion)

Blog Entry Thursday June 5 2025Download