Payam Numbers
(MathWorld)
(Robert Smith's page - Archived)
(OEIS A083556) Payam Numbers for Proths
3 (3)
9 (3)*3
15 (3*5)
105 (3*5)*7
165 (3*5*11)
75075 (3*5*11*13)*35
855855 (3*5*11*13*19)*21
18625035 (3*5*11*13*19)*457
27183585 (3*5*11*13*19)*667
1684200375 (3*5*11*13*19*29)*1425
198840832905 (3*5*11*13*19*29*37)*4547
305892154425 (3*5*11*13*19*29*37)*6995
532326689895 (3*5*11*13*19*29*37)*12173
139592518105755 (3*5*11*13*19*29*37*53)*60229
5 7 11 13 17 19 23 25 29 31 35 37 ... +2 no consecutive primes possible 3:2 (unavoidable)
3 27 33 45 69 75 87 ... +6 chain of 2 primes possible, but not more 7:3 (exception: dual 3 have 4! 7=3+2^2)
9 21 39 51 57 63 81 93 99 ... +6 chain of 3 primes possible, but not more 5:4 unavoidable
15 525 585 945 1425 1635 ...+30 chain of 4 primes possible, but not more 31:5 (exception: dual 15 have 6! 31=15+2^4)
1365 1395 3675 4635 5205 7365 chain of 6 primes possible, but not more 127:7
135 225 315 375 645 735 795 1005 1155 chain of 7 primes possible, but not more 17:8
1605 2115 5055 5265 6465 7065 7245 chain of 8 primes possible, but not more 73:9
105 345 435 555 765 855 975 1065 chain of 9 primes possible, but not more 11:10 unavoidable
2475 3465 8415 8745 11055 chain of 10 primes possible, but not more 23,89:11
165 1485 4125 5775 8085 12705 chain of 11 primes possible, but not more 13:12
4922775 7994415 69680325 72400185 chain of 12 primes possible, but not more 8191:13
165165 216645 246675 396825 757185 chain of 13 primes possible, but not more 43:14
1211925 1932645 2387385 3408405 chain of 14 primes possible, but not more 151:15
7054905 7848555 10027875 11156145 chain of 15 primes possible, but not more 257:16
848422575 883017135 1445581995 chain of 16 primes possible, but not more 131071:17
75075 100815 186615 250965b 255255 chain of 17 primes possible, but not more 19:18
chain of 18 primes possible, but not more 524287:19
855855 6317025 7702695 10066485 chain of 19 primes possible, but not more 41:20
17483895 28406235 33948915 39654615 chain of 20 primes possible, but not more 337:21
26857545 69650295 143090805 194605125 chain of 21 primes possible, but not more 683:22
5583435 6561555 17157855 21477885 chain of 22 primes possible, but not more 47,17841:23
41529345 82447365 151812375 330156255 chain of 23 primes possible, but not more 241:24
40877265 186372615 313528215 338144235 chain of 24 primes possible, but not more 601,1801:25
107063385 chain of 25 primes possible, but not more 2731:26
chain of 26 primes possible, but not more 262657:27
18625035 19440135 34030425 34600995 chain of 27 primes possible, but not more 29,113:28
27183585
int proth_dividing(int k, int p)
{
int n;
int x0=(k+1)%p;
int x=(2*x0-1)%p;
for(n=1; n<p; n++,x=(2*x-1)%p) {
if(x==0) return n;
if(x==x0) return 0;
}
return 0;
}
int main(int argc, char *argv[])
{
int k;
int c;
c=0;
for(k=3;c<20;k+=2) {
if(!proth_dividing(k,3)) {
if(!proth_dividing(k,7)) {
if(!proth_dividing(k,5)) {
if(!proth_dividing(k,31)) {
if(!proth_dividing(k,127)) {
if(!proth_dividing(k,17)) {
if(!proth_dividing(k,73)) {
if(!proth_dividing(k,11)) {
if(!(proth_dividing(k,23)||proth_dividing(k,89))) {
if(!proth_dividing(k,13)) {
if(!proth_dividing(k,8191)) {
if(!proth_dividing(k,43)) {
if(!proth_dividing(k,151)) {
if(!proth_dividing(k,257)) {
if(!proth_dividing(k,131071)) {
if(!proth_dividing(k,19)) {
if(!proth_dividing(k,524287)) {
if(!proth_dividing(k,41)) {
if(!proth_dividing(k,337)) {
if(!proth_dividing(k,683)) {
if(!(proth_dividing(k,47)||proth_dividing(k,17841))) {
if(!proth_dividing(k,241)) {
if(!(proth_dividing(k,601)||proth_dividing(k,1801))) {
if(!proth_dividing(k,2731)) {
if(!proth_dividing(k,262657)) {
if((proth_dividing(k,29)||proth_dividing(k,113))) {
printf("%d\n", k);
c++;
}}}}}}}}}}}}}}}}}}}}}}}}}}
}
return 0;
}
2 (3) 7(3) {3:2}
3 (3)*3 5(4) {7:3}
4 (3*5) 31(5) {5:4}
(3*5)*3 7
(3*5)*5 7
5-9 (3*5)*7 11(10) {31:5 127:6 17:8 73:9}
10-11 (3*5*11) 13(12) {23:11,89:11}
(3*5*11*13) 7
(3*5*11*13)*3 17
(3*5*11*13)*5 31 17
(3*5*11*13)*7 17
(3*5*11*13)*9 7 31
(3*5*11*13)*11 7 17 23
(3*5*11*13)*13 73
(3*5*11*13)*15 7
(3*5*11*13)*17 23
(3*5*11*13)*19 23
(3*5*11*13)*21 89
(3*5*11*13)*23 7 17
(3*5*11*13)*25 7
(3*5*11*13)*27 17
(3*5*11*13)*29 7 17 73
(3*5*11*13)*31 17 73
(3*5*11*13)*33 23
12-17 (3*5*11*13)*35 19(18) {8191:13 43:14 151:15 257:16 131071:17}
(3*5*11*13*19) 23
(3*5*11*13*19)*3
(3*5*11*13*19)*5
(3*5*11*13*19)*7
(3*5*11*13*19)*9
(3*5*11*13*19)*11
(3*5*11*13*19)*13
(3*5*11*13*19)*15
(3*5*11*13*19)*17
(3*5*11*13*19)*19
18-19 (3*5*11*13*19)*21 41(20) 241(24) 29(28) 113(28) {524287 41 337 683 47 178481 241 1801 2731 262657 113}
(3*5*11*13*19)*23
(3*5*11*13*19)*25
(3*5*11*13*19)*27
(3*5*11*13*19)*29
(3*5*11*13*19)*31
(3*5*11*13*19)*33
(3*5*11*13*19)*35
(3*5*11*13*19)*37
(3*5*11*13*19)*39
(3*5*11*13*19)*41
(3*5*11*13*19)*43
(3*5*11*13*19)*45
(3*5*11*13*19)*47
(3*5*11*13*19)*49
(3*5*11*13*19)*51
(3*5*11*13*19)*53
(3*5*11*13*19)*55
(3*5*11*13*19)*57
(3*5*11*13*19)*59
(3*5*11*13*19)*61
(3*5*11*13*19)*63
(3*5*11*13*19)*65
(3*5*11*13*19)*67
(3*5*11*13*19)*69
(3*5*11*13*19)*71
(3*5*11*13*19)*73
(3*5*11*13*19)*75
(3*5*11*13*19)*77
(3*5*11*13*19)*79
(3*5*11*13*19)*81
(3*5*11*13*19)*83
(3*5*11*13*19)*85
(3*5*11*13*19)*87
(3*5*11*13*19)*89
(3*5*11*13*19)*91
(3*5*11*13*19)*93
(3*5*11*13*19)*95
(3*5*11*13*19)*97
(3*5*11*13*19)*99
(3*5*11*13*19)*101
(3*5*11*13*19)*103
(3*5*11*13*19)*105
(3*5*11*13*19)*107
(3*5*11*13*19)*109
(3*5*11*13*19)*111
(3*5*11*13*19)*113
(3*5*11*13*19)*115
(3*5*11*13*19)*117
(3*5*11*13*19)*119
(3*5*11*13*19)*121
(3*5*11*13*19)*123
(3*5*11*13*19)*125
(3*5*11*13*19)*127
(3*5*11*13*19)*129
(3*5*11*13*19)*131
(3*5*11*13*19)*133
(3*5*11*13*19)*135
20-22 (3*5*11*13*19)*137 47(23)
(3*5*11*13*19)*139
(3*5*11*13*19)*141
(3*5*11*13*19)*
(3*5*11*13*19)*
(3*5*11*13*19)*
(3*5*11*13*19)*
(3*5*11*13*19)*
(3*5*11*13*19)*
(3*5*11*13*19)*
...
23-27 (3*5*11*13*19)*457 29(28)
18625036 = 2^2 * 1451 * 3209
37250071 es primo
74500141 = 107 * 696263
149000281 es primo
298000561 = 71 * 79 * 53129
596001121 = 1493 * 399197
1192002241 es primo
2384004481 es primo
4768008961 es primo
9536017921 = 3089 * 3087089
19072035841 = 541 * 35253301
38144071681 = 461 * 887 * 93283
76288143361 = 59 * 61 * 1091 * 19429
152576286721 = 136139 * 1120739
305152573441 es primo
610305146881 = 67 * 2861 * 3183863
1220610293761 = 1259 * 969507779
2441220587521 es primo
4882441175041 = 1607 * 18493 * 164291
9764882350081 es primo
19529764700161 = 74419 * 262429819
39059529400321 = 29 * 709 * 6833 * 278017
78119058800641 = 653 * 1321 * 4201 * 21557
156238117601281 = 3343 * 214003 * 218389
312476235202561 = 503 * 179099 * 3468613
624952470405121 es primo
1249904940810241 = 37 * 2837 * 11251 * 1058339
2499809881620481 es primo
4999619763240961 = 53 * 331 * 284992291127
Patrick Demichel
75075*2^3641+1 1101 DM 1996-05-31
Kimmo Herranen
855855*2^100005+1 30111 g335 2002-11-22
855855*2^100222+1 30176 g335 2002-11-22
855855*2^101662+1 30610 g335 2002-11-22
855855*2^105110+1 31648 g335 2002-11-22
855855*2^105517+1 31770 g335 2002-11-22
855855*2^105714+1 31830 g335 2002-11-22
855855*2^105875+1 31878 g335 2002-11-22
855855*2^119676+1 36032 g335 2002-11-22
855855*2^121166+1 36481 g335 2002-11-22
855855*2^124203+1 37395 g335 2002-11-22
855855*2^160882+1 48437 g335 2002-11-23
Douglas Stones
27183585*2^202595+1 60995 L122 2005-08-01
27183585*2^230809+1 69488 L122 2005-10-12
27183585*2^260753+1 78502 L122 2005-10-27
27183585*2^261550+1 78742 L122 2005-10-27
27183585*2^273852+1 82446 L122 2005-12-01
27183585*2^301903+1 90890 L122 2006-02-16
27183585*2^320404+1 96459 L122 2006-04-18
27183585*2^345939+1 104146 L122 2006-05-28
27183585*2^351225+1 105737 L122 2006-06-07
27183585*2^355894+1 107143 L122 2006-06-17
27183585*2^400641+1 120613 L122 2006-09-06
27183585*2^482296+1 145193 L122 2007-01-15
27183585*2^523451+1 157582 L122 2007-04-17
(PrimePuzzles) see problem 6