Da fattorizzare::   n = 10000000000000000102999499

Scriviamo curve ellittiche a caso E_a:  Y^2 = X^3 + a*X + 1 modulo  n

Per esempio:

Y^2 = X^3 + 8644390377546667257039683*X + 1  modulo  10000000000000000102999499

ecc.

Prendiamo P = [0,1] e calcoliamo  M!P  per un certo M > 0.

Le curve ellittiche  E_a:  Y^2 = X^3 + a*X + 1  modulo  p = 1000003

  a      #E_a(Zp)

400511,  1000252 = 2^2 * 11 * 127 * 179  
446396,   998478 = 2 * 3^2 * 13 * 17 * 251 
757527,  1001142 = 2 * 3^2 * 55619
950605,   998466 = 2 * 3 * 7 * 23773
738004,  1001774 = 2 * 500887
360914,  1000998 = 2 * 3^4 * 37 * 167
404182,   999830 = 2 * 5 * 13 * 7691 
669252,   999036 = 2^2 * 3^2 * 27751
159856,  1000804 = 2^2 * 7 * 31 * 1153
792355,  1001014 = 2 * 7 * 127 * 563 
178745,   999514 = 2 * 19 * 29 * 907
58690,    999560 = 2^3 * 5 * 24989
473302,   999484 = 2^2 * 249871
250305,  1000818 = 2 * 3^2 * 7 * 13^2 * 47 
293143,   999759 = 3 * 333253
426813,   999072 = 2^5 * 3^2 * 3469
846320,   998108 = 2^2 * 19 * 23 * 571
679742,   998138 = 2 * 17 * 31 * 947
737738,  1000664 = 2^3 * 7 * 107 * 167
637089,  1001703 = 3 * 31 * 10771
5482,     998587 = 101 * 9887
547163,   999481 = 7 * 17 * 37 * 227
602283,  1000195 = 5 * 7 * 17 * 41^2
715996,  1000020 = 2^2 * 3 * 5 * 7 * 2381
430777,  1001330 = 2 * 5 * 11 * 9103
446509,   998350 = 2 * 5^2 * 41 * 487
7824,    1000998 = 2 * 3^4 * 37 * 167 
411820,  1000207 = 13 * 47 * 1637
933393,   999832 = 2^3 * 124979
793210,   998118 = 2 * 3^2 * 11 * 71^2
603947,  1001260 = 2^2 * 5 * 13 * 3851
23959,   1001738 = 2 * 37 * 13537
72414,   1001458 = 2 * 500729
37992,    998200 = 2^3 * 5^2 * 7 * 23 * 31   <=====  "35-tonda"
199740,  1000320 = 2^7 * 3 * 5 * 521
1245,     998533 = 599 * 1667
431719,  1001570 = 2 * 5 * 47 * 2131 
491645,  1000296 = 2^3 * 3^3 * 11 * 421
777789,   999839 = 283 * 3533
470326,  1001127 = 3 * 307 * 1087

**    ,  1000002 = 2 * 3 * 166667


Y^2 = X^3 + 37992*X + 1  modulo  10000000000000000102999499


P = [0,1]

for(k=1,35, P=k*P; print(k,", ",P))

1, [0, 1]
2, [360848016, 9999999999993145434087562]
3, [7829714526825183914839830, 6745158634502223523583100]
4, [169715744758878173904879, 9850732993061658434111468]
5, [1451698759608181038345013, 2631202290186117105419137]
6, [755346847475472026583725, 4016539818968283430276826]
7, [4769476603651746427245643, 1368375510054014320887789]
8, [7730796687366377665241354, 8102350413722572477326316]
9, [3454663034523784958564294, 9373693158657518226687090]
10, [3553869925798725033582205, 4673121831953711970784431]
11, [32004311335628143387553, 6572681021934085394237953]
12, [2573351991390389531870120, 8567490636494004999969147]
13, [6242484872277286371174144, 5292323231374436274219591]
14, [1244846255355275224928929, 9288917627975065704109367]
15, [7338058160759505411666618, 2602020818704625042386085]
16, [8182143631916781490259028, 9960892768644865101079827]
17, [1894507027568336780153716, 9390777263788851922711383]
18, [3663330248837872225077458, 5630391189198171240595051]
19, [7470553347067819237599, 3703497171451914223570398]
20, [9004727640004763232800315, 2671777545369693762949699]
21, [3793543816015212594347824, 1606644572908146545185974]
22, [4860209534547335034496709, 8339723108927533125401926]
23, [3533270993020414201495810, 6240144477431697778137981]
24, [2881567099261018327564830, 5400811881124710409234383]
25, [8378203677961581386764014, 2189202295795650999316395]
26, [8492426286858215843961748, 3065860762973349647748381]
27, [3232272377639700737447146, 6817961284319673920557197]
28, [1385438055416206121302467, 6726319705236352743857040]
29, [1484697406376005739055781, 3917504683216905467331028]
30, [2366344867430841007216502, 203238978925802150092285]
  ***   impossible inverse modulo: mod(1000003, 10000000000000000102999499)
? 

10000000000000000102999499 = 1000003 * 9999970000089999833