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  1. #include <stdio.h> 
  2. #include <math.h>
  3.  
  4.  
  5. int a, b; 
  6. int mm, nn;
  7. int t; 
  8.  
  9. float pi = 3.1415926535897932384626433832795; 
  10. float pi7 = 0.4487989505128276054946633404685; 
  11. float r7 = 2.6457513110645905905016157536393; 
  12.  
  13. float sn2, sn4, sn8; 
  14. float cs2, cs4, cs8; 
  15. float tn2, tn4, tn8; 
  16.  
  17. float alpha, beta,  gamma; 
  18.  
  19.  
  20. #define MAX  128
  21. typedef char Str[MAX]; 
  22.  
  23. ///////////////////////////////
  24. int ct = 1; 
  25.  
  26.  
  27.  
  28.  
  29. int cvt(float x) {
  30.    if (x > 0.0)
  31.        return (int)(x+0.9); 
  32.    if (x < 0.0)
  33.        return (int)(x-0.9); 
  34.  
  35.    return 0; 
  36. }
  37.  
  38.  
  39. float xabs(float x) {
  40.   if (x < 0.0) 
  41.      return -x; 
  42.   return x; 
  43. } 
  44.  
  45. float croot(float num) {
  46.   int flag = 1.0;
  47.   float x; 
  48.   if (num < 0.0) {
  49.     flag = -1.0;
  50.     num = -num;
  51.   }
  52.   x = flag * exp(log(num)/3.0); 
  53.  
  54. // printf("croot: num = %f, x = %f\n", num, x); 
  55.  
  56.   return x;
  57. }
  58.  
  59. float xsqrt(float num) {
  60.   int flag = 1.0;
  61.   float x; 
  62.   if (num < 0.0) {
  63.     flag = -1.0;
  64.     num = -num;
  65.   }
  66.   x = flag * exp(log(num)/2.0); 
  67.  
  68. // printf("croot: num = %f, x = %f\n", num, x); 
  69.  
  70.   return x;
  71. }
  72.  
  73. float pwrf(float f, int p) {
  74.    int i, neg; 
  75.    float x = 1.0;
  76.  
  77.    if (p == 0) 
  78.      return 1.0;
  79.  
  80.    neg = 0;
  81.    if (p < 0) {
  82.       neg = 1;
  83.       p = -p;
  84.    }
  85.  
  86.    for (i=0; i< p; i++) 
  87.      x = x*f;
  88.  
  89.    if (neg == 1) 
  90.       x = 1.0/x;
  91.  
  92.    return x;
  93. }
  94.  
  95. int isInt(float x) {
  96.    float y = 1.0*cvt(x); 
  97.  
  98.    if (xabs(x-y) < 0.01) 
  99.       return 1; 
  100.  
  101.    return 0; 
  102. }
  103.  
  104. //////////////////////////////
  105. int toInt(float x) {
  106.    float y; 
  107.    int u;
  108.  
  109.    if (x > 0.1) {
  110.        y = x+0.1; 
  111.    }
  112.    else if (x < 0.1) { 
  113.        y = x-0.1; 
  114.    }
  115.    else 
  116.        y = 0.0;
  117.  
  118.    u = (int)(y);
  119.  
  120.  
  121.  
  122.    return u; 
  123. }
  124.  
  125.  
  126.  
  127.  
  128. // p >= 0 
  129. int pwri(int f, int p) {
  130.    int i; 
  131.    int x = 1;
  132.  
  133.    if (p == 0) 
  134.      return 1;
  135.  
  136.  
  137.    for (i=0; i< p; i++) 
  138.      x = x*f;
  139.  
  140.    return x;
  141. }
  142.  
  143.  
  144. void make_eq(int a, int b, int c, Str eq){
  145.  
  146.    sprintf(eq, "x^3");  
  147.    if (a < -1) {
  148.       sprintf(eq, "%s - %d x^2", eq, -a); 
  149.    }
  150.    else if (a == -1) {
  151.       sprintf(eq, "%s - x^2", eq); 
  152.    }
  153.    else if (a > 1) {
  154.       sprintf(eq, "%s + %d x^2", eq, a); 
  155.    }
  156.    else if (a == 1) {
  157.       sprintf(eq, "%s + x^2", eq); 
  158.    }
  159.  
  160.    if (b < -1) {
  161.       sprintf(eq, "%s - %d x", eq, -b); 
  162.    }
  163.    else if (b == -1) {
  164.       sprintf(eq, "%s - x^2", eq); 
  165.    }
  166.    else if (b > 1) {
  167.       sprintf(eq, "%s + %d x^2", eq, b); 
  168.    }
  169.    else if (b == 1) {
  170.       sprintf(eq, "%s + x^2", eq); 
  171.    }
  172.  
  173.  
  174.    if (c < -1) {
  175.       sprintf(eq, "%s - %d", eq, -c); 
  176.    }
  177.    else if (c == -1) {
  178.       sprintf(eq, "%s - 1", eq); 
  179.    }
  180.    else if (c > 1) {
  181.       sprintf(eq, "%s + %d", eq, c); 
  182.    }
  183.    else if (c == 1) {
  184.       sprintf(eq, "%s + 1", eq); 
  185.    }
  186.  
  187.  
  188.     sprintf(eq, "%s = 0", eq); 
  189.  
  190.  
  191. }
  192.  
  193.  
  194. //////////////////////////////////////////////
  195.  
  196. int tt(int k) {
  197. int x; 
  198.    x = k*k*k - 3*(a+b+3)*k-(a*b + 6*(a + b) + 9);
  199.    return x; 
  200. }
  201.  
  202.  
  203. void print_abg(){
  204.  
  205. printf("\\alpha = \\frac{(2\\cos(2\\theta)^%d}{(2\\cos(4\\theta)^%d},\n", mm, nn); 
  206. printf("\\beta  = \\frac{(2\\cos(4\\theta)^%d}{(2\\cos(8\\theta)^%d},\n", mm, nn); 
  207. printf("\\gamma = \\frac{(2\\cos(8\\theta)^%d}{(2\\cos(2\\theta)^%d}.\n", mm, nn); 
  208.  
  209. }
  210.  
  211.  
  212. void process(int sel) {
  213. int k, t;
  214. int x, y; 
  215. Str eq; 
  216.  
  217. if (sel == 1) {
  218.    if ((a+b+3) == 0) {
  219.        x = a*b-9; 
  220.  
  221.        make_eq(-a, b, -1, eq); 
  222.  
  223.  
  224.  
  225. printf("\\begin{theorem} \\label{tmn_%d_%d} \n", mm, nn); 
  226. printf("Let \n"); 
  227. printf("$$\n"); 
  228. print_abg(); 
  229. printf("$$\n"); 
  230.  
  231.  
  232. printf("Then $ \\{ \\alpha, \\beta, \\gamma \\} $ are roots of the equation:\n"); 
  233.  
  234. printf("\\begin{equation} \\label{emn_%d_%d} \n", mm, nn); 
  235. printf("%s\n", eq); 
  236. printf("\\end{equation}  \n"); 
  237. printf("which satisfies Rammnujan condition. \n"); 
  238. printf("And \n"); 
  239.  
  240. if (x >= 0) {
  241. printf("\\begin{equation} \\label{fmn_%d_%d} \n", mm, nn); 
  242. printf("\\sqrt[3]{\\alpha} + \\sqrt[3]{\\beta} + \\sqrt[3]{\\gamma} = \n"); 
  243. printf("\\sqrt[3]{%d + 3\\sqrt[3]{%d}}  \n", a+6, x); 
  244. printf("\\end{equation}  \n"); 
  245. printf("\\begin{equation} \\label{gmn_%d_%d} \n", mm, nn); 
  246. printf("\\frac{1}{\\sqrt[3]{\\alpha}} + \\frac{1}{\\sqrt[3]{\\beta}} + \\frac{1}{\\sqrt[3]{\\gamma}} = \n"); 
  247. printf("\\sqrt[3]{%d + 3\\sqrt[3]{%d}}  \n", b+6, x); 
  248. printf("\\end{equation}  \n"); 
  249. }
  250.  
  251. else {
  252. printf("\\begin{equation} \\label{fmn_%d_%d} \n", mm, nn); 
  253. printf("\\sqrt[3]{\\alpha} + \\sqrt[3]{\\beta} + \\sqrt[3]{\\gamma} = \n"); 
  254. printf("\\sqrt[3]{%d - 3\\sqrt[3]{%d}}  \n", a+6, -x); 
  255. printf("\\end{equation}  \n"); 
  256. printf("\\begin{equation} \\label{gmn_%d_%d} \n", mm, nn); 
  257. printf("\\frac{1}{\\sqrt[3]{\\alpha}} + \\frac{1}{\\sqrt[3]{\\beta}} + \\frac{1}{\\sqrt[3]{\\gamma}} = \n"); 
  258. printf("\\sqrt[3]{%d - 3\\sqrt[3]{%d}}  \n", b+6, -x); 
  259. printf("\\end{equation}  \n"); 
  260. }
  261.  
  262. printf("\\end{theorem}\n\n"); 
  263.  
  264.  
  265.       return; 
  266.    }
  267. }
  268.  
  269. if (sel == 2) {
  270.    for (k = -1000; k<= 1000; k++) {
  271.        if (tt(k) == 0) {
  272.           t = k; 
  273.  
  274. //          printf("(%3d,%3d}: a = %6d, b = %6d, t = %6d  \n", mm, nn, a, b, t); 
  275.  
  276.           // sum a = \sqrt[3]{a+6+3t},
  277.           // sum b = \sqrt[3]{b+6+3t},
  278.           x = a+6+3*t;
  279.           y = b+6+3*t;
  280.  
  281.        make_eq(-a, b, -1, eq); 
  282.  
  283.  
  284. printf("\\begin{theorem} \\label{tmn_%d_%d} \n", mm, nn); 
  285. printf("Let \n"); 
  286. printf("$$\n"); 
  287. print_abg(); 
  288. printf("$$\n"); 
  289.  
  290. printf("Then $ \\{ \\alpha, \\beta, \\gamma \\} $ are roots of the equation:\n"); 
  291.  
  292. printf("\\begin{equation} \\label{emn_%d_%d} \n", mm, nn); 
  293. printf("%s\n", eq); 
  294. printf("\\end{equation}  \n"); 
  295. printf("The associatd Rammnujan equation has integer solution $ %d $. \n", t); 
  296. printf("And \n"); 
  297.  
  298. if (x >= 0) {
  299. printf("\\begin{equation} \\label{fmn_%d_%d} \n", mm, nn); 
  300. printf("\\sqrt[3]{\\alpha} + \\sqrt[3]{\\beta} + \\sqrt[3]{\\gamma} = \n"); 
  301. printf("\\sqrt[3]{%d} \n", x); 
  302. printf("\\end{equation}  \n"); 
  303. }
  304.  
  305. else {
  306. printf("\\begin{equation} \\label{fmn_%d_%d} \n", mm, nn); 
  307. printf("\\sqrt[3]{\\alpha} + \\sqrt[3]{\\beta} + \\sqrt[3]{\\gamma} = \n"); 
  308. printf("-\\sqrt[3]{%d} \n", -x); 
  309. printf("\\end{equation}  \n"); 
  310. }
  311.  
  312. if (y >= 0) {
  313. printf("\\begin{equation} \\label{gmn_%d_%d} \n", mm, nn); 
  314. printf("\\frac{1}{\\sqrt[3]{\\alpha}} + \\frac{1}{\\sqrt[3]{\\beta}} + \\frac{1}{\\sqrt[3]{\\gamma}} = \n"); 
  315. printf(" \\sqrt[3]{%d} \n", y); 
  316. printf("\\end{equation}  \n"); 
  317. }
  318.  
  319. else {
  320. printf("\\begin{equation} \\label{gmn_%d_%d} \n", mm, nn); 
  321. printf("\\frac{1}{\\sqrt[3]{\\alpha}} + \\frac{1}{\\sqrt[3]{\\beta}} + \\frac{1}{\\sqrt[3]{\\gamma}} = \n"); 
  322. printf("-\\sqrt[3]{%d} \n", -y); 
  323. printf("\\end{equation}  \n"); 
  324. }
  325. printf("\\end{theorem}\n\n"); 
  326.  
  327.  
  328. /*
  329.           printf("     sum a = %f, expect: %f\n", 
  330.                croot(1.0*x), croot(alpha) + croot(beta) + croot(gamma));  
  331.           printf("     sum b = %f, expect: %f\n",
  332.                croot(1.0*y), croot(1.0/alpha) + croot(1.0/beta) + croot(1.0/gamma)); 
  333.           if (x >= 0) 
  334.               printf("     sum a =  \\sqrt[3]{%d} \n", x); 
  335.           else
  336.               printf("     sum a =  -\\sqrt[3]{%d} \n", -x); 
  337.  
  338.           if (y >= 0) 
  339.               printf("     sum b =  \\sqrt[3]{%d} \n", y); 
  340.           else
  341.               printf("     sum b =  -\\sqrt[3]{%d} \n", -y); 
  342. */
  343.  
  344.            return; 
  345.        }
  346.    }
  347. }
  348.  
  349. };
  350.  
  351. ////////////////////////////////////////////////////////
  352.  
  353.  
  354. void init () {
  355.    sn2 = sin(2*pi7);
  356.    sn4 = sin(4*pi7);
  357.    sn8 = sin(8*pi7);
  358.  
  359.    cs2 = cos(2*pi7);
  360.    cs4 = cos(4*pi7);
  361.    cs8 = cos(8*pi7);
  362.  
  363.    tn2 = tan(2*pi7);
  364.    tn4 = tan(4*pi7);
  365.    tn8 = tan(8*pi7);
  366.  
  367.  
  368.  
  369.  
  370. }
  371.  
  372.  
  373.  
  374.  
  375.  
  376.  
  377. void comp_ram(int sel) {
  378.  
  379.  
  380. /*
  381. float  pf, qf, rf, root, r1f, r2f; 
  382. int p, q, rr, r2, r12, r22; 
  383.  
  384. printf("alpha  = %f\n", alpha); 
  385. printf("beta  = %f\n", beta); 
  386. printf("gammma  = %f\n", gamma); 
  387.  
  388.  
  389.  
  390.  
  391. printf("***** (%d,%d) *****\n", mm, nn); 
  392.  
  393.  
  394.  
  395. printf("alpha+beta+gamma = %f\n", alpha +beta+gamma); 
  396. printf("alpha*beta+beta*gamma+gamma*alpha = %f\n", alpha *beta+beta*gamma+gamma*alpha ); 
  397. printf("alpha*beta*gamma = %f\n", alpha *beta*gamma); 
  398.  
  399.  
  400. printf("alpha^(1/3) + beta^(1/3) + gamma^(1/3) = %f\n", 
  401. croot(alpha) + croot(beta) + croot(gamma)); 
  402.  
  403. printf("(1/alpha)^(1/3) + (1/beta)^(1/3) + 1/(gamma)^(1/3) = %f\n", 
  404. croot(1.0/alpha) + croot(1.0/beta) + croot(1.0/gamma)); 
  405.  
  406. */
  407.  
  408. a = cvt(alpha +beta+gamma); 
  409. b = cvt(alpha *beta+beta*gamma+gamma*alpha); 
  410.  
  411.  
  412.  
  413.  
  414.  
  415. /*
  416.  
  417. // printf("a = %d, b = %d\n", a, b); 
  418.  
  419. //  x = k*k*k - 3*(a+b+3)*k - (a*b + 6*(a + b) + 9);
  420.  
  421. printf("Ramanujan eq:  x^3 - %d x - %d = 0 \n", 
  422.  3*(a+b+3), a*b + 6*(a + b) + 9); 
  423.  
  424. // x^3 + px = q = 0 
  425. // Cardano formula 
  426. // root = (-q/2 + (q^2/4+p^3/27)^(1/2))^(1/3) + (-q/2 - (q^2/4+p^3/27)^(1/2))^(1/3)
  427. // r = (q^2/4+p^3/27)^(1/2)
  428. p = -(3*(a+b+3)); 
  429. q = -(a*b + 6*(a + b) + 9);
  430.  
  431. printf("p = %d, q = %d\n", p, q); 
  432. pf = 1.0*p;
  433. qf = 1.0*q;
  434.  
  435. rf = sqrt( pwrf(qf,2)/4.0 + pwrf(pf,3)/27.0 ); 
  436. r2 = q*q + 4*p*p*p /27; 
  437. rr = cvt(sqrt(1.0*r2));  
  438.  
  439. printf(" rr = sqrt(%d)/2 = %d/2\n", r2, rr );
  440. printf(" rf = %f\n", rf);
  441.  
  442. r12 = (-q + rr)/2;
  443. r22 = (-q - rr)/2;
  444.  
  445. printf("r12 = %d\n", r12); 
  446. printf("r22 = %d\n", r22); 
  447.  
  448.  
  449. r1f = croot(-qf/2.0 + rf ); 
  450. r2f = croot(-qf/2.0 - rf ); 
  451.  
  452. printf(" r12 = (%d)^(1/3) \n", r12);
  453. printf(" r22 = (%d)^(1/3) \n", r22);
  454. printf("root = (%d)^(1/3) + (%d)^(1/3)\n", r12, r22); 
  455.  
  456. root = r1f + r2f; 
  457. printf("root = %f, %f + %f \n", root, r1f, r2f); 
  458.  
  459. */ 
  460.  
  461.  
  462. process(sel); 
  463.  
  464.  
  465. }
  466.  
  467. void set_cos_mn(int m, int n) {
  468.  
  469.  
  470. alpha = pwrf(2.0*cs2, m)/(pwrf(2.0*cs4, n));
  471. beta  = pwrf(2.0*cs4, m)/(pwrf(2.0*cs8, n));
  472. gamma = pwrf(2.0*cs8, m)/(pwrf(2.0*cs2, n));
  473.  
  474.  
  475. }
  476.  
  477.  
  478.  
  479. ///////////////////////////////////////////////
  480.  
  481.  
  482.  
  483. int main(int argc, char *argv[]) {
  484.    int i, j; 
  485.    int nterm = 100; 
  486.    init(); 
  487.  
  488.  
  489.    for (i=0; i<nterm; i++) {
  490.       for (j=0; j<nterm; j++) {
  491.            mm = i;
  492.            nn = j; 
  493.  
  494.           set_cos_mn(i, j); 
  495.           comp_ram(1); 
  496.  
  497.        }
  498.    }
  499.  
  500.  
  501.  
  502.  
  503. printf("End of JOB (%d,%d)\n", mm, nn); 
  504. return 1; 
  505.  
  506. }
  507.  
  508.  
Runtime error #stdin #stdout 0s 9432KB
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\begin{theorem} \label{tmn_0_1} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^0}{(2\cos(4\theta)^1},
\beta  = \frac{(2\cos(4\theta)^0}{(2\cos(8\theta)^1},
\gamma = \frac{(2\cos(8\theta)^0}{(2\cos(2\theta)^1}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_0_1} 
x^3 + 2 x^2 - x^2 - 1 = 0
\end{equation}  
which satisfies Rammnujan condition. 
And 
\begin{equation} \label{fmn_0_1} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{4 - 3\sqrt[3]{7}}  
\end{equation}  
\begin{equation} \label{gmn_0_1} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
\sqrt[3]{5 - 3\sqrt[3]{7}}  
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_1_0} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^1}{(2\cos(4\theta)^0},
\beta  = \frac{(2\cos(4\theta)^1}{(2\cos(8\theta)^0},
\gamma = \frac{(2\cos(8\theta)^1}{(2\cos(2\theta)^0}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_1_0} 
x^3 + x^2 - 2 x - 1 = 0
\end{equation}  
which satisfies Rammnujan condition. 
And 
\begin{equation} \label{fmn_1_0} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{5 - 3\sqrt[3]{7}}  
\end{equation}  
\begin{equation} \label{gmn_1_0} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
\sqrt[3]{4 - 3\sqrt[3]{7}}  
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_1_2} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^1}{(2\cos(4\theta)^2},
\beta  = \frac{(2\cos(4\theta)^1}{(2\cos(8\theta)^2},
\gamma = \frac{(2\cos(8\theta)^1}{(2\cos(2\theta)^2}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_1_2} 
x^3 - 5 x^2 - 8 x - 1 = 0
\end{equation}  
which satisfies Rammnujan condition. 
And 
\begin{equation} \label{fmn_1_2} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{11 - 3\sqrt[3]{49}}  
\end{equation}  
\begin{equation} \label{gmn_1_2} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
\sqrt[3]{-2 - 3\sqrt[3]{49}}  
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_1_3} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^1}{(2\cos(4\theta)^3},
\beta  = \frac{(2\cos(4\theta)^1}{(2\cos(8\theta)^3},
\gamma = \frac{(2\cos(8\theta)^1}{(2\cos(2\theta)^3}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_1_3} 
x^3 + 15 x^2 + 12 x^2 - 1 = 0
\end{equation}  
which satisfies Rammnujan condition. 
And 
\begin{equation} \label{fmn_1_3} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{-9 - 3\sqrt[3]{189}}  
\end{equation}  
\begin{equation} \label{gmn_1_3} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
\sqrt[3]{18 - 3\sqrt[3]{189}}  
\end{equation}  
\end{theorem}

\begin{theorem} \label{tmn_3_8} 
Let 
$$
\alpha = \frac{(2\cos(2\theta)^3}{(2\cos(4\theta)^8},
\beta  = \frac{(2\cos(4\theta)^3}{(2\cos(8\theta)^8},
\gamma = \frac{(2\cos(8\theta)^3}{(2\cos(2\theta)^8}.
$$
Then $ \{ \alpha, \beta, \gamma \} $ are roots of the equation:
\begin{equation} \label{emn_3_8} 
x^3 - 1259 x^2 - 1262 x - 1 = 0
\end{equation}  
which satisfies Rammnujan condition. 
And 
\begin{equation} \label{fmn_3_8} 
\sqrt[3]{\alpha} + \sqrt[3]{\beta} + \sqrt[3]{\gamma} = 
\sqrt[3]{1265 - 3\sqrt[3]{1588867}}  
\end{equation}  
\begin{equation} \label{gmn_3_8} 
\frac{1}{\sqrt[3]{\alpha}} + \frac{1}{\sqrt[3]{\beta}} + \frac{1}{\sqrt[3]{\gamma}} = 
\sqrt[3]{-1256 - 3\sqrt[3]{1588867}}  
\end{equation}  
\end{theorem}

End of JOB (99,99)
https://ideone.com/kRbBLI
language:
C (gcc 8.3)
created:
4 years ago
visibility:
public

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