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 prime7989 2012-07-10 01:53

Mersenne Psuedo Primes

Mp=2^p-1 where p=11 is a Mersenne pseudo prime using FLT with base a=11.
Does anyone know of any other Mersenne pseudo primes?
Thank you,
Allan Menezes

 Batalov 2012-07-10 02:49

Well, all Mp are all pseudo primes in base a=2... (except primes are really primes)

In a>2?

 LaurV 2012-07-10 03:27

They all are pseudo primes to some bases (a LOT of bases). If you mean for a=p, then 11 is the only case.

pari/gp:

[CODE]
[COLOR=Blue](10:25:43) gp > default(primelimit,10^9)[/COLOR]
[COLOR=Blue](10:25:49) gp > p=[B][COLOR=Red]11[/COLOR][/B]; m=1<<p-1; n=m-1; forprime(a=2,n,if(Mod(a,m)^n==1,print1(a",")))[/COLOR]
2,11,67,73,97,139,167,179,223,251,263,269,271,283,311,317,331,367,401,443,449,461,467,509,523,601,607,619,631,673,701,751,757,769,797,809,823,929,947,
971,977,983,1069,1093,1153,1201,1291,1303,1319,1327,1367,1399,1423,1481,1511,1601,1613,1627,1669,1693,1699,1723,1741,1847,1861,1867,1871,1873,1877,190
1,1913,1933,1997,2003,2039,
[COLOR=Blue](10:25:56) gp > p=[COLOR=Red][B]23[/B][/COLOR]; m=1<<p-1; n=m-1; forprime(a=2,n,if(Mod(a,m)^n==1,print1(a",")))[/COLOR]
2,83663,178417,178489,178513,178609,267721,368117,469907,527251,531347,534931,535571,536467,600979,624683,891893,892421,893429,1232983,1249111,1249363
,1257559,1380439,1598137,1606201,1606321,1606331,1606457,1606841,1639097,1650949,1737401,1832219,1874051,1955099,1963259,1967387,2007911,2130617,22756
33,2319997,2409493,2546143,2660831,2673119,2676959,2677151,2677183,2677217,2677343,2677471,2685407,2944937,3034049,3034169,3050561,3099713,3223813,339
1123,3391651,3485957,3739909,3746053,3747077,3747589,3748037,3748099,3748133,3748229,3748357,3780869,3792721,3915427,4096871,4105319,4106087,4170599,4
294699,4462009,4462021,4462033,4462537,4464073,4494793,4593097,4786219,4796677,4818923,4819499,4823083,4841297,4986313,5044877,5086709,5153639,5175917
,5176973,5220569,5270767,5354429,5438093,5531887,5532847,5532907,5533039,5627729,5722547,5885777,5889809,5889857,5889869,5889881,5889889,5912183,59344
93,6057199,6246833,6247859,6341653,6508979,6603733,6604309,6611989,6829687,6895223,6960763,6960791,6960823,6960887,6961271,6962807,7091831,7222903,730
9529,7315673,7317689,7317719,7317977,7412539,7485047,7674619,7674691,7696993,7763923,8029597,8031581,8031641,8031643,8031677,8031773,8032669,8299367,
[COLOR=Blue](10:26:19) gp > p=[B][COLOR=Red]29[/COLOR][/B]; m=1<<p-1; n=m-1; forprime(a=2,n,if(Mod(a,m)^n==1,print1(a",")))[/COLOR]
2,349,827,4597,4817,16229,27259,28517,50221,58451,61583,61837,63127,76423,78317,115303,129083,137873,153313,175633,181739,197453,205157,206813,231551,
231709,251483,262583,265271,267713,270551,308107,331277,334963,344653,356261,358153,362951,370687,386119,393073,403181,405527,409529,417451,463339,495
109,513481,536057,539303,571759,579587,591023,601411,615289,632083,637129,648047,660559,662149,665113,675817,681493,687923,703123,716399,717989,741593
,768793,779189,782011,783721,784249,792359,794239,799679,856333,856903,872789,875323,875419,876569,895343,897251,922699,959689,979757,996011,1008659,1
020137,1065847,1095791,1098037,1106821,1119269,1124509,1125907,1135019,1150397,1157773,1180727,1181309,1181329,1201489,1203437,1234769,1235281,1236467
,1253741,1255591,1257043,1275067,1284739,1284991,1315603,1323599,1328603,1340789,1355843,1364963,1405693,1408669,1427753,1459277,1471919,1478369,14923
57,1493839,1550359,1553093,1592653,1598089,1612069,1613057,1622641,1626649,1626923,1641613,1657937,1691411,1717379,1718933,1763539,1795763,1816963,182
5787,1828397,1842079,1849723,1864661,1877683,1881533,1883191,1892197,1908707,1908923,1925837,1929149,1979893,1980529,1997257,2001313,2016577,2048239,2
054873,2061503,2068109,2070203,2101213,2106437,2112107,2147911,2157091,2160461,2165327,2184317,2187247,2213209,2237401,2244881,2256923,2258651,2269457
,2277703,2285779,2291629,2293723,2294533,2300071,2309389,2315069,2330641,2350123,2364833,2365999,2376961,2377241,2400197,2401547,2403161,2404541,24211
49,2421407,2432869,2433689,2436211,2440213,2443789,2461163,2495833,2496227,2503351,2503433,2504881,2512229,2521751,2579273,2604167,2621693,2631919,263
5099,2640581,2648047,2667571,2682497,2725517,2743673,2743859,2778329,2830591,2831953,2862413,2867869,2933687,2935147,2939219,2939479,2954681,3007723,3
011707,3026633,3034351,3038701,3060793,3082241,3146861,3170399,3186571,3205789,3226147,3228343,3236201,3258667,3266569,3274823,3277657,3285083,3290257
,3313883,3364937,3365251,3365381,3373043,3386371,3417553,3433511,3435589,3484889,3509677,3523829,3596239,3606293,3640999,3643217,3656453,3664327,36672
19,3673507,3674029,3691603,3698363,3701077,3715991,3719693,3720163,3720427,3741739,3767779,3784457,3796997,3812261,3818209,3822649,3827881,3837179,384
1753,3846163,3859081,3885997,3893213,3906827,3939457,3952561,3957271,3964963,3971887,3977497,4004267,4015471,4030363,4047647,4057567,4062413,4064693,4
137031,4161461,4168253,4214879,4216897,4247101,4252081,4261211,4265999,4267763,4277029,4277951,4280191,4285219,4287359,4315097,4324231,4324861,4329211
,4340621,4344653,4362367,4364249,4377839,4379273,4387919,4391027,4419593,4423379,4428223,4461763,4475813,4529389,4537151,4567247,4568989,4574279,45750
83,4575323,4577101,4587007,4621571,4673407,4678549,4680769,4726999,4731577,4748399,4749989,4760627,4770443,4785901,4812251,4850617,4875329,4877651,490
8713,4916441,4927259,4933783,4962259,4966937,4985399,4986647,5044021,5049241,5055803,5055859,5059091,5066081,5071673,5079947,5087413,5113037,5124997,5
127037,5132417,5133013,5135917,5138941,5143291,5152177,5167559,5273759,5288869,5291123,5306789,5319029,5333557,5335763,5343713,5352701,5354123,5354311
,5363257,5423773,5426411,5448889,5462837,5479889,5480491,5483657,5486137,5487457,5487701,5502863,5550217,5552341,5555653,5568791,5574091,5592049,56164
07,5630813,5633149,5701403,5713069,5739553,5745251,5752303,5767217,5774047,5796137,5796991,5816117,5822741,5833783,5844211,5845523,5850473,5893171,589
4353,5906881,5914471,5924489,5937577,5953019,5961587,5962441,5990521,6007913,6014027,6035801,6075617,6088561,6098471,6122167,6135047,6153361,6181993,6
182317,6201799,6206423,6213751,6221767,6234077,6270547,6277189,6309509,6337321,6373867,6422761,6432989,6445381,6451463,6500531,6522031,6531781,6548669
,6584749,6588223,6598523,6618349,6632063,6640409,6663227,6705649,6707087,6734401,6743449,6776281,6779771,6783611,6797927,6821543,6833117,6835259,68649
11,6873437,6896117,6906917,6919117,6923669,6935479,6940007,6941731,6944303,6954383,6960091,6966469,6979003,7015927,7081553,7113109,7130573,7135939,714
1007,7181761,7186987,7192019,7204279,7238071,7241747,7247557,7272833,7277593,7277663,7280149,7282411,7301929,7319857,7329277,7330943,7347359,7360831,7
394041,7417517,7433749,7443043,7450637,7466483,7473377,7496057,7512217,7549907,7580917,7601807,7624121,7633547,7647131,7652321,7655183,7657543,7657807
,7682711,7702111,7713281,7743907,7770667,7792693,7793719,
*** at top-level: ...orprime(a=2,n,if(Mod(a,m)^n==1,print1(a",")))
*** ^--------------------
*** _^_: user interrupt after 422 ms.

*** Break loop: type <Return> to continue; 'break' to go back to GP
break>
[/CODE]

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