Are there infinitely many prime p satistying the following conditions?

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 $$p$$?
• 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. – FredH Mar 26 at 14:16
• Perhaps you could comment on Will Jagy's work and on the comment of @Fred, and let us know what more you want. – Gerry Myerson Apr 2 at 2:40
• Previously posted to, but closed at, MO, mathoverflow.net/questions/326348/… – did you ever take my advice to read up on Bunyakovsky's conjecture? – Gerry Myerson Apr 2 at 2:43

3 Answers

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

• Can we prove that there are infinite such primes? – Zuo Ye Mar 26 at 23:36
• @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 $\pmod p$ – Will Jagy Mar 27 at 0:06
• Thanks for your detailed explanation! I don't know much about number theory and you answer help me a lot. – Zuo Ye Mar 27 at 0:09
• @ZuoYe good. Important that you do some numerical experiments to get a feel for this. Do you know any computer languages? – Will Jagy Mar 27 at 0:13
• 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. – Zuo Ye Mar 27 at 0:15

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
?
?
?
?


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