Problems & Puzzles: Puzzles

Puzzle 1037. (m,n) producing five twin primes. Fausto Morales sent the following nice puzzle: Find pairs of integers m and n, such that m +/- 1,  n +/- 1,  m(n-1) +/- 1, mn +/- 1, and m(n+1) +/- 1 are all primes.   For example, the pair with m = 108 and n = 6300 produces the following family of twin pairs:    {(107,109), (6299,6301), (680291,680293), (680399,680401), (680507,680509)}   Q1: Find more examples. Q2: Are there same-n solutions with two or more different values of m?

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During the week 1-7 May, 2021, contributions came from Giorgos Kalogeropoulos, Simon Cavegn, Ivan Ianakiev, Hakan Summakoglu, Oscar Volpatti and Metin Sariyar

***

Giorgos wrote:

For every n we can find many different values m. 

Here are some pairs {n,m1}, {n,m2}...{n,mk} for ascending n presented in the form:

n -> {m1,m2,m3...}

 

6->{514530,14118900,19035000,27234720,27519840,43462260,91124280,94951740,120092280,136564710,138099780,
156265830...}
12->{20384910,27221040,42197190,57548190,59277540,66060540,106950060,116546010,170517690,267202320...}
18->{324450,12593070,81493230,96746160,104556270,302061270,460048050,658037100,741250230,943971210}
30->{1428,38380608,198977352,239427762,318669078,458121930,473960298,615587238,730105530,832670412,
878314248,882490182...}
42->{9196140,10268370,16360110,81146970,115038360,123087120,144173460,164769090,168696630,178658940,
181449270...}
60->{95088,5704188,7965720,8221668,8458758,8835372,102150510,121641702,157871952,267095010...}
72->{324870,10276350,63530460,65787540,220199700,394683240,616872480,664014120,770359590,840647220...}
150->{174762,1723722,5854800,39473742,52094322,178099152,256530918,276865050,284532738,329728938,
426110328,438778158,610039920...}
240->{3871602,41327328,211440138,213624222,259368690,459722172,521377668,670987548,834414798,879306120,
928314618...}
420->{4398132,7895682,10253232,11601642,14922030,17653638,32062440,59852130,76968378,79128168,82754802,
84134052,97364412,117949032...}
462->{3252870,16898670,17931450,18639390,21685800,26169360,28049310,50965590,98952390,103479750,125915430,
128047560,128903700,130090110...}
570->{2177238,5199810,12319398,58775808,77685930,122912748,140004018,166783848,265552350,367124142...}
660->{4080510,30007278,161724990,265761930,302841420,303840810,388114818,497372148,540769068,551341518...}
 

and the list goes on....

Here is the list for n:

for every n in the list there exist many m (infinite?)

{6,12,18,30,42,60,72,150,240,420,462,570,660,858,882,1050,1092,1290,1302,1320,1428,1788,1932,1998,2268,2310,
2730,2802,2970,3000,3120,3300,3360,3462,3468,3558,3852,4002,4158,5100,5418,5478,5502,5868,5880,6090,6270,
6300,6450,6762,6870,7332,7350,7560,7590,8010,8820,8970,9012,9042,9240,9282,9420,9462,9858,10038......}
 

 

Finally, the pair {n,m} with the largest n that I found is 

{n, m} = {100000001823380820, 12}    

 

***

Simon wrote:

Q1
{(1427,1429), (29,31), (41411,41413), (42839,42841), (44267,44269)}
{(4799,4801), (3359,3361), (16123199,16123201), (16127999,16128001), (16132799,16132801)}
{(107,109), (6299,6301), (680291,680293), (680399,680401), (680507,680509)}
{(239,241), (8819,8821), (2116559,2116561), (2116799,2116801), (2117039,2117041)}
{(1721,1723), (9419,9421), (16219517,16219519), (16221239,16221241), (16222961,16222963)}
{(1319,1321), (25031,25033), (33040919,33040921), (33042239,33042241), (33043559,33043561)}
{(12611,12613), (33809,33811), (426399107,426399109), (426411719,426411721), (426424331,426424333)}
{(9437,9439), (43889,43891), (414224381,414224383), (414233819,414233821), (414243257,414243259)}
{(34157,34159), (51239,51241), (1750221761,1750221763), (1750255919,1750255921), (1750290077,1750290079)}

 

Fast Algorithm:
0. Calculate h = multiplication of first 90 odd primes. (About 90 seems to be the sweet spot for speed in this number range with gnu multiple precision library)
1. Use sieve of Eratosthenes to find twin primes, store twin primes in a list.
2. For each new twin prime found: Test with all previous twin primes if they make an (m,n) pair in both orders: (previous, new) or (new, previous).
2a. Caclulate the 3 other twin numbers from (n, m)
2b. For each of the 6 numbers do a fast preliminary prime test: Only when GCD(h, number) == 1, the number may be prime.
2c. Do a better (slower) prime test for the 6 numbers.

 

Q2 Same-n solutions

An interesting n is 232050, it gives way more same-n solutions than normal in the low number range. Here just the first two:

{(1181879,1181881), (232049,232051), (274254072119,274254072121), (274255253999,274255254001), (274256435879,274256435881)}
{(1357547,1357549), (232049,232051), (315017655851,315017655853), (315019013399,315019013401), (315020370947,315020370949)}

 

Here I picked the first twin prime for n: 4

{(79303559,79303561), (3,5), (237910679,237910681), (317214239,317214241), (396517799,396517801)}
{(232770719,232770721), (3,5), (698312159,698312161), (931082879,931082881), (1163853599,1163853601)}
{(706001939,706001941), (3,5), (2118005819,2118005821), (2824007759,2824007761), (3530009699,3530009701)}
{(707861489,707861491), (3,5), (2123584469,2123584471), (2831445959,2831445961), (3539307449,3539307451)}
{(718257539,718257541), (3,5), (2154772619,2154772621), (2873030159,2873030161), (3591287699,3591287701)}
{(863502359,863502361), (3,5), (2590507079,2590507081), (3454009439,3454009441), (4317511799,4317511801)}
{(1092611519,1092611521), (3,5), (3277834559,3277834561), (4370446079,4370446081), (5463057599,5463057601)}
{(1412224169,1412224171), (3,5), (4236672509,4236672511), (5648896679,5648896681), (7061120849,7061120851)}
{(2123995229,2123995231), (3,5), (6371985689,6371985691), (8495980919,8495980921), (10619976149,10619976151)}
{(2746877489,2746877491), (3,5), (8240632469,8240632471), (10987509959,10987509961), (13734387449,13734387451)}
{(5256920129,5256920131), (3,5), (15770760389,15770760391), (21027680519,21027680521), (26284600649,26284600651)}
{(5799131309,5799131311), (3,5), (17397393929,17397393931), (23196525239,23196525241), (28995656549,28995656551)}
{(7049343839,7049343841), (3,5), (21148031519,21148031521), (28197375359,28197375361), (35246719199,35246719201)}
{(8184297659,8184297661), (3,5), (24552892979,24552892981), (32737190639,32737190641), (40921488299,40921488301)}
{(9227340149,9227340151), (3,5), (27682020449,27682020451), (36909360599,36909360601), (46136700749,46136700751)}
{(10854211829,10854211831), (3,5), (32562635489,32562635491), (43416847319,43416847321), (54271059149,54271059151)}
{(11547094019,11547094021), (3,5), (34641282059,34641282061), (46188376079,46188376081), (57735470099,57735470101)}
{(12042754079,12042754081), (3,5), (36128262239,36128262241), (48171016319,48171016321), (60213770399,60213770401)}
{(12303159329,12303159331), (3,5), (36909477989,36909477991), (49212637319,49212637321), (61515796649,61515796651)}
{(13423330199,13423330201), (3,5), (40269990599,40269990601), (53693320799,53693320801), (67116650999,67116651001)}
{(14436223619,14436223621), (3,5), (43308670859,43308670861), (57744894479,57744894481), (72181118099,72181118101)}
{(14472985169,14472985171), (3,5), (43418955509,43418955511), (57891940679,57891940681), (72364925849,72364925851)}
{(15091494599,15091494601), (3,5), (45274483799,45274483801), (60365978399,60365978401), (75457472999,75457473001)}
{(15162111299,15162111301), (3,5), (45486333899,45486333901), (60648445199,60648445201), (75810556499,75810556501)}
{(15873729689,15873729691), (3,5), (47621189069,47621189071), (63494918759,63494918761), (79368648449,79368648451)}
{(15902518169,15902518171), (3,5), (47707554509,47707554511), (63610072679,63610072681), (79512590849,79512590851)}
{(16140996899,16140996901), (3,5), (48422990699,48422990701), (64563987599,64563987601), (80704984499,80704984501)}
{(17896291769,17896291771), (3,5), (53688875309,53688875311), (71585167079,71585167081), (89481458849,89481458851)}
{(18836552069,18836552071), (3,5), (56509656209,56509656211), (75346208279,75346208281), (94182760349,94182760351)}
{(20811535079,20811535081), (3,5), (62434605239,62434605241), (83246140319,83246140321), (104057675399,104057675401)}
{(21752240789,21752240791), (3,5), (65256722369,65256722371), (87008963159,87008963161), (108761203949,108761203951)}
{(24840295199,24840295201), (3,5), (74520885599,74520885601), (99361180799,99361180801), (124201475999,124201476001)}
{(26205493649,26205493651), (3,5), (78616480949,78616480951), (104821974599,104821974601), (131027468249,131027468251)}
{(29448343259,29448343261), (3,5), (88345029779,88345029781), (117793373039,117793373041), (147241716299,147241716301)}
{(29626885259,29626885261), (3,5), (88880655779,88880655781), (118507541039,118507541041), (148134426299,148134426301)}
{(30077403509,30077403511), (3,5), (90232210529,90232210531), (120309614039,120309614041), (150387017549,150387017551)}
{(30501306989,30501306991), (3,5), (91503920969,91503920971), (122005227959,122005227961), (152506534949,152506534951)}
{(31031497139,31031497141), (3,5), (93094491419,93094491421), (124125988559,124125988561), (155157485699,155157485701)}
{(33179688569,33179688571), (3,5), (99539065709,99539065711), (132718754279,132718754281), (165898442849,165898442851)}
{(34032048329,34032048331), (3,5), (102096144989,102096144991), (136128193319,136128193321), (170160241649,170160241651)}
{(36478101659,36478101661), (3,5), (109434304979,109434304981), (145912406639,145912406641), (182390508299,182390508301)}
{(36771756839,36771756841), (3,5), (110315270519,110315270521), (147087027359,147087027361), (183858784199,183858784201)}
{(36918289379,36918289381), (3,5), (110754868139,110754868141), (147673157519,147673157521), (184591446899,184591446901)}
{(38346292319,38346292321), (3,5), (115038876959,115038876961), (153385169279,153385169281), (191731461599,191731461601)}
{(39873157169,39873157171), (3,5), (119619471509,119619471511), (159492628679,159492628681), (199365785849,199365785851)}
{(40595460569,40595460571), (3,5), (121786381709,121786381711), (162381842279,162381842281), (202977302849,202977302851)}
{(40819274159,40819274161), (3,5), (122457822479,122457822481), (163277096639,163277096641), (204096370799,204096370801)}
{(44219675219,44219675221), (3,5), (132659025659,132659025661), (176878700879,176878700881), (221098376099,221098376101)}
{(45645946499,45645946501), (3,5), (136937839499,136937839501), (182583785999,182583786001), (228229732499,228229732501)}
{(50173515419,50173515421), (3,5), (150520546259,150520546261), (200694061679,200694061681), (250867577099,250867577101)}
{(50229022199,50229022201), (3,5), (150687066599,150687066601), (200916088799,200916088801), (251145110999,251145111001)}
{(53336586659,53336586661), (3,5), (160009759979,160009759981), (213346346639,213346346641), (266682933299,266682933301)}
{(55151680919,55151680921), (3,5), (165455042759,165455042761), (220606723679,220606723681), (275758404599,275758404601)}
{(55781208839,55781208841), (3,5), (167343626519,167343626521), (223124835359,223124835361), (278906044199,278906044201)}
{(60362924159,60362924161), (3,5), (181088772479,181088772481), (241451696639,241451696641), (301814620799,301814620801)}
{(60825126179,60825126181), (3,5), (182475378539,182475378541), (243300504719,243300504721), (304125630899,304125630901)}
{(62332530539,62332530541), (3,5), (186997591619,186997591621), (249330122159,249330122161), (311662652699,311662652701)}
{(62997018419,62997018421), (3,5), (188991055259,188991055261), (251988073679,251988073681), (314985092099,314985092101)}
{(63301471589,63301471591), (3,5), (189904414769,189904414771), (253205886359,253205886361), (316507357949,316507357951)}
{(65878530899,65878530901), (3,5), (197635592699,197635592701), (263514123599,263514123601), (329392654499,329392654501)}
{(66234113609,66234113611), (3,5), (198702340829,198702340831), (264936454439,264936454441), (331170568049,331170568051)}
{(66241768949,66241768951), (3,5), (198725306849,198725306851), (264967075799,264967075801), (331208844749,331208844751)}
{(66859374119,66859374121), (3,5), (200578122359,200578122361), (267437496479,267437496481), (334296870599,334296870601)}
{(67781231699,67781231701), (3,5), (203343695099,203343695101), (271124926799,271124926801), (338906158499,338906158501)}
{(69616845269,69616845271), (3,5), (208850535809,208850535811), (278467381079,278467381081), (348084226349,348084226351)}
{(71986684769,71986684771), (3,5), (215960054309,215960054311), (287946739079,287946739081), (359933423849,359933423851)}
{(73384987829,73384987831), (3,5), (220154963489,220154963491), (293539951319,293539951321), (366924939149,366924939151)}
{(73526192459,73526192461), (3,5), (220578577379,220578577381), (294104769839,294104769841), (367630962299,367630962301)}
{(73958610389,73958610391), (3,5), (221875831169,221875831171), (295834441559,295834441561), (369793051949,369793051951)}
{(74667676439,74667676441), (3,5), (224003029319,224003029321), (298670705759,298670705761), (373338382199,373338382201)}
{(82189577189,82189577191), (3,5), (246568731569,246568731571), (328758308759,328758308761), (410947885949,410947885951)}
{(82464880679,82464880681), (3,5), (247394642039,247394642041), (329859522719,329859522721), (412324403399,412324403401)}
{(83406679019,83406679021), (3,5), (250220037059,250220037061), (333626716079,333626716081), (417033395099,417033395101)}
{(85153424999,85153425001), (3,5), (255460274999,255460275001), (340613699999,340613700001), (425767124999,425767125001)}
{(85423600289,85423600291), (3,5), (256270800869,256270800871), (341694401159,341694401161), (427118001449,427118001451)}
{(85509998909,85509998911), (3,5), (256529996729,256529996731), (342039995639,342039995641), (427549994549,427549994551)}
{(90370248149,90370248151), (3,5), (271110744449,271110744451), (361480992599,361480992601), (451851240749,451851240751)}
{(91246174739,91246174741), (3,5), (273738524219,273738524221), (364984698959,364984698961), (456230873699,456230873701)}
{(96053078939,96053078941), (3,5), (288159236819,288159236821), (384212315759,384212315761), (480265394699,480265394701)}
{(96156724019,96156724021), (3,5), (288470172059,288470172061), (384626896079,384626896081), (480783620099,480783620101)}
{(96309701669,96309701671), (3,5), (288929105009,288929105011), (385238806679,385238806681), (481548508349,481548508351)}
{(98341247339,98341247341), (3,5), (295023742019,295023742021), (393364989359,393364989361), (491706236699,491706236701)}
{(100168466579,100168466581), (3,5), (300505399739,300505399741), (400673866319,400673866321), (500842332899,500842332901)}
{(100427316989,100427316991), (3,5), (301281950969,301281950971), (401709267959,401709267961), (502136584949,502136584951)}
{(100803789239,100803789241), (3,5), (302411367719,302411367721), (403215156959,403215156961), (504018946199,504018946201)}
{(100941510389,100941510391), (3,5), (302824531169,302824531171), (403766041559,403766041561), (504707551949,504707551951)}
{(102843058499,102843058501), (3,5), (308529175499,308529175501), (411372233999,411372234001), (514215292499,514215292501)}
{(103262195189,103262195191), (3,5), (309786585569,309786585571), (413048780759,413048780761), (516310975949,516310975951)}
{(103515261569,103515261571), (3,5), (310545784709,310545784711), (414061046279,414061046281), (517576307849,517576307851)}
{(104949466439,104949466441), (3,5), (314848399319,314848399321), (419797865759,419797865761), (524747332199,524747332201)}
{(105441560069,105441560071), (3,5), (316324680209,316324680211), (421766240279,421766240281), (527207800349,527207800351)}
{(106249904039,106249904041), (3,5), (318749712119,318749712121), (424999616159,424999616161), (531249520199,531249520201)}
{(110026594649,110026594651), (3,5), (330079783949,330079783951), (440106378599,440106378601), (550132973249,550132973251)}
{(110587631489,110587631491), (3,5), (331762894469,331762894471), (442350525959,442350525961), (552938157449,552938157451)}
{(111136754189,111136754191), (3,5), (333410262569,333410262571), (444547016759,444547016761), (555683770949,555683770951)}
{(112521609479,112521609481), (3,5), (337564828439,337564828441), (450086437919,450086437921), (562608047399,562608047401)}
{(112730714249,112730714251), (3,5), (338192142749,338192142751), (450922856999,450922857001), (563653571249,563653571251)}
{(115687580189,115687580191), (3,5), (347062740569,347062740571), (462750320759,462750320761), (578437900949,578437900951)}
{(116001972239,116001972241), (3,5), (348005916719,348005916721), (464007888959,464007888961), (580009861199,580009861201)}
{(117919033259,117919033261), (3,5), (353757099779,353757099781), (471676133039,471676133041), (589595166299,589595166301)}
{(122525443319,122525443321), (3,5), (367576329959,367576329961), (490101773279,490101773281), (612627216599,612627216601)}
{(123464234249,123464234251), (3,5), (370392702749,370392702751), (493856936999,493856937001), (617321171249,617321171251)}
{(123621449489,123621449491), (3,5), (370864348469,370864348471), (494485797959,494485797961), (618107247449,618107247451)}
{(125659049669,125659049671), (3,5), (376977149009,376977149011), (502636198679,502636198681), (628295248349,628295248351)}
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{(1102730268359,1102730268361), (3,5), (3308190805079,3308190805081), (4410921073439,4410921073441), (5513651341799,5513651341801)}
{(1104208952489,1104208952491), (3,5), (3312626857469,3312626857471), (4416835809959,4416835809961), (5521044762449,5521044762451)}
{(1110996743009,1110996743011), (3,5), (3332990229029,3332990229031), (4443986972039,4443986972041), (5554983715049,5554983715051)}
{(1111218632369,1111218632371), (3,5), (3333655897109,3333655897111), (4444874529479,4444874529481), (5556093161849,5556093161851)}
{(1113767216099,1113767216101), (3,5), (3341301648299,3341301648301), (4455068864399,4455068864401), (5568836080499,5568836080501)}
{(1118209760639,1118209760641), (3,5), (3354629281919,3354629281921), (4472839042559,4472839042561), (5591048803199,5591048803201)}
{(1120620028499,1120620028501), (3,5), (3361860085499,3361860085501), (4482480113999,4482480114001), (5603100142499,5603100142501)}
{(1123071167189,1123071167191), (3,5), (3369213501569,3369213501571), (4492284668759,4492284668761), (5615355835949,5615355835951)}
{(1124537199449,1124537199451), (3,5), (3373611598349,3373611598351), (4498148797799,4498148797801), (5622685997249,5622685997251)}
{(1131580823729,1131580823731), (3,5), (3394742471189,3394742471191), (4526323294919,4526323294921), (5657904118649,5657904118651)}
{(1131935312969,1131935312971), (3,5), (3395805938909,3395805938911), (4527741251879,4527741251881), (5659676564849,5659676564851)}
{(1139619430949,1139619430951), (3,5), (3418858292849,3418858292851), (4558477723799,4558477723801), (5698097154749,5698097154751)}
{(1143798809999,1143798810001), (3,5), (3431396429999,3431396430001), (4575195239999,4575195240001), (5718994049999,5718994050001)}
{(1144578843869,1144578843871), (3,5), (3433736531609,3433736531611), (4578315375479,4578315375481), (5722894219349,5722894219351)}
{(1145811305099,1145811305101), (3,5), (3437433915299,3437433915301), (4583245220399,4583245220401), (5729056525499,5729056525501)}
{(1145936211419,1145936211421), (3,5), (3437808634259,3437808634261), (4583744845679,4583744845681), (5729681057099,5729681057101)}
{(1147000831619,1147000831621), (3,5), (3441002494859,3441002494861), (4588003326479,4588003326481), (5735004158099,5735004158101)}
{(1150452775289,1150452775291), (3,5), (3451358325869,3451358325871), (4601811101159,4601811101161), (5752263876449,5752263876451)}
{(1154064275699,1154064275701), (3,5), (3462192827099,3462192827101), (4616257102799,4616257102801), (5770321378499,5770321378501)}
{(1155600492689,1155600492691), (3,5), (3466801478069,3466801478071), (4622401970759,4622401970761), (5778002463449,5778002463451)}
{(1160345915189,1160345915191), (3,5), (3481037745569,3481037745571), (4641383660759,4641383660761), (5801729575949,5801729575951)}

 

***

Ivan wrote:

I found the following 44 {m,n} pairs:

{{42, 322770}, {108, 6300}, {240, 8820}, {240, 630168}, {420,
   139368}, {570, 775740}, {858, 867510}, {1320, 25032}, {1320,
   162288}, {1428, 30}, {1722, 9420}, {3330, 745710}, {4800,
   3360}, {5232, 714840}, {6090, 524958}, {6552, 758730}, {7590,
   148302}, {9438, 43890}, {11940, 809802}, {12612, 33810}, {13398,
   180750}, {13398, 372060}, {15288, 435570}, {16230, 52542}, {17388,
   574620}, {18060, 178068}, {19890, 343770}, {19992, 237090}, {21378,
    867510}, {21840, 552090}, {24180, 985278}, {25410,
   216918}, {26730, 87150}, {26880, 970788}, {30558, 659610}, {34158,
   51240}, {37548, 526740}, {42222, 317730}, {51480, 277578}, {51870,
   601542}, {52710, 8010}, {55440, 34842}, {56532, 93480}, {57348,
   548520}}
 

 

The pairs {858, 867510} and {21378, 867510} show that the answer to Q2 is YES.

 

***

Hakan wrote:

Q1: I found 107 sets for twin primes < 10^10 :

m = 12 and n = 1248240 : {(11,13), (1248239,1248241), (14978867,14978869), (14978879,14978881), (14978891,14978893)}
m = 12 and n = 21680610 : {(11,13), (21680609,21680611), (260167307,260167309), (260167319,260167321), (260167331,260167333)}
m = 12 and n = 34677090 : {(11,13), (34677089,34677091), (416125067,416125069), (416125079,416125081), (416125091,416125093)}
m = 12 and n = 37853130 : {(11,13), (37853129,37853131), (454237547,454237549), (454237559,454237561), (454237571,454237573)}
m = 12 and n = 88912530 : {(11,13), (88912529,88912531), (1066950347,1066950349), (1066950359,1066950361), (1066950371,1066950373)}
m = 12 and n = 94113390 : {(11,13), (94113389,94113391), (1129360667,1129360669), (1129360679,1129360681), (1129360691,1129360693)}
m = 12 and n = 168744450 : {(11,13), (168744449,168744451), (2024933387,2024933389), (2024933399,2024933401), (2024933411,2024933413)}
m = 12 and n = 221202030 : {(11,13), (221202029,221202031), (2654424347,2654424349), (2654424359,2654424361), (2654424371,2654424373)}
m = 12 and n = 563943870 : {(11,13), (563943869,563943871), (6767326427,6767326429), (6767326439,6767326441), (6767326451,6767326453)}
m = 12 and n = 716165940 : {(11,13), (716165939,716165941), (8593991267,8593991269), (8593991279,8593991281), (8593991291,8593991293)}
m = 12 and n = 818228040 : {(11,13), (818228039,818228041), (9818736467,9818736469), (9818736479,9818736481), (9818736491,9818736493)}
m = 18 and n = 17411730 : {(17,19), (17411729,17411731), (313411121,313411123), (313411139,313411141), (313411157,313411159)}
m = 18 and n = 289180920 : {(17,19), (289180919,289180921), (5205256541,5205256543), (5205256559,5205256561), (5205256577,5205256579)}
m = 18 and n = 445353090 : {(17,19), (445353089,445353091), (8016355601,8016355603), (8016355619,8016355621), (8016355637,8016355639)}
m = 18 and n = 539273070 : {(17,19), (539273069,539273071), (9706915241,9706915243), (9706915259,9706915261), (9706915277,9706915279)}
m = 18 and n = 551020470 : {(17,19), (551020469,551020471), (9918368441,9918368443), (9918368459,9918368461), (9918368477,9918368479)}
m = 30 and n = 10498488 : {(29,31), (10498487,10498489), (314954609,314954611), (314954639,314954641), (314954669,314954671)}
m = 30 and n = 20450892 : {(29,31), (20450891,20450893), (613526729,613526731), (613526759,613526761), (613526789,613526791)}
m = 30 and n = 41851152 : {(29,31), (41851151,41851153), (1255534529,1255534531), (1255534559,1255534561), (1255534589,1255534591)}
m = 30 and n = 68227698 : {(29,31), (68227697,68227699), (2046830909,2046830911), (2046830939,2046830941), (2046830969,2046830971)}
m = 30 and n = 102461898 : {(29,31), (102461897,102461899), (3073856909,3073856911), (3073856939,3073856941), (3073856969,3073856971)}
m = 30 and n = 111616890 : {(29,31), (111616889,111616891), (3348506669,3348506671), (3348506699,3348506701), (3348506729,3348506731)}
m = 30 and n = 158574528 : {(29,31), (158574527,158574529), (4757235809,4757235811), (4757235839,4757235841), (4757235869,4757235871)}
m = 42 and n = 322770 : {(41,43), (322769,322771), (13556297,13556299), (13556339,13556341), (13556381,13556383)}
m = 42 and n = 2639700 : {(41,43), (2639699,2639701), (110867357,110867359), (110867399,110867401), (110867441,110867443)}
m = 42 and n = 19893750 : {(41,43), (19893749,19893751), (835537457,835537459), (835537499,835537501), (835537541,835537543)}
m = 42 and n = 27081450 : {(41,43), (27081449,27081451), (1137420857,1137420859), (1137420899,1137420901), (1137420941,1137420943)}
m = 42 and n = 35805660 : {(41,43), (35805659,35805661), (1503837677,1503837679), (1503837719,1503837721), (1503837761,1503837763)}
m = 42 and n = 39625770 : {(41,43), (39625769,39625771), (1664282297,1664282299), (1664282339,1664282341), (1664282381,1664282383)}
m = 42 and n = 63255810 : {(41,43), (63255809,63255811), (2656743977,2656743979), (2656744019,2656744021), (2656744061,2656744063)}
m = 42 and n = 68520450 : {(41,43), (68520449,68520451), (2877858857,2877858859), (2877858899,2877858901), (2877858941,2877858943)}
m = 42 and n = 77405820 : {(41,43), (77405819,77405821), (3251044397,3251044399), (3251044439,3251044441), (3251044481,3251044483)}
m = 42 and n = 80576340 : {(41,43), (80576339,80576341), (3384206237,3384206239), (3384206279,3384206281), (3384206321,3384206323)}
m = 42 and n = 92609550 : {(41,43), (92609549,92609551), (3889601057,3889601059), (3889601099,3889601101), (3889601141,3889601143)}
m = 42 and n = 123633510 : {(41,43), (123633509,123633511), (5192607377,5192607379), (5192607419,5192607421), (5192607461,5192607463)}
m = 42 and n = 141181350 : {(41,43), (141181349,141181351), (5929616657,5929616659), (5929616699,5929616701), (5929616741,5929616743)}
m = 42 and n = 151911450 : {(41,43), (151911449,151911451), (6380280857,6380280859), (6380280899,6380280901), (6380280941,6380280943)}
m = 42 and n = 202664880 : {(41,43), (202664879,202664881), (8511924917,8511924919), (8511924959,8511924961), (8511925001,8511925003)}
m = 60 and n = 5980128 : {(59,61), (5980127,5980129), (358807619,358807621), (358807679,358807681), (358807739,358807741)}
m = 60 and n = 32250372 : {(59,61), (32250371,32250373), (1935022259,1935022261), (1935022319,1935022321), (1935022379,1935022381)}
m = 60 and n = 41120310 : {(59,61), (41120309,41120311), (2467218539,2467218541), (2467218599,2467218601), (2467218659,2467218661)}
m = 60 and n = 69067698 : {(59,61), (69067697,69067699), (4144061819,4144061821), (4144061879,4144061881), (4144061939,4144061941)}
m = 60 and n = 89737872 : {(59,61), (89737871,89737873), (5384272259,5384272261), (5384272319,5384272321), (5384272379,5384272381)}
m = 102 and n = 26570040 : {(101,103), (26570039,26570041), (2710143977,2710143979), (2710144079,2710144081), (2710144181,2710144183)}
m = 102 and n = 71198820 : {(101,103), (71198819,71198821), (7262279537,7262279539), (7262279639,7262279641), (7262279741,7262279743)}
m = 102 and n = 80138100 : {(101,103), (80138099,80138101), (8174086097,8174086099), (8174086199,8174086201), (8174086301,8174086303)}
m = 108 and n = 6300 : {(107,109), (6299,6301), (680291,680293), (680399,680401), (680507,680509)}
m = 150 and n = 20045592 : {(149,151), (20045591,20045593), (3006838649,3006838651), (3006838799,3006838801), (3006838949,3006838951)}
m = 150 and n = 55867812 : {(149,151), (55867811,55867813), (8380171649,8380171651), (8380171799,8380171801), (8380171949,8380171951)}
m = 198 and n = 1280160 : {(197,199), (1280159,1280161), (253471481,253471483), (253471679,253471681), (253471877,253471879)}
m = 198 and n = 5909820 : {(197,199), (5909819,5909821), (1170144161,1170144163), (1170144359,1170144361), (1170144557,1170144559)}
m = 198 and n = 15563100 : {(197,199), (15563099,15563101), (3081493601,3081493603), (3081493799,3081493801), (3081493997,3081493999)}
m = 198 and n = 23586990 : {(197,199), (23586989,23586991), (4670223821,4670223823), (4670224019,4670224021), (4670224217,4670224219)}
m = 240 and n = 8820 : {(239,241), (8819,8821), (2116559,2116561), (2116799,2116801), (2117039,2117041)}
m = 240 and n = 630168 : {(239,241), (630167,630169), (151240079,151240081), (151240319,151240321), (151240559,151240561)}
m = 282 and n = 22293810 : {(281,283), (22293809,22293811), (6286854137,6286854139), (6286854419,6286854421), (6286854701,6286854703)}
m = 312 and n = 12617220 : {(311,313), (12617219,12617221), (3936572327,3936572329), (3936572639,3936572641), (3936572951,3936572953)}
m = 420 and n = 139368 : {(419,421), (139367,139369), (58534139,58534141), (58534559,58534561), (58534979,58534981)}
m = 420 and n = 1419828 : {(419,421), (1419827,1419829), (596327339,596327341), (596327759,596327761), (596328179,596328181)}
m = 420 and n = 5567328 : {(419,421), (5567327,5567329), (2338277339,2338277341), (2338277759,2338277761), (2338278179,2338278181)}
m = 420 and n = 7485810 : {(419,421), (7485809,7485811), (3144039779,3144039781), (3144040199,3144040201), (3144040619,3144040621)}
m = 420 and n = 10657848 : {(419,421), (10657847,10657849), (4476295739,4476295741), (4476296159,4476296161), (4476296579,4476296581)}
m = 420 and n = 15205638 : {(419,421), (15205637,15205639), (6386367539,6386367541), (6386367959,6386367961), (6386368379,6386368381)}
m = 462 and n = 2189460 : {(461,463), (2189459,2189461), (1011530057,1011530059), (1011530519,1011530521), (1011530981,1011530983)}
m = 462 and n = 4025430 : {(461,463), (4025429,4025431), (1859748197,1859748199), (1859748659,1859748661), (1859749121,1859749123)}
m = 570 and n = 775740 : {(569,571), (775739,775741), (442171229,442171231), (442171799,442171801), (442172369,442172371)}
m = 660 and n = 3441102 : {(659,661), (3441101,3441103), (2271126659,2271126661), (2271127319,2271127321), (2271127979,2271127981)}
m = 858 and n = 867510 : {(857,859), (867509,867511), (744322721,744322723), (744323579,744323581), (744324437,744324439)}
m = 858 and n = 7610820 : {(857,859), (7610819,7610821), (6530082701,6530082703), (6530083559,6530083561), (6530084417,6530084419)}
m = 882 and n = 6634890 : {(881,883), (6634889,6634891), (5851972097,5851972099), (5851972979,5851972981), (5851973861,5851973863)}
m = 1092 and n = 1299210 : {(1091,1093), (1299209,1299211), (1418736227,1418736229), (1418737319,1418737321), (1418738411,1418738413)}
m = 1302 and n = 4293630 : {(1301,1303), (4293629,4293631), (5590304957,5590304959), (5590306259,5590306261), (5590307561,5590307563)}
m = 1302 and n = 4542060 : {(1301,1303), (4542059,4542061), (5913760817,5913760819), (5913762119,5913762121), (5913763421,5913763423)}
m = 1320 and n = 25032 : {(1319,1321), (25031,25033), (33040919,33040921), (33042239,33042241), (33043559,33043561)}
m = 1320 and n = 162288 : {(1319,1321), (162287,162289), (214218839,214218841), (214220159,214220161), (214221479,214221481)}
m = 1428 and n = 6320130 : {(1427,1429), (6320129,6320131), (9025144211,9025144213), (9025145639,9025145641), (9025147067,9025147069)}
m = 1722 and n = 9420 : {(1721,1723), (9419,9421), (16219517,16219519), (16221239,16221241), (16222961,16222963)}
m = 1950 and n = 3457188 : {(1949,1951), (3457187,3457189), (6741514649,6741514651), (6741516599,6741516601), (6741518549,6741518551)}
m = 2130 and n = 4436250 : {(2129,2131), (4436249,4436251), (9449210369,9449210371), (9449212499,9449212501), (9449214629,9449214631)}
m = 2130 and n = 4445028 : {(2129,2131), (4445027,4445029), (9467907509,9467907511), (9467909639,9467909641), (9467911769,9467911771)}
m = 2310 and n = 2358738 : {(2309,2311), (2358737,2358739), (5448682469,5448682471), (5448684779,5448684781), (5448687089,5448687091)}
m = 2310 and n = 4177908 : {(2309,2311), (4177907,4177909), (9650965169,9650965171), (9650967479,9650967481), (9650969789,9650969791)}
m = 2310 and n = 4299900 : {(2309,2311), (4299899,4299901), (9932766689,9932766691), (9932768999,9932769001), (9932771309,9932771311)}
m = 3330 and n = 745710 : {(3329,3331), (745709,745711), (2483210969,2483210971), (2483214299,2483214301), (2483217629,2483217631)}
m = 3360 and n = 1749072 : {(3359,3361), (1749071,1749073), (5876878559,5876878561), (5876881919,5876881921), (5876885279,5876885281)}
m = 3582 and n = 1880340 : {(3581,3583), (1880339,1880341), (6735374297,6735374299), (6735377879,6735377881), (6735381461,6735381463)}
m = 5232 and n = 714840 : {(5231,5233), (714839,714841), (3740037647,3740037649), (3740042879,3740042881), (3740048111,3740048113)}
m = 56532 and n = 93480 : {(56531,56533), (93479,93481), (5284554827,5284554829), (5284611359,5284611361), (5284667891,5284667893)}
m = 6090 and n = 524958 : {(6089,6091), (524957,524959), (3196988129,3196988131), (3196994219,3196994221), (3197000309,3197000311)}
m = 6552 and n = 758730 : {(6551,6553), (758729,758731), (4971192407,4971192409), (4971198959,4971198961), (4971205511,4971205513)}
m = 7590 and n = 148302 : {(7589,7591), (148301,148303), (1125604589,1125604591), (1125612179,1125612181), (1125619769,1125619771)}
m = 9438 and n = 43890 : {(9437,9439), (43889,43891), (414224381,414224383), (414233819,414233821), (414243257,414243259)}
m = 11940 and n = 809802 : {(11939,11941), (809801,809803), (9669023939,9669023941), (9669035879,9669035881), (9669047819,9669047821)}
m = 12612 and n = 33810 : {(12611,12613), (33809,33811), (426399107,426399109), (426411719,426411721), (426424331,426424333)}
m = 13398 and n = 180750 : {(13397,13399), (180749,180751), (2421675101,2421675103), (2421688499,2421688501), (2421701897,2421701899)}
m = 13398 and n = 372060 : {(13397,13399), (372059,372061), (4984846481,4984846483), (4984859879,4984859881), (4984873277,4984873279)}
m = 15288 and n = 435570 : {(15287,15289), (435569,435571), (6658978871,6658978873), (6658994159,6658994161), (6659009447,6659009449)}
m = 16230 and n = 52542 : {(16229,16231), (52541,52543), (852740429,852740431), (852756659,852756661), (852772889,852772891)}
m = 17388 and n = 574620 : {(17387,17389), (574619,574621), (9991475171,9991475173), (9991492559,9991492561), (9991509947,9991509949)}
m = 18060 and n = 178068 : {(18059,18061), (178067,178069), (3215890019,3215890021), (3215908079,3215908081), (3215926139,3215926141)}
m = 19890 and n = 343770 : {(19889,19891), (343769,343771), (6837565409,6837565411), (6837585299,6837585301), (6837605189,6837605191)}
m = 19992 and n = 237090 : {(19991,19993), (237089,237091), (4739883287,4739883289), (4739903279,4739903281), (4739923271,4739923273)}
m = 25410 and n = 216918 : {(25409,25411), (216917,216919), (5511860969,5511860971), (5511886379,5511886381), (5511911789,5511911791)}
m = 26730 and n = 87150 : {(26729,26731), (87149,87151), (2329492769,2329492771), (2329519499,2329519501), (2329546229,2329546231)}
m = 34158 and n = 51240 : {(34157,34159), (51239,51241), (1750221761,1750221763), (1750255919,1750255921), (1750290077,1750290079)}
m = 65700 and n = 85932 : {(65699,65701), (85931,85933), (5645666699,5645666701), (5645732399,5645732401), (5645798099,5645798101)}
m = 72222 and n = 92400 : {(72221,72223), (92399,92401), (6673240577,6673240579), (6673312799,6673312801), (6673385021,6673385023)}
     
Q2: It can be found so many sets for same n values too. I found 49 sets for n = 6 and twin primes < 10^10 :

m = 514530 and n = 6 : {(514529,514531), (5,7), (2572649,2572651), (3087179,3087181), (3601709,3601711)}
m = 14118900 and n = 6 : {(14118899,14118901), (5,7), (70594499,70594501), (84713399,84713401), (98832299,98832301)}
m = 19035000 and n = 6 : {(19034999,19035001), (5,7), (95174999,95175001), (114209999,114210001), (133244999,133245001)}
m = 27234720 and n = 6 : {(27234719,27234721), (5,7), (136173599,136173601), (163408319,163408321), (190643039,190643041)}
m = 27519840 and n = 6 : {(27519839,27519841), (5,7), (137599199,137599201), (165119039,165119041), (192638879,192638881)}
m = 43462260 and n = 6 : {(43462259,43462261), (5,7), (217311299,217311301), (260773559,260773561), (304235819,304235821)}
m = 91124280 and n = 6 : {(91124279,91124281), (5,7), (455621399,455621401), (546745679,546745681), (637869959,637869961)}
m = 94951740 and n = 6 : {(94951739,94951741), (5,7), (474758699,474758701), (569710439,569710441), (664662179,664662181)}
m = 120092280 and n = 6 : {(120092279,120092281), (5,7), (600461399,600461401), (720553679,720553681), (840645959,840645961)}
m = 136564710 and n = 6 : {(136564709,136564711), (5,7), (682823549,682823551), (819388259,819388261), (955952969,955952971)}
m = 138099780 and n = 6 : {(138099779,138099781), (5,7), (690498899,690498901), (828598679,828598681), (966698459,966698461)}
m = 156265830 and n = 6 : {(156265829,156265831), (5,7), (781329149,781329151), (937594979,937594981), (1093860809,1093860811)}
m = 166882980 and n = 6 : {(166882979,166882981), (5,7), (834414899,834414901), (1001297879,1001297881), (1168180859,1168180861)}
m = 183488520 and n = 6 : {(183488519,183488521), (5,7), (917442599,917442601), (1100931119,1100931121), (1284419639,1284419641)}
m = 194198580 and n = 6 : {(194198579,194198581), (5,7), (970992899,970992901), (1165191479,1165191481), (1359390059,1359390061)}
m = 223149780 and n = 6 : {(223149779,223149781), (5,7), (1115748899,1115748901), (1338898679,1338898681), (1562048459,1562048461)}
m = 265130850 and n = 6 : {(265130849,265130851), (5,7), (1325654249,1325654251), (1590785099,1590785101), (1855915949,1855915951)}
m = 294710400 and n = 6 : {(294710399,294710401), (5,7), (1473551999,1473552001), (1768262399,1768262401), (2062972799,2062972801)}
m = 320407740 and n = 6 : {(320407739,320407741), (5,7), (1602038699,1602038701), (1922446439,1922446441), (2242854179,2242854181)}
m = 326629560 and n = 6 : {(326629559,326629561), (5,7), (1633147799,1633147801), (1959777359,1959777361), (2286406919,2286406921)}
m = 332204070 and n = 6 : {(332204069,332204071), (5,7), (1661020349,1661020351), (1993224419,1993224421), (2325428489,2325428491)}
m = 342661380 and n = 6 : {(342661379,342661381), (5,7), (1713306899,1713306901), (2055968279,2055968281), (2398629659,2398629661)}
m = 352174830 and n = 6 : {(352174829,352174831), (5,7), (1760874149,1760874151), (2113048979,2113048981), (2465223809,2465223811)}
m = 380645610 and n = 6 : {(380645609,380645611), (5,7), (1903228049,1903228051), (2283873659,2283873661), (2664519269,2664519271)}
m = 422889990 and n = 6 : {(422889989,422889991), (5,7), (2114449949,2114449951), (2537339939,2537339941), (2960229929,2960229931)}
m = 423915840 and n = 6 : {(423915839,423915841), (5,7), (2119579199,2119579201), (2543495039,2543495041), (2967410879,2967410881)}
m = 460251510 and n = 6 : {(460251509,460251511), (5,7), (2301257549,2301257551), (2761509059,2761509061), (3221760569,3221760571)}
m = 495674130 and n = 6 : {(495674129,495674131), (5,7), (2478370649,2478370651), (2974044779,2974044781), (3469718909,3469718911)}
m = 554224440 and n = 6 : {(554224439,554224441), (5,7), (2771122199,2771122201), (3325346639,3325346641), (3879571079,3879571081)}
m = 562088280 and n = 6 : {(562088279,562088281), (5,7), (2810441399,2810441401), (3372529679,3372529681), (3934617959,3934617961)}
m = 577738740 and n = 6 : {(577738739,577738741), (5,7), (2888693699,2888693701), (3466432439,3466432441), (4044171179,4044171181)}
m = 700516560 and n = 6 : {(700516559,700516561), (5,7), (3502582799,3502582801), (4203099359,4203099361), (4903615919,4903615921)}
m = 716694060 and n = 6 : {(716694059,716694061), (5,7), (3583470299,3583470301), (4300164359,4300164361), (5016858419,5016858421)}
m = 749633460 and n = 6 : {(749633459,749633461), (5,7), (3748167299,3748167301), (4497800759,4497800761), (5247434219,5247434221)}
m = 780723300 and n = 6 : {(780723299,780723301), (5,7), (3903616499,3903616501), (4684339799,4684339801), (5465063099,5465063101)}
m = 796918050 and n = 6 : {(796918049,796918051), (5,7), (3984590249,3984590251), (4781508299,4781508301), (5578426349,5578426351)}
m = 806586660 and n = 6 : {(806586659,806586661), (5,7), (4032933299,4032933301), (4839519959,4839519961), (5646106619,5646106621)}
m = 816662700 and n = 6 : {(816662699,816662701), (5,7), (4083313499,4083313501), (4899976199,4899976201), (5716638899,5716638901)}
m = 844133670 and n = 6 : {(844133669,844133671), (5,7), (4220668349,4220668351), (5064802019,5064802021), (5908935689,5908935691)}
m = 944444550 and n = 6 : {(944444549,944444551), (5,7), (4722222749,4722222751), (5666667299,5666667301), (6611111849,6611111851)}
m = 964465080 and n = 6 : {(964465079,964465081), (5,7), (4822325399,4822325401), (5786790479,5786790481), (6751255559,6751255561)}
m = 985392870 and n = 6 : {(985392869,985392871), (5,7), (4926964349,4926964351), (5912357219,5912357221), (6897750089,6897750091)}
m = 992022180 and n = 6 : {(992022179,992022181), (5,7), (4960110899,4960110901), (5952133079,5952133081), (6944155259,6944155261)}
m = 1010587650 and n = 6 : {(1010587649,1010587651), (5,7), (5052938249,5052938251), (6063525899,6063525901), (7074113549,7074113551)}
m = 1185225960 and n = 6 : {(1185225959,1185225961), (5,7), (5926129799,5926129801), (7111355759,7111355761), (8296581719,8296581721)}
m = 1237225050 and n = 6 : {(1237225049,1237225051), (5,7), (6186125249,6186125251), (7423350299,7423350301), (8660575349,8660575351)}
m = 1307658660 and n = 6 : {(1307658659,1307658661), (5,7), (6538293299,6538293301), (7845951959,7845951961), (9153610619,9153610621)}
m = 1341640680 and n = 6 : {(1341640679,1341640681), (5,7), (6708203399,6708203401), (8049844079,8049844081), (9391484759,9391484761)}
m = 1423943460 and n = 6 : {(1423943459,1423943461), (5,7), (7119717299,7119717301), (8543660759,8543660761), (9967604219,9967604221)}

***

Oscar wrote:

There are many sequences of same-n solutions.

n = 4:

m = 79303560, 232770720, 706001940, 707861490, 718257540, 863502360...

n = 6: 

m = 514530, 14118900, 19035000, 27234720, 27519840, 43462260...

n = 12:

m = 20384910, 27221040, 42197190, 57548190, 59277540, 66060540...

etc

 

There are many sequences of same-m solutions too.

m = 12:

n = 1248240, 21680610, 34677090, 37853130, 88912530, 94113390...

m = 18:

n = 17411730, 289180920, 445353090, 539273070, 551020470, 721612920...

m = 30:

n = 10498488, 20450892, 41851152, 68227698, 102461898, 111616890... 

etc

 

It can be conjectured that there are infinitely many such sequences, and that each of them has infinite length.

Let X be the set of all numbers placed between the members of a twin pair:

X = {4, 6, 12, 18, 30, 42, 60, 72...}

We strongly believe that X has infinitely many elements, but it's still a conjecture.

Of course m and n must both belong to X, but they can't be equal: if m = n = x, with x-1 and x+1 both prime, then m*n-1 = x*x-1 = (x-1)*(x+1) is a semiprime.

For fixed n, we have eight linear polynomials in m with coprime coefficients, so Dirichlet's theorem ensures that each of them is prime infinitely often.

Dickson's conjecture states they are ALL prime infinitely often, unless there is some prime p which divides at least one of them for every index m.

But there can be no such "common divisor": for a given prime p, the choice m = p*t makes all eight numbers coprime with p.

For fixed m, we also have eight linear polynomials in n with coprime coefficients.

In this case, the prime 5 is a common divisor for m = 4 and for m = 6, there is no such common divisor for m >=12.

***

Metin wrote:

Q1:

There are many pairs, some I found are;
 

{m,n} => {108,6300}, {240,8820}, {1722,9420}, {1320,25032}, {12612,33810}, {9438,43890},{34158,51240}, {16230,52542}, {65700,85932}, {26730,87150}, {72222,92400},{56532,93480},...

 

Q2: 

Same n results; n=6300=> {108,6300}, {2405832,6300}, {64667922,6300},{86144058,6300}, {86879628,6300}, {95107122,6300}, {155360172,6300},{181735092,6300}

                          n=8820 =>  {240,8820}, {8007870,8820}, {53472582,8820},{77427042,8820}, {80634402,8820}, {89409912,8820}, {123643650,8820}

 

Same m results; m=240=> {240,8820}, {240,630168}, {240,60129762}, {240,61697748}

                               m=1722=> {1722,9420}, {1722,13915830}, {1722,29384190},{1722,43804050}

 

***