perplexus.info :: Numbers : Does it continue? 4: Euclid's next prime

Does it continue? 4: Euclid's next prime (Posted on 2017-09-21)

Before trying the problem "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."

Let P(n)= The product of the first n primes.

P(n)+1 is like Euclid's method to show there are infinitely many primes, and may or not be prime itself. Now look at the difference between P(n) and the next prime after P(n)+1.

n=1, 5-(2)=3
n=2, 11-(2x3)=5
n=3, 37-(2x3x5)=7
n=4, 223-(2x3x5x7)=13
n=5, 2333-(2x3x5x7x11)=23
n=6, 30047-(2x3x5x7x11x13)=17
n=7, 510529-(2x3x5x7x11x13x17)=19
n=8, 9699713-(2x3x5x7x11x13x17x19)=23

Are these differences always prime?

No Solution Yet Submitted by Jer

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Initial thought and computer exploration

| Comment 1 of 3

Offhand I would not see any reason for this to be the case, but the program below fails to find a counterexample all the way through n=72.

5 open "euclnxpr.txt" for output as #2

10 N=1:Prod=2:P=2

20 repeat

30 inc N:P=nxtprm(P):Prod=Prod*P:Nout=N

40 Nx=fnNxprime(Prod+2)

50 if fnPrime(Nx-Prod)<>1 then print N,Nx,Prod,Nx-Prod,fnPrime(Nx-Prod)

51 print #2,N,Nx,Prod,Nx-Prod,fnPrime(Nx-Prod)

60 until N>71

70 close #2

9999 end

10000 fnOddfact(N)

10010 local K=0,P

10030 while N@2=0

10040 N=N

10050 K=K+1

10060 wend

10070 P=pack(N,K)

10080 return(P)

10090 '

10100 fnPrime(N)

10110 local I,X,J,Y,Q,K,T,Ans

10120 if N@2=0 then Ans=0:goto *EndPrime

10125 O=fnOddfact(N-1)

10130 Q=member(O,1)

10140 K=member(O,2)

10150 I=0

10160 repeat

10170 repeat

10180 X=fnLrand(N)

10190 until X>1

10200 J=0

10210 Y=modpow(X,Q,N)

10220 loop

10230 if or{and{J=0,Y=1},Y=N-1} then goto *ProbPrime

10240 if and{J>0,Y=1} then goto *NotPrime

10250 J=J+1

10260 if J=K then goto *NotPrime

10270 Y=(Y*Y)@N

10280 endloop

10290 *ProbPrime

10300 I=I+1

10310 until I>50

10320 Ans=1

10330 goto *EndPrime

10340 *NotPrime

10350 Ans=0

10360 *EndPrime

10370 return(Ans)

10380 '

10400 fnLrand(N)

10410 local R

10415 N=int(N)

10420 R=(int(rnd*10^(alen(N)+2)))@N

10430 return(R)

10440 '

10500 fnNxprime(X)

10510 if X@2=0 then X=X+1

10520 while fnPrime(X)=0

10530 X=X+2

10540 wend

10550 return(X)

10560 '

finds only primes through the time it was manually stopped at n = 72.

The figures are:

n   next  product  difference  1 indicates
    prime                      it is prime

2 11 6 5 1

3 37 30 7 1

4 223 210 13 1

5 2333 2310 23 1

6 30047 30030 17 1

7 510529 510510 19 1

8 9699713 9699690 23 1

9 223092907 223092870 37 1

10 6469693291 6469693230 61 1

11 200560490197 200560490130 67 1

12 7420738134871 7420738134810 61 1

13 304250263527281 304250263527210 71 1

14 13082761331670077 13082761331670030 47 1

15 614889782588491517 614889782588491410 107 1

16 32589158477190044789 32589158477190044730 59 1

17 1922760350154212639131 1922760350154212639070 61 1

18 117288381359406970983379 117288381359406970983270 109 1

19 7858321551080267055879179 7858321551080267055879090 89 1

20 557940830126698960967415493 557940830126698960967415390 103 1

21 40729680599249024150621323549 40729680599249024150621323470 79 1

22 3217644767340672907899084554281 3217644767340672907899084554130 151 1

23 267064515689275851355624017992987 267064515689275851355624017992790 197 1

24 23768741896345550770650537601358411 23768741896345550770650537601358310 101 1

25 2305567963945518424753102147331756173 2305567963945518424753102147331756070 103 1

26 232862364358497360900063316880507363303 232862364358497360900063316880507363070 233 1

27 23984823528925228172706521638692258396433 23984823528925228172706521638692258396210 223 1

28 2566376117594999414479597815340071648394597 2566376117594999414479597815340071648394470 127 1

29 279734996817854936178276161872067809674997453 279734996817854936178276161872067809674997230 223 1

30 31610054640417607788145206291543662493274687181 31610054640417607788145206291543662493274686990 191 1

31 4014476939333036189094441199026045136645885247893 4014476939333036189094441199026045136645885247730 163 1

32 525896479052627740771371797072411912900610967452859 525896479052627740771371797072411912900610967452630 229 1

33 72047817630210000485677936198920432067383702541010953 72047817630210000485677936198920432067383702541010310 643 1

34 10014646650599190067509233131649940057366334653200433329 10014646650599190067509233131649940057366334653200433090 239 1

35 1492182350939279320058875736615841068547583863326864530567 1492182350939279320058875736615841068547583863326864530410 157 1

36 225319534991831177328890236228992001350685163362356544092077 225319534991831177328890236228992001350685163362356544091910 167 1

37 35375166993717494840635767087951744212057570647889977422430309 35375166993717494840635767087951744212057570647889977422429870 439 1

38 5766152219975951659023630035336134306565384015606066319856069049 5766152219975951659023630035336134306565384015606066319856068810 239 1

39 962947420735983927056946215901134429196419130606213075415963491469 962947420735983927056946215901134429196419130606213075415963491270 199 1

40 166589903787325219380851695350896256250980509594874862046961683989901 166589903787325219380851695350896256250980509594874862046961683989710 191 1

41 29819592777931214269172453467810429868925511217482600306406141434158289 29819592777931214269172453467810429868925511217482600306406141434158090 199 1

42 5397346292805549782720214077673687806275517530364350655459511599582614673 5397346292805549782720214077673687806275517530364350655459511599582614290 383 1

43 1030893141925860008499560888835674370998623848299590975192766715520279329623 1030893141925860008499560888835674370998623848299590975192766715520279329390 233 1

44 198962376391690981640415251545285153602734402721821058212203976095413910573021 198962376391690981640415251545285153602734402721821058212203976095413910572270 751 1

45 39195588149163123383161804554421175259738677336198748467804183290796540382737503 39195588149163123383161804554421175259738677336198748467804183290796540382737190 313 1

46 7799922041683461553249199106329813876687996789903550945093032474868511536164701583 7799922041683461553249199106329813876687996789903550945093032474868511536164700810 773 1

47 1645783550795210387735581011435590727981167322669649249414629852197255934130751871517 1645783550795210387735581011435590727981167322669649249414629852197255934130751870910 607 1

48 367009731827331916465034565550136732339800312955331782619462457039988073311157667213243 367009731827331916465034565550136732339800312955331782619462457039988073311157667212930 313 1

49 83311209124804345037562846379881038241134671040860314654617977748077292641632790457335493 83311209124804345037562846379881038241134671040860314654617977748077292641632790457335110 383 1

50 19078266889580195013601891820992757757219839668357012055907516904309700014933909014729740483 19078266889580195013601891820992757757219839668357012055907516904309700014933909014729740190 293 1

51 4445236185272185438169240794291312557432222642727183809026451438704160103479600800432029464713 4445236185272185438169240794291312557432222642727183809026451438704160103479600800432029464270 443 1

52 1062411448280052319722448549835623701226301211611796930357321893850294264731624591303255041960861 1062411448280052319722448549835623701226301211611796930357321893850294264731624591303255041960530 331 1

53 256041159035492609053110100510385311995538591998443060216114576417920917800321526504084465112488013 256041159035492609053110100510385311995538591998443060216114576417920917800321526504084465112487730 283 1

54 64266330917908644872330635228106713310880186591609208114244758680898150367880703152525200743234420507 64266330917908644872330635228106713310880186591609208114244758680898150367880703152525200743234420230 277 1

55 16516447045902521732188973253623425320896207954043566485360902980990824644545340710198976591011245999381 16516447045902521732188973253623425320896207954043566485360902980990824644545340710198976591011245999110 271 1

56 4343825573072363215565699965702960859395702691913457985649917484000586881515424606782330843435957697766331 4343825573072363215565699965702960859395702691913457985649917484000586881515424606782330843435957697765930 401 1

57 1168489079156465704987173290774096471177444024124720198139827803196157871127649219224446996884272620699035477 1168489079156465704987173290774096471177444024124720198139827803196157871127649219224446996884272620699035170 307 1

58 316660540451402206051523961799780143689087330537799173695893334666158783075592938409825136155637880209438531401 316660540451402206051523961799780143689087330537799173695893334666158783075592938409825136155637880209438531070 331 1

59 87714969705038411076272137418539099801877190558970371113762453702525982911939243939521562715111692818014473106769 87714969705038411076272137418539099801877190558970371113762453702525982911939243939521562715111692818014473106390 379 1

60 24647906487115793512432470614609487044327490547070674282967249490409801198254927547005559122946385681862066942896081 24647906487115793512432470614609487044327490547070674282967249490409801198254927547005559122946385681862066942895590 491 1

61 6975357535853769564018389183934484833544679824821000822079731605785973739106144495802573231793827147966964944839452301 6975357535853769564018389183934484833544679824821000822079731605785973739106144495802573231793827147966964944839451970 331 1

62 2043779758005154482257388030892804056228591188672553240869361360495290305558100337270153956915591354354320728837959427521 2043779758005154482257388030892804056228591188672553240869361360495290305558100337270153956915591354354320728837959427210 311 1

63 627440385707582426053018125484090845262177494922473844946893937672054123806336803541937264773086545786776463753253544153867 627440385707582426053018125484090845262177494922473844946893937672054123806336803541937264773086545786776463753253544153470 397 1

64 195133959955058134502488637025552252876537200920889365778484014616008832503770745901542489344429915739687480227261852231729501 195133959955058134502488637025552252876537200920889365778484014616008832503770745901542489344429915739687480227261852231729170 331 1

65 61076929465933196099278943388997855150356143888238371488665496574810764573680243467182799164806563626522181311132959748531230563 61076929465933196099278943388997855150356143888238371488665496574810764573680243467182799164806563626522181311132959748531230210 353 1

66 19361386640700823163471425054312320082662897612571563761906962414215012369856637179096947335243680669607531475629148240284399976989 19361386640700823163471425054312320082662897612571563761906962414215012369856637179096947335243680669607531475629148240284399976570 419 1

67 6408618978071972467109041692977377947361419109761187605191204559105169094422546906281089567965658301640092918433248067534136392245091 6408618978071972467109041692977377947361419109761187605191204559105169094422546906281089567965658301640092918433248067534136392244670 421 1

68 2159704595610254721415747050533376368260798239989520222949435936418441984820398307416727184404426847652711313512004598759003964186454673 2159704595610254721415747050533376368260798239989520222949435936418441984820398307416727184404426847652711313512004598759003964186453790 883 1

69 749417494676758388331264226535081599786496989276363517363454269937199368732678212673604332988336116135490825788665595769374375572699465677 749417494676758388331264226535081599786496989276363517363454269937199368732678212673604332988336116135490825788665595769374375572699465130 547 1

70 261546705642188677527611215060743478325487449257450867559845540208082579687704696223087912212929304531286298200244292923511657074872113331751 261546705642188677527611215060743478325487449257450867559845540208082579687704696223087912212929304531286298200244292923511657074872113330370 1381 1

71 92325987091692603167246758916442447848897069587880156248625475693453150629759757766750033011164044499544063264686235401999614947429856005621067 92325987091692603167246758916442447848897069587880156248625475693453150629759757766750033011164044499544063264686235401999614947429856005620610 457 1

72 33145029365917644537041586451002838777754047982048976093256545773949681076083753038263261851007891975336318712022358509317861766127318306017799447 33145029365917644537041586451002838777754047982048976093256545773949681076083753038263261851007891975336318712022358509317861766127318306017798990 457 1

Edit: a version of the program has verified through n=135, and is continuing.

Edited on September 21, 2017, 4:47 pm

By now it has verified through 151, and I'm terminating the execution.

Edited on September 21, 2017, 7:20 pm

Posted by Charlie on 2017-09-21 13:05:22

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