Erdös-Straus conjecture of 1948, integer part function and RSA numbers

Erdös-Straus conjecture of 1948, integer part function and RSA numbers.

Advertisements

Erdös-Straus conjecture of 1948, integer part function and RSA numbers

New result about Erdös-Straus decompositions
Theorem 1 : (2014)
Every rational number 4/n, with n an integer greater than or equal to 2, can be written as a sum of three unit fractions if :
( (4*b-1)*([b*n/(4*b-1)] + c) – b*n ) divise ( b*n*([b*n/(4*b-1)] + c) )
With b and c nonzero positives integers.

4/n = 1/x + 1/y +1/z ; positive integer >=2 ; x,y and z non zero integers
With:
x = b*n; y = [b*n/(4*b-1)] + c
and
z= ( b*n*([b*n/(4*b-1)] + c) ) / ((4*b-1)*([b*n/(4*b-1)] + c) – b*n )
For c = 1, wehave:
x = b*n; y = [b*n/(4*b-1)] + 1
and
z= ( b*n*([b*n/(4*b-1)] + 1) ) / ((4*b-1)*([b*n/(4*b-1)] + 1) – b*n )
Examples: 4/5 = 1/5 + 1/2 + 1/10 ; with b = 1 and c = 1
4/17 = 1/17 + 1/6 + 1/102 ; with b = 1 and c = 1
4/33 = 1/33 + 1/12 + 1/132; with b = 1 and c = 1
4/73 = 1/219 + 1/20 + 1/4380 ; with b = 3 and c = 1
4/97 = 1/388 + 1/26 + 1/5044 ; with b = 4 and c = 1
4/1801 = 1/5403 + 1/492 + 1/295364 ; with b = 3 and c = 1
4/2521 = 1/30252 + 1/644 +1/1217643 ; with b = 12 and c = 1

[ ] means the integer part function

#include
#include
#include
#define MAX_Q 150
int main()
{
int n,b,q,y,x,z,c;
float t;
FILE*f = fopen(“liste_erdos-strauspositives.txt”, “a”);
for(q = 1; q <= MAX_Q; q++)
{
n =24*q+1;
int ok=0;
int e=0;
for(c = 1; c <= 100; c++)
{
for(b = 1; b 0)

//if((n%(z/y) == 0) || (n%(x/y) == 0))
{
ok=1;
e=1;
printf(“\tq = %d,”, q);
printf(“\tn = %d,”, n);
printf(“\tb = %d,”, b);
printf(“\tc = %d,”, c);
printf(“\ty = %d”, y);
printf(“\tx = %d”, x);
printf(“\tz = %d”, z);
printf(“\tt = %f,”, t);
printf(“\n”);
fprintf(f, “\tq = %d”, q);
fprintf(f, “\tn = %d”, n);
fprintf(f, “\tb = %d”, b);
fprintf(f, “\tc = %d”, c);
fprintf(f, “\tx = %d”, x);
fprintf(f, “\ty = %d”, y);
fprintf(f, “\tz = %d”, z);
fprintf(f, “\tt = %f”, t);
fprintf(f, “\n”);
//getch();
//return 0;
break;
}
}
}
}
if (ok==0)
printf(“**** echec pour n = %d \n”,n);
}
//printf(“%d\n echec”,n);
fclose(f);
getch ();
return 0;
}

q = 1 n = 25 b = 4 c = 1 x = 100 y = 7 z = 140 t = 140.000000
q = 2 n = 49 b = 2 c = 1 x = 98 y = 15 z = 210 t = 210.000000
q = 3 n = 73 b = 2 c = 1 x = 146 y = 21 z = 3066 t = 3066.000000
q = 4 n = 97 b = 2 c = 1 x = 194 y = 28 z = 2716 t = 2716.000000
q = 5 n = 121 b = 3 c = 1 x = 363 y = 34 z = 1122 t = 1122.000000
q = 6 n = 145 b = 2 c = 1 x = 290 y = 42 z = 3045 t = 3045.000000
q = 7 n = 169 b = 10 c = 1 x = 1690 y = 44 z = 2860 t = 2860.000000
q = 8 n = 193 b = 4 c = 1 x = 772 y = 52 z = 5018 t = 5018.000000
q = 9 n = 217 b = 2 c = 1 x = 434 y = 63 z = 3906 t = 3906.000000
q = 10 n = 241 b = 2 c = 1 x = 482 y = 69 z = 33258 t = 33258.000000
q = 11 n = 265 b = 2 c = 1 x = 530 y = 76 z = 20140 t = 20140.000000
q = 12 n = 289 b = 13 c = 1 x = 3757 y = 74 z = 16354 t = 16354.000000
q = 13 n = 313 b = 2 c = 1 x = 626 y = 90 z = 14085 t = 14085.000000
q = 14 n = 337 b = 3 c = 1 x = 1011 y = 92 z = 93012 t = 93012.000000
q = 15 n = 361 b = 5 c = 1 x = 1805 y = 96 z = 9120 t = 9120.000000
q = 16 n = 385 b = 2 c = 1 x = 770 y = 111 z = 12210 t = 12210.000000
q = 17 n = 409 b = 2 c = 1 x = 818 y = 117 z = 95706 t = 95706.000000
q = 18 n = 433 b = 2 c = 1 x = 866 y = 124 z = 53692 t = 53692.000000
q = 19 n = 457 b = 4 c = 1 x = 1828 y = 122 z = 111508 t = 111508.000000
q = 20 n = 481 b = 2 c = 1 x = 962 y = 138 z = 33189 t = 33189.000000
q = 21 n = 505 b = 2 c = 1 x = 1010 y = 145 z = 29290 t = 29290.000000
q = 22 n = 529 b = 6 c = 1 x = 3174 y = 139 z = 19182 t = 19182.000000
q = 23 n = 553 b = 2 c = 1 x = 1106 y = 159 z = 25122 t = 25122.000000
q = 24 n = 577 b = 2 c = 1 x = 1154 y = 165 z = 190410 t = 190410.000000
q = 25 n = 601 b = 2 c = 1 x = 1202 y = 172 z = 103372 t = 103372.000000
q = 26 n = 625 b = 4 c = 1 x = 2500 y = 167 z = 83500 t = 83500.000000
q = 27 n = 649 b = 2 c = 1 x = 1298 y = 186 z = 60357 t = 60357.000000
q = 28 n = 673 b = 4 c = 1 x = 2692 y = 180 z = 60570 t = 60570.000000
q = 29 n = 697 b = 4 c = 1 x = 2788 y = 186 z = 259284 t = 259284.000000
q = 30 n = 721 b = 2 c = 1 x = 1442 y = 207 z = 42642 t = 42642.000000
q = 31 n = 745 b = 2 c = 1 x = 1490 y = 213 z = 317370 t = 317370.000000
q = 32 n = 769 b = 2 c = 1 x = 1538 y = 220 z = 169180 t = 169180.000000
q = 33 n = 793 b = 4 c = 1 x = 3172 y = 212 z = 84058 t = 84058.000000
q = 34 n = 817 b = 2 c = 1 x = 1634 y = 234 z = 95589 t = 95589.000000
q = 35 n = 841 b = 22 c = 1 x = 18502 y = 213 z = 135894 t = 135894.000000
q = 36 n = 865 b = 3 c = 1 x = 2595 y = 236 z = 612420 t = 612420.000000
q = 37 n = 889 b = 2 c = 1 x = 1778 y = 255 z = 64770 t = 64770.000000
q = 38 n = 913 b = 2 c = 1 x = 1826 y = 261 z = 476586 t = 476586.000000
q = 39 n = 937 b = 2 c = 1 x = 1874 y = 268 z = 251116 t = 251116.000000
q = 40 n = 961 b = 8 c = 1 x = 7688 y = 249 z = 61752 t = 61752.000000
q = 41 n = 985 b = 2 c = 1 x = 1970 y = 282 z = 138885 t = 138885.000000
q = 42 n = 1009 b = 3 c = 1 x = 3027 y = 276 z = 92828 t = 92828.000000
q = 43 n = 1033 b = 3 c = 1 x = 3099 y = 282 z = 291306 t = 291306.000000
q = 44 n = 1057 b = 2 c = 1 x = 2114 y = 303 z = 91506 t = 91506.000000
q = 45 n = 1081 b = 2 c = 1 x = 2162 y = 309 z = 668058 t = 668058.000000
q = 46 n = 1105 b = 2 c = 1 x = 2210 y = 316 z = 349180 t = 349180.000000
q = 47 n = 1129 b = 3 c = 1 x = 3387 y = 308 z = 1043196 t = 1043196.000000
q = 48 n = 1153 b = 2 c = 1 x = 2306 y = 330 z = 190245 t = 190245.000000
q = 49 n = 1177 b = 3 c = 1 x = 3531 y = 322 z = 103362 t = 103362.000000
q = 50 n = 1201 b = 8 c = 1 x = 9608 y = 310 z = 1489240 t = 1489240.000000
q = 51 n = 1225 b = 2 c = 1 x = 2450 y = 351 z = 122850 t = 122850.000000
q = 52 n = 1249 b = 2 c = 1 x = 2498 y = 357 z = 891786 t = 891786.000000
q = 53 n = 1273 b = 2 c = 1 x = 2546 y = 364 z = 463372 t = 463372.000000
q = 54 n = 1297 b = 3 c = 1 x = 3891 y = 354 z = 459138 t = 459138.000000
q = 55 n = 1321 b = 2 c = 1 x = 2642 y = 378 z = 249669 t = 249669.000000
q = 56 n = 1345 b = 2 c = 1 x = 2690 y = 385 z = 207130 t = 207130.000000
q = 57 n = 1369 b = 28 c = 1 x = 38332 y = 346 z = 179228 t = 179228.000000
q = 58 n = 1393 b = 2 c = 1 x = 2786 y = 399 z = 158802 t = 158802.000000
q = 59 n = 1417 b = 2 c = 1 x = 2834 y = 405 z = 1147770 t = 1147770.000000
q = 60 n = 1441 b = 2 c = 1 x = 2882 y = 412 z = 593692 t = 593692.000000
q = 61 n = 1465 b = 3 c = 1 x = 4395 y = 400 z = 351600 t = 351600.000000
q = 62 n = 1489 b = 2 c = 1 x = 2978 y = 426 z = 317157 t = 317157.000000
q = 63 n = 1513 b = 4 c = 1 x = 6052 y = 404 z = 305626 t = 305626.000000
q = 64 n = 1537 b = 3 c = 1 x = 4611 y = 420 z = 215180 t = 215180.000000
q = 65 n = 1561 b = 2 c = 1 x = 3122 y = 447 z = 199362 t = 199362.000000
q = 66 n = 1585 b = 2 c = 1 x = 3170 y = 453 z = 1436010 t = 1436010.000000
q = 67 n = 1609 b = 2 c = 1 x = 3218 y = 460 z = 740140 t = 740140.000000
q = 68 n = 1633 b = 4 c = 1 x = 6532 y = 436 z = 355994 t = 355994.000000
q = 69 n = 1657 b = 2 c = 1 x = 3314 y = 474 z = 392709 t = 392709.000000
q = 70 n = 1681 b = 31 c = 1 x = 52111 y = 424 z = 538904 t = 538904.000000
q = 71 n = 1705 b = 3 c = 1 x = 5115 y = 466 z = 216690 t = 216690.000000
q = 72 n = 1729 b = 2 c = 1 x = 3458 y = 495 z = 244530 t = 244530.000000
q = 73 n = 1753 b = 2 c = 1 x = 3506 y = 501 z = 1756506 t = 1756506.000000
q = 74 n = 1777 b = 2 c = 1 x = 3554 y = 508 z = 902716 t = 902716.000000
q = 75 n = 1801 b = 3 c = 1 x = 5403 y = 492 z = 295364 t = 295364.000000
q = 76 n = 1825 b = 2 c = 1 x = 3650 y = 522 z = 476325 t = 476325.000000
q = 77 n = 1849 b = 11 c = 1 x = 20339 y = 474 z = 224202 t = 224202.000000
q = 78 n = 1873 b = 4 c = 1 x = 7492 y = 500 z = 468250 t = 468250.000000
q = 79 n = 1897 b = 2 c = 1 x = 3794 y = 543 z = 294306 t = 294306.000000
q = 80 n = 1921 b = 2 c = 1 x = 3842 y = 549 z = 2109258 t = 2109258.000000
q = 81 n = 1945 b = 2 c = 1 x = 3890 y = 556 z = 1081420 t = 1081420.000000
q = 82 n = 1969 b = 3 c = 1 x = 5907 y = 538 z = 288906 t = 288906.000000
q = 83 n = 1993 b = 2 c = 1 x = 3986 y = 570 z = 568005 t = 568005.000000
q = 84 n = 2017 b = 4 c = 1 x = 8068 y = 538 z = 2170292 t = 2170292.000000
q = 85 n = 2041 b = 6 c = 1 x = 12246 y = 533 z = 502086 t = 502086.000000
q = 86 n = 2065 b = 2 c = 1 x = 4130 y = 591 z = 348690 t = 348690.000000
q = 87 n = 2089 b = 2 c = 1 x = 4178 y = 597 z = 2494266 t = 2494266.000000
q = 88 n = 2113 b = 2 c = 1 x = 4226 y = 604 z = 1276252 t = 1276252.000000
q = 89 n = 2137 b = 4 c = 1 x = 8548 y = 570 z = 2436180 t = 2436180.000000
q = 90 n = 2161 b = 2 c = 1 x = 4322 y = 618 z = 667749 t = 667749.000000
q = 91 n = 2185 b = 2 c = 1 x = 4370 y = 625 z = 546250 t = 546250.000000
q = 92 n = 2209 b = 12 c = 1 x = 26508 y = 565 z = 318660 t = 318660.000000
q = 93 n = 2233 b = 2 c = 1 x = 4466 y = 639 z = 407682 t = 407682.000000
q = 94 n = 2257 b = 2 c = 1 x = 4514 y = 645 z = 2911530 t = 2911530.000000
q = 95 n = 2281 b = 2 c = 1 x = 4562 y = 652 z = 1487212 t = 1487212.000000
q = 96 n = 2305 b = 4 c = 1 x = 9220 y = 615 z = 1134060 t = 1134060.000000
q = 97 n = 2329 b = 2 c = 1 x = 4658 y = 666 z = 775557 t = 775557.000000
q = 98 n = 2353 b = 3 c = 1 x = 7059 y = 642 z = 1510626 t = 1510626.000000
q = 99 n = 2377 b = 4 c = 1 x = 9508 y = 634 z = 3014036 t = 3014036.000000
q = 100 n = 2401 b = 2 c = 1 x = 4802 y = 687 z = 471282 t = 471282.000000
q = 101 n = 2425 b = 2 c = 1 x = 4850 y = 693 z = 3361050 t = 3361050.000000
q = 102 n = 2449 b = 2 c = 1 x = 4898 y = 700 z = 1714300 t = 1714300.000000
q = 103 n = 2473 b = 4 c = 1 x = 9892 y = 660 z = 816090 t = 816090.000000
q = 104 n = 2497 b = 2 c = 1 x = 4994 y = 714 z = 891429 t = 891429.000000
q = 105 n = 2521 b = 12 c = 1 x = 30252 y = 644 z = 1217643 t = 1217643.000000
q = 106 n = 2545 b = 4 c = 1 x = 10180 y = 679 z = 1382444 t = 1382444.000000
q = 107 n = 2569 b = 2 c = 1 x = 5138 y = 735 z = 539490 t = 539490.000000
q = 108 n = 2593 b = 2 c = 1 x = 5186 y = 741 z = 3842826 t = 3842826.000000
q = 109 n = 2617 b = 2 c = 1 x = 5234 y = 748 z = 1957516 t = 1957516.000000
q = 110 n = 2641 b = 5 c = 1 x = 13205 y = 696 z = 483720 t = 483720.000000
q = 111 n = 2665 b = 2 c = 1 x = 5330 y = 762 z = 1015365 t = 1015365.000000
q = 112 n = 2689 b = 6 c = 1 x = 16134 y = 702 z = 943839 t = 943839.000000
q = 113 n = 2713 b = 3 c = 1 x = 8139 y = 740 z = 6022860 t = 6022860.000000
q = 114 n = 2737 b = 2 c = 1 x = 5474 y = 783 z = 612306 t = 612306.000000
q = 115 n = 2761 b = 2 c = 1 x = 5522 y = 789 z = 4356858 t = 4356858.000000
q = 116 n = 2785 b = 2 c = 1 x = 5570 y = 796 z = 2216860 t = 2216860.000000
q = 117 n = 2809 b = 40 c = 1 x = 112360 y = 707 z = 1498840 t = 1498840.000000
q = 118 n = 2833 b = 2 c = 1 x = 5666 y = 810 z = 1147365 t = 1147365.000000
q = 119 n = 2857 b = 3 c = 1 x = 8571 y = 780 z = 742820 t = 742820.000000
q = 120 n = 2881 b = 3 c = 1 x = 8643 y = 786 z = 2264466 t = 2264466.000000
q = 121 n = 2905 b = 2 c = 1 x = 5810 y = 831 z = 689730 t = 689730.000000
q = 122 n = 2929 b = 2 c = 1 x = 5858 y = 837 z = 4903146 t = 4903146.000000
q = 123 n = 2953 b = 2 c = 1 x = 5906 y = 844 z = 2492332 t = 2492332.000000
q = 124 n = 2977 b = 3 c = 1 x = 8931 y = 812 z = 7251972 t = 7251972.000000
q = 125 n = 3001 b = 2 c = 1 x = 6002 y = 858 z = 1287429 t = 1287429.000000
q = 126 n = 3025 b = 2 c = 1 x = 6050 y = 865 z = 1046650 t = 1046650.000000
q = 127 n = 3049 b = 11 c = 1 x = 33539 y = 780 z = 26160420 t = 26160420.000000
q = 128 n = 3073 b = 2 c = 1 x = 6146 y = 879 z = 771762 t = 771762.000000
q = 129 n = 3097 b = 2 c = 1 x = 6194 y = 885 z = 5481690 t = 5481690.000000
q = 130 n = 3121 b = 2 c = 1 x = 6242 y = 892 z = 2783932 t = 2783932.000000
q = 131 n = 3145 b = 3 c = 1 x = 9435 y = 858 z = 2698410 t = 2698410.000000
q = 132 n = 3169 b = 2 c = 1 x = 6338 y = 906 z = 1435557 t = 1435557.000000
q = 133 n = 3193 b = 4 c = 1 x = 12772 y = 852 z = 1360218 t = 1360218.000000
q = 134 n = 3217 b = 4 c = 1 x = 12868 y = 858 z = 5520372 t = 5520372.000000
q = 135 n = 3241 b = 2 c = 1 x = 6482 y = 927 z = 858402 t = 858402.000000
q = 136 n = 3265 b = 2 c = 1 x = 6530 y = 933 z = 6092490 t = 6092490.000000
q = 137 n = 3289 b = 2 c = 1 x = 6578 y = 940 z = 3091660 t = 3091660.000000
q = 138 n = 3313 b = 4 c = 1 x = 13252 y = 884 z = 1464346 t = 1464346.000000
q = 139 n = 3337 b = 2 c = 1 x = 6674 y = 954 z = 1591749 t = 1591749.000000
q = 140 n = 3361 b = 25 c = 2 x = 84025 y = 850 z = 571370 t = 571370.000000
q = 141 n = 3385 b = 3 c = 1 x = 10155 y = 924 z = 1042580 t = 1042580.000000
q = 142 n = 3409 b = 2 c = 1 x = 6818 y = 975 z = 949650 t = 949650.000000
q = 143 n = 3433 b = 2 c = 1 x = 6866 y = 981 z = 6735546 t = 6735546.000000
q = 144 n = 3457 b = 2 c = 1 x = 6914 y = 988 z = 3415516 t = 3415516.000000
q = 145 n = 3481 b = 15 c = 1 x = 52215 y = 886 z = 784110 t = 784110.000000
q = 146 n = 3505 b = 2 c = 1 x = 7010 y = 1002 z = 1756005 t = 1756005.000000
q = 147 n = 3529 b = 6 c = 1 x = 21174 y = 921 z = 2166806 t = 2166806.000000
q = 148 n = 3553 b = 3 c = 1 x = 10659 y = 970 z = 939930 t = 939930.000000
q = 149 n = 3577 b = 2 c = 1 x = 7154 y = 1023 z = 1045506 t = 1045506.000000
q = 150 n = 3601 b = 2 c = 1 x = 7202 y = 1029 z = 7410858 t = 7410858.000000

Conjecture :
Given n a positive integer number >= 2 and x,y, z three nonzero positive integer numbers so that:
4/n = 1/x + 1/y +1/z
Given b and c nonzero positives integers numbers so that :
x = b*n; y = [b*n/(4*b-1)] + c
and
z = ( b*n*([b*n/(4*b-1)] + c) ) / ( (4*b-1)*([b*n/(4*b-1)] + c) – b*n )
Given p and q primes numbers so that :
n = p*q
n is an RSA number
I conjecture that we can find b and c so that:
z/y = p
and/or
x/y = q

0
0
Rate This