求证:\(\frac{2^m-2}{m-1}\ne a\)

太阳

已知:整数\(a＞0\)，\(m＞0\)，\(2^c-1＝m\)，素数\(c＞0\)，合数\(2^c-1\)
求证:\(\frac{2^m-2}{m-1}\ne a\)
命题1是正确，保证下面这个命题也是正确
已知:整数\(a＞0\)，\(2^c-1＝m\)，\(\frac{2^m-2}{m-1}＝a\)，素数\(c＞0\)，\(m＞0\)，\(p＞0\)
求证:\(2^m-1＝p\)

太阳

已知:整数\(a＞0\)，\(\frac{2^c-2}{c-1}＝a\)，奇数\(c＞0\)，素数\(p＞0\)
求证:\(c＝p\)
这个命题是错误，一个反例，\(c＝55\)，\(c\)是伪素数，保证命题是正确吗？
伪素数
341,561,645,1105,1387,1729,1905,2047,2465,2701,2821,3277,4033,4369,4371,4681,5461,6601,7957,8321,8481,8911,10261,10585,11305,12801,13741,13747,13981,14491,15709,15841,16705,18705,18721,19951,23001,23377,25761,29341,30121,30889,31417,31609,31621,33153,34945,35333,39865,41041,41665,42799,46657,49141,49981,52633,55245,57421,60701,60787,62745,63973,65077,65281,68101,72885,74665,75361,80581,83333,83665,85489,87249,88357,88561,90751,91001,93961,101101,104653,107185,113201,115921,121465,123251,126217,129889,129921,130561,137149,149281,150851,154101,157641,158369,161038,162193,162401,164737,172081,176149,181901,188057,188461,194221,196021,196093,204001,206601,208465,212421,215265,215326,215749,219781,220729,223345,226801,228241,233017,241001,249841,252601,253241,256999,258511,264773,266305,271951,272251,275887,276013,278545,280601,282133,284581,285541,289941,294271,294409,314821,318361,323713,332949,334153,340561,341497,348161,357761,367081,387731,390937,396271,399001,401401,410041,422659,423793,427233,435671,443719,448921,449065,451905,452051,458989,464185,476971,481573,486737,488881,489997,493697,493885,512461,513629,514447,526593,530881,534061,552721,556169,563473,574561,574861,580337,582289,587861,588745,604117,611701,617093,622909,625921,635401,642001,647089,653333,656601,657901,658801,665281,665333,665401,670033,672487,679729,680627,683761,688213,710533,711361,721801,722201,722261,729061,738541,741751,742813,743665,745889,748657,757945,769567,769757,786961,800605,818201,825265,831405,838201,838861,841681,847261,852481,852841,873181,875161,877099,898705,915981,916327,934021,950797,976873,983401,997633,1004653,1016801,1018921,1023121,1024651,1033669,1050985,1052503,1052929,1053761,1064053,1073021,1082401,1082809,1092547,1093417,1104349,1106785,1109461,1128121,1132657,1139281,1141141,1145257,1152271,1157689,1168513,1193221,1194649,1207361,1246785,1251949,1252697,1275681,1277179,1293337,1302451,1306801,1325843,1333333,1357441,1357621,1373653,1394185,1397419,1398101,1419607,1433407,1441091,1457773,1459927,1461241,1463749,1472065,1472353,1472505,1485177,1489665,1493857,1500661,1507561,1507963,1509709,1520905,1529185,1530787,1533601,1533961,1534541,1537381,1549411,1569457,1579249,1584133,1608465,1615681,1620385,1643665,1678541,1690501,1711381,1719601,1730977,1735841,1746289,1755001,1773289,1801969,1809697,1811573,1815465,1826203,1827001,1830985,1837381,1839817,1840357,1857241,1876393,1892185,1896961,1907851,1908985,1909001,1937881,1969417,1987021,1993537,1994689,2004403,2008597,2035153,2077545,2081713,2085301,2089297,2100901,2113665,2113921,2121301,2134277,2142141,2144521,2162721,2163001,2165801,2171401,2181961,2184571,2205967,2213121,2232865,2233441,2261953,2264369,2269093,2284453,2288661,2290641,2299081,2304167,2313697,2327041,2350141,2387797,2414001,2419385,2433601,2434651,2455921,2487941,2491637,2503501,2508013,2510569,2513841,2528921,2531845,2537641,2568226,2603381,2609581,2615977,2617451,2626177,2628073,2649029,2649361,2670361,2704801,2719981,2722681,2746477,2746589,2748023,2757241,2773981,2780731,2793351,2797921,2811271,2827801,2867221,2880361,2882265,2899801,2909197,2921161,2940337,2944261,2953711,2976487,2977217,2987167,3020361,3020626,3048841,3057601,3059101,3073357,3090091,3094273,3116107,3125281,3146221,3165961,3181465,3186821,3224065,3225601,3235699,3316951,3336319,3337849,3345773,3363121,3370641,3375041,3375487,3400013,3413533,3429037,3435565,3471071,3539101,3542533,3567481,3568661,3581761,3605429,3656449,3664585,3679201,3726541,3746289,3755521,3763801,3779185,3814357,3828001,3898129,3911197,3916261,3936691,3985921,4005001,4014361,4025905,4038673,4069297,4072729,4082653,4097791,4101637,4151869,4154161,4154977,4181921,4188889,4209661,4229601,4259905,4314967,4335241,4360621,4361389,4363261,4371445,4415251,4463641,4469471,4480477,4502485,4504501,4513841,4535805,4567837,4613665,4650049,4670029,4682833,4698001,4706821,4714201,4767841,4806061,4827613,4835209,4863127,4864501,4868701,4869313,4877641,4895065,4903921,4909177,4917331,4917781,4922413,4974971,4984001,5016191,5031181,5034601,5044033,5049001,5095177,5131589,5133201,5148001,5173169,5173601,5176153,5187637,5193721,5250421,5256091,5258701,5271841,5284333,5310721,5351537,5400489,5423713,5444489,5456881,5481451,5489121,5489641,5524693,5529745,5545145,5551201,5560809,5575501,5590621,5599765,5632705,5672041,5681809,5733649,5758273,5766001,5804821,5859031,5872361,5919187,5968261,5968873,5977153,6027193,6049681,6054985,6118141,6122551,6135585,6140161,6159301,6183601,6189121,6212361,6226193,6233977,6235345,6236257,6236473,6242685,6255341,6278533,6309901,6313681,6334351,6350941,6368689,6386993,6474691,6539527,6617929,6628385,6631549,6658669,6732817,6733693,6749021,6779137,6787327,6836233,6840001,6868261,6886321,6912079,6952037,6955541,6973057,6973063,6998881,7008001,7017193,7040001,7177105,7207201,7215481,7232321,7233265,7259161,7273267,7295851,7306261,7306561,7414333,7416289,7428421,7429117,7455709,7462001,7516153,7519441,7546981,7656721,7674967,7693401,7724305,7725901,7759937,7803769,7808593,7814401,7820201,7866046,7883731,7995169,8012845,8036033,8041345,8043841,8095447,8134561,8137585,8137633,8180461,8209657,8231653,8239477,8280229,8321671,8322945,8341201,8355841,8362201,8384513,8388607,8462233,8534233,8640661,8646121,8650951,8656705,8719309,8719921,8725753,8727391,8745277,8812273,8830801,8902741,8916251,8927101,8992201,9006401,9037729,9040013,9046297,9056501,9063105,9069229,9073513,9084223,9106141,9115426,9131401,9143821,9223401,9224391,9273547,9345541,9371251,9439201,9480461,9494101,9533701,9564169,9567673,9582145,9585541,9588151,9591661,9613297,9638785,9692453,9724177,9729301,9774181,9816465,9834781,9863461,9890881,9908921,9920401,9995671

太阳

已知:整数\(a＞0\)，\(\frac{2^c-2}{c-1}＝a\)，奇数\(c＞0\)，素数\(p＞0\)
求证:\(c＝p\)
这个命题是错误，一个反例，\(c＝55\)，\(c\)是伪素数，保证命题是正确吗？
合数\(2^c-1＝m\)
假设\(\frac{2^m-1}{m-1}＝t\)，整数\(t＞0\)
必定有\(\frac{2^c-1}{c-1}\ne y\)，整数\(y＞0\)
\(c\)是合数，或\(c\)是伪素数，\(2^c-1\)是合数
取\(c＝55\)，\(\frac{2^{55}-2}{54}＝667199944795629\)
\(2^{55}-1＝36028797018963967，m＝36028797018963967\)
必定有\(\frac{2^{36028797018963967}-2}{36028797018963966}\ne a\)
1楼主帖是正确

太阳

已知:整数\(a＞0\)，\(\frac{2^c-2}{c-1}＝a\)，奇数\(c＞0\)，素数\(p＞0\)
求证:\(c＝p\)
这个命题是错误，\(c\)是合数
第一种情况
\(c\)是合数，\(2^c-1＝m\)
\(\frac{2^c-1}{c-1}\ne t\)，整数\(t＞0\)
必定有\(\frac{2^m-2}{m-1}\ne y\)，整数\(y＞0\)
第二种情况
\(c\)是合数，\(2^c-1＝m\)
\(\frac{2^c-1}{c-1}＝t\)，整数\(t＞0\)
必定有\(\frac{2^m-2}{m-1}\ne y\)，整数\(y＞0\)
例1:\(c＝55\)，\(\frac{2^{55}-2}{54}＝667199944795629\)
\(2^{55}-1＝36028797018963967，m＝36028797018963967\)
必定有\(\frac{2^{36028797018963967}-2}{36028797018963966}\ne a\)
1楼主帖是正确

太阳

已知:整数\(a＞0\)，\(m＞0\)，\(2^c-1＝m\)，素数\(c＞0\)，合数\(2^c-1\)
求证:\(\frac{2^m-2}{m-1}\ne a\)
已知:整数\(a＞0\)，\(m＞0\)，\(2^c-1＝m\)，奇数\(c＞0\)，合数\(2^c-1\)
求证:\(\frac{2^m-2}{m-1}\ne a\)
已知:整数\(a＞0\)，\(2^c-1＝m\)，\(\frac{2^m-2}{m-1}＝a\)，素数\(c＞0\)，\(m＞0\)，\(p＞0\)
求证:\(2^m-1＝p\)
第一种情况
\(c\)是合数，\(2^c-1＝m\)
\(\frac{2^c-1}{c-1}\ne t\)，整数\(t＞0\)
必定有\(\frac{2^m-2}{m-1}\ne y\)，整数\(y＞0\)
第二种情况
\(c\)是合数，\(2^c-1＝m\)
\(\frac{2^c-1}{c-1}＝t\)，整数\(t＞0\)
必定有\(\frac{2^m-2}{m-1}\ne y\)，整数\(y＞0\)
例1:\(c＝55\)，\(\frac{2^{55}-2}{54}＝667199944795629\)
\(2^{55}-1＝36028797018963967，m＝36028797018963967\)
必定有\(\frac{2^{36028797018963967}-2}{36028797018963966}\ne a\)
根据两种情况判断，命题3是正确