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I encountered a number theory problem(I don't know much about number theory) when doing my research:

  1. I want to know whether or not there are infinitely many primes $p$ satisfying $\gcd\left(\dfrac{p-1}{6},6\right)=1$, such that $6$ is a cubic residue mod $p$, but $2$ and $3$ are not cubic residues mod $p$? If there are, can we give a expression of $p$?
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  • 3
    $\begingroup$ It has never been proved that any quadratic polynomial takes on infinitely many prime values. The conjecture that $x^2 + 1$ does has resisted proof for a long time. The Bunyakovsky conjecture suggests that your polynomial does produce infinitely many primes, but it is of course unproved. $\endgroup$ – FredH Mar 26 at 14:16
  • $\begingroup$ Perhaps you could comment on Will Jagy's work and on the comment of @Fred, and let us know what more you want. $\endgroup$ – Gerry Myerson Apr 2 at 2:40
  • $\begingroup$ Previously posted to, but closed at, MO, mathoverflow.net/questions/326348/… – did you ever take my advice to read up on Bunyakovsky's conjecture? $\endgroup$ – Gerry Myerson Apr 2 at 2:43
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The cleanest way to do this is to take primes given by integer $x,y$ in $$ 7 x^2 + 6 xy + 36 y^2, $$ allowing $x,y$ positive or negative as need be. After that, restrict from that list to $p \equiv 7, 31 \pmod {36}.$

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./primego
Input three coefficients a b c for positive f(x,y)= a x^2 + b x y + c y^2 
7 6 36
Discriminant  -972

Modulus for arithmetic progressions? 
36
Maximum number represented? 
10000

      7     37    139    163    181    241    313    337    349    379
    409    421    541    571    607    631    751    859    877    937
   1033   1087   1123   1171   1291   1297   1447   1453   1483   1693
   1741   1747   2011   2161   2239   2311   2371   2473   2539   2647
   2677   2707   2719   2857   3169   3361   3433   3511   3547   3559
   3571   3613   3637   3727   3877   3919   3931   4003   4021   4111
   4201   4219   4261   4297   4357   4363   4441   4507   4561   4603
   4801   4831   4861   4903   4987   4999   5023   5107   5119   5431
   5479   5563   5683   5689   5743   5749   5827   5857   5869   5881
   5923   6073   6343   6379   6397   6469   6571   6577   6733   6781
   6823   6907   6949   7129   7159   7237   7243   7759   7789   7879
   7993   8017   8269   8311   8329   8431   8467   8641   8821   8887
   9007   9127   9151   9199   9241   9283   9319   9397   9433   9547
   9601   9679   9733   9871

    1    7   13   19   25   31

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$

CHOOSE ONLY 7, 31 mod 36:

jagy@phobeusjunior:~$ ./mse
      7    139    571    607    751    859   1087   1123   1291   1447
   1483   2011   2239   2311   2371   2707   3559   3571   3919   3931
   4003   4111   4219   4363   4507   4603   4831   4903   4999   5107
   5119   5431   5479   5683   5827   6343   6379   6907   7159   7243
   7879   8311   8431   8467   8887   9007   9151   9283   9319   9547
   9679   9871

jagy@phobeusjunior:~$ 
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  • $\begingroup$ Can we prove that there are infinite such primes? $\endgroup$ – Zuo Ye Mar 26 at 23:36
  • $\begingroup$ @ZuoYe yes, it is Chebotarev density, as others have pointed out. I just gave a specific way to represent them. A good reference is Primes of the Form $x^2 + n y^2$ by David A. Cox. See Theorem 9.12, in the first edition it is on page 188. I recommend you use a computer, take some integer pairs $x,y$ and calculate $p = 7 x^2 + 6 xy + 36 y^2,$ confirm that you really did get a prime, then check whether $2,3,6$ are cubic residues $\pmod p$ $\endgroup$ – Will Jagy Mar 27 at 0:06
  • $\begingroup$ Thanks for your detailed explanation! I don't know much about number theory and you answer help me a lot. $\endgroup$ – Zuo Ye Mar 27 at 0:09
  • $\begingroup$ @ZuoYe good. Important that you do some numerical experiments to get a feel for this. Do you know any computer languages? $\endgroup$ – Will Jagy Mar 27 at 0:13
  • $\begingroup$ I did get some primes by compuer that satisfies the requirements. I just didn't know how to prove there are infinite such primes before I read your reply. $\endgroup$ – Zuo Ye Mar 27 at 0:15
1
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EXAMPLE, using gp-Pari

PARI/GP is free software, covered by the GNU General Public License,
 and comes WITHOUT ANY WARRANTY WHATSOEVER.

Type ? for help, \q to quit.
Type ?12 for how to get moral (and possibly technical) support.

parisize = 4000000, primelimit = 500000
? x = 983
%1 = 983
? y = 1000
%2 = 1000
? p = 7 * x^2 + 6*x*y + 36 * y^2
%3 = 48662023
? factor(p)
%4 = 
[48662023 1]

? factormod( t^3 - 2, p )
%5 = 
[Mod(1, 48662023)*t^3 + Mod(48662021, 48662023) 1]

? 
? factormod( t^3 - 3, p )
%6 = 
[Mod(1, 48662023)*t^3 + Mod(48662020, 48662023) 1]

? 
? factormod( t^3 - 6, p )
%7 = 
[ Mod(1, 48662023)*t + Mod(6114873, 48662023) 1]

[Mod(1, 48662023)*t + Mod(14273400, 48662023) 1]

[Mod(1, 48662023)*t + Mod(28273750, 48662023) 1]

? 
? 
? 
? p % 36
%8 = 31
? 
? 
? 
? 
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1
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here are a bunch of primes you can check, with the x,y values in $7x^2 +6xy+36y^2 = p$

jagy@phobeusjunior:~$ ./mse
  p:  35706607   x: 1    y:  -996
  p:  35826379   x: 5    y:  -998
  p:  35958343   x: 7    y:  -1000
  p:  36116527   x: 19    y:  1000
  p:  36231583   x: 37    y:  1000
  p:  36378367   x: 59    y:  1000
  p:  36401371   x: 83    y:  998
  p:  36415699   x: 85    y:  998
  p:  36444523   x: 89    y:  998
  p:  36447127   x: 109    y:  996
  p:  36714019   x: 125    y:  998
  p:  36906127   x: 131    y:  1000
  p:  36921823   x: 133    y:  1000
  p:  37065607   x: 151    y:  1000
  p:  37163983   x: 163    y:  1000
  p:  37213927   x: 169    y:  1000
  p:  37366783   x: 187    y:  1000
  p:  37431259   x: 211    y:  998
  p:  37467139   x: 215    y:  998
  p:  37631371   x: 233    y:  998
  p:  37705819   x: 241    y:  998
  p:  37966063   x: 253    y:  1000
  p:  38062183   x: 263    y:  1000
  p:  38199103   x: 277    y:  1000
  p:  38238727   x: 281    y:  1000
  p:  38318647   x: 289    y:  1000
  p:  38520571   x: 323    y:  998
  p:  38731687   x: 329    y:  1000
  p:  38752927   x: 331    y:  1000
  p:  38798563   x: 349    y:  998
  p:  38842171   x: 353    y:  998
  p:  39063571   x: 373    y:  998
  p:  39108523   x: 377    y:  998
  p:  39324823   x: 383    y:  1000
  p:  39531607   x: 401    y:  1000
  p:  39617779   x: 421    y:  998
  p:  39758071   x: 445    y:  996
  p:  40129171   x: 463    y:  998
  p:  40355899   x: 481    y:  998
  p:  40659343   x: 493    y:  1000
  p:  40875979   x: 521    y:  998
  p:  41106103   x: 527    y:  1000
  p:  41294767   x: 541    y:  1000
  p:  41376463   x: 547    y:  1000
  p:  41431207   x: 551    y:  1000
  p:  41596783   x: 563    y:  1000
  p:  41613619   x: 575    y:  998
  p:  41726371   x: 583    y:  998
  p:  41877223   x: 583    y:  1000
  p:  42105607   x: 599    y:  1000
  p:  42244843   x: 619    y:  998
  p:  42303571   x: 623    y:  998
  p:  42392083   x: 629    y:  998
  p:  42573127   x: 631    y:  1000
  p:  42722167   x: 641    y:  1000
  p:  42842203   x: 659    y:  998
  p:  42933739   x: 665    y:  998
  p:  43116223   x: 667    y:  1000
  p:  43149283   x: 679    y:  998
  p:  43336219   x: 691    y:  998
  p:  43677463   x: 703    y:  1000
  p:  43772767   x: 709    y:  1000
  p:  43973899   x: 731    y:  998
  p:  44421007   x: 749    y:  1000
  p:  44553343   x: 757    y:  1000
  p:  44720023   x: 767    y:  1000
  p:  44751271   x: 787    y:  996
  p:  44921911   x: 797    y:  996
  p:  45574423   x: 817    y:  1000
  p:  45689359   x: 841    y:  996
  p:  46175407   x: 851    y:  1000
  p:  46237171   x: 863    y:  998
  p:  46500127   x: 869    y:  1000
  p:  46755823   x: 883    y:  1000
  p:  46896763   x: 899    y:  998
  p:  47088607   x: 901    y:  1000
  p:  47498299   x: 931    y:  998
  p:  47727571   x: 943    y:  998
  p:  47804443   x: 947    y:  998
  p:  48153139   x: 965    y:  998
  p:  48662023   x: 983    y:  1000
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