Contact
CoCalc Logo Icon
StoreFeaturesDocsShareSupport News Sign UpSign In
| Download
Views: 418

The issue I was trying to explore computationally is whether the largest prime factor of 3^n-2^n, for n>1, occurs only once in the prime factorization. Seems SageMath might be a better bet.

-- Tweet

f = list(factor(3^6-2^6)) f sorted(f)[-1]
[(5, 1), (7, 1), (19, 1)] (19, 1)
for n in srange(2, 200): f = factor(3^n-2^n) largest = sorted(f)[-1] interesting = '*' if largest[1] > 1 else '' print "n: {0:>5d}, largest: {1[0]:>60d}, frequency: {1[1]:>3d} {2}".format(n, largest, interesting)
n: 2, largest: 5, frequency: 1 n: 3, largest: 19, frequency: 1 n: 4, largest: 13, frequency: 1 n: 5, largest: 211, frequency: 1 n: 6, largest: 19, frequency: 1 n: 7, largest: 71, frequency: 1 n: 8, largest: 97, frequency: 1 n: 9, largest: 1009, frequency: 1 n: 10, largest: 211, frequency: 1 n: 11, largest: 331, frequency: 1 n: 12, largest: 61, frequency: 1 n: 13, largest: 29927, frequency: 1 n: 14, largest: 463, frequency: 1 n: 15, largest: 3571, frequency: 1 n: 16, largest: 401, frequency: 1 n: 17, largest: 129009091, frequency: 1 n: 18, largest: 1009, frequency: 1 n: 19, largest: 745181, frequency: 1 n: 20, largest: 4621, frequency: 1 n: 21, largest: 6217, frequency: 1 n: 22, largest: 35839, frequency: 1 n: 23, largest: 2002867877, frequency: 1 n: 24, largest: 5521, frequency: 1 n: 25, largest: 39756701, frequency: 1 n: 26, largest: 29927, frequency: 1 n: 27, largest: 397760329, frequency: 1 n: 28, largest: 369181, frequency: 1 n: 29, largest: 68629840493971, frequency: 1 n: 30, largest: 3571, frequency: 1 n: 31, largest: 617671248800299, frequency: 1 n: 32, largest: 14177, frequency: 1 n: 33, largest: 24939553, frequency: 1 n: 34, largest: 129009091, frequency: 1 n: 35, largest: 18906721, frequency: 1 n: 36, largest: 1009, frequency: 1 n: 37, largest: 50642213021, frequency: 1 n: 38, largest: 745181, frequency: 1 n: 39, largest: 134473349011, frequency: 1 n: 40, largest: 4621, frequency: 1 n: 41, largest: 13869481, frequency: 1 n: 42, largest: 92233, frequency: 1 n: 43, largest: 636618868367, frequency: 1 n: 44, largest: 2414250301, frequency: 1 n: 45, largest: 204521695561, frequency: 1 n: 46, largest: 2002867877, frequency: 1 n: 47, largest: 77681973839, frequency: 1 n: 48, largest: 41432641, frequency: 1 n: 49, largest: 589955028851094677, frequency: 1 n: 50, largest: 39756701, frequency: 1 n: 51, largest: 904841491, frequency: 1 n: 52, largest: 13761229, frequency: 1 n: 53, largest: 19383245658672820642055731, frequency: 1 n: 54, largest: 397760329, frequency: 1 n: 55, largest: 143083295851, frequency: 1 n: 56, largest: 3001769, frequency: 1 n: 57, largest: 8191816751977, frequency: 1 n: 58, largest: 68629840493971, frequency: 1 n: 59, largest: 14130386091162273752461387579, frequency: 1 n: 60, largest: 24001, frequency: 1 n: 61, largest: 6643288986915481, frequency: 1 n: 62, largest: 617671248800299, frequency: 1 n: 63, largest: 278775379, frequency: 1 n: 64, largest: 1607133116929, frequency: 1 n: 65, largest: 319716931, frequency: 1 n: 66, largest: 24939553, frequency: 1 n: 67, largest: 19487546413505693209, frequency: 1 n: 68, largest: 1282861452271981, frequency: 1 n: 69, largest: 2002867877, frequency: 1 n: 70, largest: 392943879511, frequency: 1 n: 71, largest: 111998979662707645844109121, frequency: 1 n: 72, largest: 27793, frequency: 1 n: 73, largest: 357153955800796052994163, frequency: 1 n: 74, largest: 604408111852283, frequency: 1 n: 75, largest: 16751715789451, frequency: 1 n: 76, largest: 103911691734684541, frequency: 1 n: 77, largest: 24634250588641, frequency: 1 n: 78, largest: 134473349011, frequency: 1 n: 79, largest: 6743424997709927, frequency: 1 n: 80, largest: 1783433557073281, frequency: 1 n: 81, largest: 48110884801223689, frequency: 1 n: 82, largest: 223574641, frequency: 1 n: 83, largest: 47890252291257128126044781281592543, frequency: 1 n: 84, largest: 8639401, frequency: 1 n: 85, largest: 58797880998898354362629971, frequency: 1 n: 86, largest: 636618868367, frequency: 1 n: 87, largest: 68629840493971, frequency: 1 n: 88, largest: 184879309516177, frequency: 1 n: 89, largest: 706132008101135602203621405289, frequency: 1 n: 90, largest: 204521695561, frequency: 1 n: 91, largest: 29602264147, frequency: 1 n: 92, largest: 12798522145252969, frequency: 1 n: 93, largest: 485158402948122523, frequency: 1 n: 94, largest: 11170373961701, frequency: 1 n: 95, largest: 13720405721466007835103211, frequency: 1 n: 96, largest: 245953921, frequency: 1 n: 97, largest: 59813046375181769306016700165290169537, frequency: 1 n: 98, largest: 589955028851094677, frequency: 1 n: 99, largest: 12884961778437920314402513, frequency: 1 n: 100, largest: 104189401, frequency: 1 n: 101, largest: 1546132562196033990574082188840405015112916155251, frequency: 1 n: 102, largest: 904841491, frequency: 1 n: 103, largest: 13276260528852139084537022924276558050559, frequency: 1 n: 104, largest: 82104167839721, frequency: 1 n: 105, largest: 15953660849491, frequency: 1 n: 106, largest: 19383245658672820642055731, frequency: 1 n: 107, largest: 33661629004279868278205583994326979, frequency: 1 n: 108, largest: 397760329, frequency: 1 n: 109, largest: 8274026360284002523, frequency: 1 n: 110, largest: 28909244547481, frequency: 1 n: 111, largest: 4185699723047415259, frequency: 1 n: 112, largest: 1009493350615396529, frequency: 1 n: 113, largest: 3760808297896862725084484133384890659, frequency: 1 n: 114, largest: 8191816751977, frequency: 1 n: 115, largest: 1244003531637747395162400674351, frequency: 1 n: 116, largest: 68629840493971, frequency: 1 n: 117, largest: 455049775436257, frequency: 1 n: 118, largest: 14130386091162273752461387579, frequency: 1 n: 119, largest: 42908016367933271190455539986407, frequency: 1 n: 120, largest: 2201414671278481, frequency: 1 n: 121, largest: 8294725495836621986715516373121185537, frequency: 1 n: 122, largest: 209061986047163521, frequency: 1 n: 123, largest: 132972762370027, frequency: 1 n: 124, largest: 2268330947579252221, frequency: 1 n: 125, largest: 515397931586890805724262529672368870694338434001, frequency: 1 n: 126, largest: 194208941809, frequency: 1 n: 127, largest: 3957859287933228402790714823803, frequency: 1 n: 128, largest: 36629538145348481, frequency: 1 n: 129, largest: 56748336081377426535170587, frequency: 1 n: 130, largest: 149166625778220961, frequency: 1 n: 131, largest: 1210399177182288006201752262354382648158190136861552303421773, frequency: 1 n: 132, largest: 40834167001, frequency: 1 n: 133, largest: 60286285357068323908884890185142379901668431, frequency: 1 n: 134, largest: 19487546413505693209, frequency: 1 n: 135, largest: 6485668152547332660657841, frequency: 1 n: 136, largest: 1282861452271981, frequency: 1 n: 137, largest: 41856855393835922319755115319026643, frequency: 1 n: 138, largest: 1266021208542598810111, frequency: 1 n: 139, largest: 3326254807627778280849547, frequency: 1 n: 140, largest: 3330924131977261, frequency: 1 n: 141, largest: 131479459318792575326390806103446996508857, frequency: 1 n: 142, largest: 111998979662707645844109121, frequency: 1 n: 143, largest: 8608106912640683724041725103, frequency: 1 n: 144, largest: 25169360989905069409, frequency: 1 n: 145, largest: 482760636131855627222316551, frequency: 1 n: 146, largest: 357153955800796052994163, frequency: 1 n: 147, largest: 603113129008172292789931, frequency: 1 n: 148, largest: 31330151333999377, frequency: 1 n: 149, largest: 35826894687127246804985548437808941002293995812091691, frequency: 1 n: 150, largest: 16751715789451, frequency: 1 n: 151, largest: 1553871519815521574742646118801067039341, frequency: 1 n: 152, largest: 131233622361283804841, frequency: 1 n: 153, largest: 100798517038294835716321, frequency: 1 n: 154, largest: 2805980251094900881, frequency: 1 n: 155, largest: 2218167055670022437734191976031585121031497411717039341, frequency: 1 n: 156, largest: 134473349011, frequency: 1 n: 157, largest: 969330861796869994456492130438574359, frequency: 1 n: 158, largest: 6743424997709927, frequency: 1 n: 159, largest: 40668809718666344663272025438059, frequency: 1 n: 160, largest: 17911723223817601, frequency: 1 n: 161, largest: 3546539189212321013014718087, frequency: 1 n: 162, largest: 119401875833029529220991, frequency: 1 n: 163, largest: 283519263045670202950666700299218504457, frequency: 1 n: 164, largest: 102329189594547549657540565413396038701, frequency: 1 n: 165, largest: 29231679455089165223401, frequency: 1 n: 166, largest: 798167678837469920188160718521149336927, frequency: 1 n: 167, largest: 284602839755962600307038183361142274453177384697761703968640951718869, frequency: 1 n: 168, largest: 1066416705314353, frequency: 1 n: 169, largest: 117995171693834454246644003933945645889052923318362305633969, frequency: 1 n: 170, largest: 5051752485270228777139427773111, frequency: 1 n: 171, largest: 8117487912611031245026653397581662995837458361, frequency: 1 n: 172, largest: 67285658924969260260836929, frequency: 1 n: 173, largest: 25395520580004769792062089975593383731943129452467071848046383813, frequency: 1 n: 174, largest: 672870264554455488882904591, frequency: 1 n: 175, largest: 203315521506434771079581843014801, frequency: 1 n: 176, largest: 30576308069075829315234744136241, frequency: 1 n: 177, largest: 14130386091162273752461387579, frequency: 1 n: 178, largest: 706132008101135602203621405289, frequency: 1 n: 179, largest: 19238611434292026195045887544024913, frequency: 1 n: 180, largest: 204521695561, frequency: 1 n: 181, largest: 180177770641636417112546690128193917944439092140520910441370256179661913, frequency: 1 n: 182, largest: 41204133559341077510740829, frequency: 1 n: 183, largest: 995988534023541908441821015651, frequency: 1 n: 184, largest: 14961296951860360853213161489, frequency: 1 n: 185, largest: 144283286596886504823039088995815387788435308012527125576301, frequency: 1 n: 186, largest: 485158402948122523, frequency: 1 n: 183, largest: 995988534023541908441821015651, frequency: 1 n: 184, largest: 14961296951860360853213161489, frequency: 1 n: 180, largest: 204521695561, frequency: 1 n: 181, largest: 180177770641636417112546690128193917944439092140520910441370256179661913, frequency: 1 n: 182, largest: 41204133559341077510740829, frequency: 1 n: 187, largest: 1459790465782658058025281463338798630403528636354133521, frequency: 1 n: 188, largest: 581485707003066089360821, frequency: 1 n: 189, largest: 5224174896652377110521061329, frequency: 1 n: 190, largest: 13720405721466007835103211, frequency: 1 n: 191, largest: 17529097123947879503762992973014217865235532554975171, frequency: 1 n: 192, largest: 3433675861649849282330674248961, frequency: 1 n: 193, largest: 157116815095122696291789672145814943987605497895096234870661710074857006307174092298131047, frequency: 1 n: 194, largest: 59813046375181769306016700165290169537, frequency: 1 n: 195, largest: 333547824886531319388057241, frequency: 1 n: 196, largest: 6274085743404921460537, frequency: 1 n: 197, largest: 14158605703498662092605666552664253711127693911176505757607, frequency: 1 n: 198, largest: 12884961778437920314402513, frequency: 1 n: 199, largest: 432042242155762716591315528631609, frequency: 1