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

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One thought on “Erdös-Straus conjecture of 1948, integer part function and RSA numbers

  1. Erdös-Straus conjecture of 1948, integer part function and RSA numbers
    Ibrahima GUEYE1
    1Senegalese Mathematician amator (Phd Human Medecine UCAD ; Master 1 of Biomathematics and Bioinformatics UCAD-Cheikh Anta Diop University of Dakar- )

    Abstract :
    In this article we show the link between integer part function and Erdös-Straus conjecture of 1948.
    Keywords: Erdös-Straus conjecture; integer part function
    MCS 2010: 11A41
    Introduction:
    Erdös- Straus conjecture of 1948
    The life of Paul Erdős (March 26, 1913 in Budapest, Hungary; September 20, 1996 in Warsaw, Poland) was entirely devoted to his research. Living in destitution, he had no wife, no job, not even a house he lived with an old suitcase and a bag of orange plastic supermarket. The only possession that mattered to him was his little book [4]. He was a prolific researcher in any discipline, with more than 1,500 research articles published. In particular, many of these articles was to study his favorite fields (graph theory, number theory, combinatorics) from different angles, and to continually improve the elegance of the demonstrations.
    One of the favorite maxims of Erdős was: “Sometimes you have to complicate a problem to simplify the solution.”
    Another famous quote often attributed incorrectly to Erdős, but in reality from the Alfred Renyi [1]: “A mathematician is a machine that turns coffee into theorems.”
    The Erdős-Straus conjecture implies that any rational number 4/n, with n an integer greater than or equal to two, can be written as a sum of three unit fractions, that is to say that there are three natural numbers not zero x, y and z such that:

    4/n = 1/x +1/y +1/z

    La conjecture d’Erdös-Straus was prooved by SOW Thierno M. and his article was accepted as a « short communication » by ICM « International Congress of Mathematics » Seoul 2014. He has given an very quick algorithm of factorization RSA numbers (we will see) [6].

    Integer part function
    The integer part (or truncation) of a real number corresponds to rounding towards zero. Thus, the integer part of a positive number is its floor, and the integer part of a negative number is its ceiling [7].

    RSA numbers
    In mathematics, the RSA numbers are a set of large semiprimes (numbers with exactly two prime factors) that are part of theRSA Factoring Challenge. The challenge was to find the prime factors but it was declared inactive in 2007. It was created by RSA Laboratories in March 1991 to encourage research into computational number theory and the practical difficulty offactoring large integers. The RSA challenge officially ended in 2007 but people are still attempting to find the factorizations. [8].
    Given a RSA number n.

    n = p*q
    The problem is to find p and q, when n in knowed.
    If j = (p + q) then values of some basics arithmetic functions are :
    d(n) = 2

    where is the Euler indicator.
    Phi(n) = (p – 1)*(q – 1) = n + 1 – s
    Mû(n) = (p + 1)*(q + 1) = n + 1 + s
    Where Phi is the Euler indicator.
    The RSA algorithm was publicly described in 1977 by Ron Rivest, Adi Shamir, and Leonard Adleman at MIT; the letters RSA are the initials of their surnames, listed in the same order as on the paper [5,9].

    Conjecture 1 : Erdös-Straus conjecture [2,3]
    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 (also called Egyptian fractions) , that is to say that there are three nonzero integers x, y and z such that:
    4/n = 1/x + 1/y +1/z
    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.

    Conjecture 2 :
    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

    Proof of Theorem 1 :
    4/n = 1/x +1/y +1/z
    Given b a nonzero positive integer so that :
    x = b*n
    4/n = 1/(b*n) +1/y +1/z
    4/n – 1/(b*n) = 1/y +1/z
    (4*b – 1)/(b*n) = 1/y +1/z
    1/y +1/z = (4*b – 1)/(b*n)
    Given c a nonzero positive integer so that :
    1/y +1/z = (4*b – 1)*([b*n/(4*b-1)] + c) / ( (b*n)*([b*n/(4*b-1)] + c) )
    1/y+1/z = ((4*b-1)*([b*n/(4*b-1)]+c)–b*N+b*N) / ((b*n)*([b*n/(4*b-1)]+c))
    1/y+1/z =1/([b*n/(4*b-1)]+c) + ((4*b-1)*([b*n/(4*b-1)]+c)–b*N) / ((b*n)*([b*n/(4*b-1)]+c))
    If :
    y = ([b*n/(4*b-1)] + c)
    z = ( b*n*([b*n/(4*b-1)] + c) ) / ( (4*b-1)*([b*n/(4*b-1)] + c) – b*n )
    then :
    4/n = 1/x + 1/y +1/z
    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 )
    This was to be demonstrated
    For c = 1, we have:
    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
    Conjecture 2: example
    4/33 = 1/33 + 1/12 + 1/132 ; with b = 1; c = 1; x = 33; y = 12 and z = 132
    q = 3 and p = z/y = 11
    A program in C++ :
    #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;
    }
    The results is:
    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

    With c = 1

    References:
    1- Bruce Schechter, My Brain Is Open : The Mathematical Journeys of Paul Erdos, 1998, p. 155
    2- GUEYE I., MIZONY M.,Towards the proof of Erdös-Straus conjecture, Bulletin of Society for mathematical services & standards (B SO MA S S), Vol. I No. 2 (2012), pp. 141-150
    3- GUEYE I., MIZONY M., Recent progress about the conjecture of Erdös-Straus, Bulletin of Society for mathematical services & standards (B SO MA S S), Vol. I No. 2 (2012), pp. 6-14
    4- Paul Hoffman , Paul Erdös : 1913-1996 : l’homme qui n’aimait que les nombres , Éditions Belin, 2000 (ISBN 2-7011-2539-1)
    5- R. Rivest, A. Shamir, L. Adleman. A Method for Obtaining Digital Signatures and Public-Key Cryptosystems]. Communications of the ACM, Vol. 21 (2), pp.120–126. 1978

    6- SOW Thierno M., Stealth Elliptic Curves and The Quantum Fields, accepted as a short communication at the Seoul ICM 2014
    7- http://oeis.org/wiki/Integer_part
    8- http://fr.wikipedia.org/wiki/Nombre_RSA
    9- http://fr.wikipedia.org/wiki/Chiffrement_RSA

    Contacts : Emails: ibrahimaeygue@yahoo.fr ; ibrahimaeygue@gmail.com
    Adress: Ouakam, Quarter Gouye Sor, Dakar – Senegal
    HLM Gibraltar 1 Villa N°71 Dakar – Senegal
    Rabat, quarter El Menzeh , Maroc (adress of life)
    Phone : Fixes : 00221338209052 (Senegal) ; 00221338221746(Senegal)
    Portable :

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