A014753 Primes p==1 (mod 6) such that 3 and -3 are both cubes (one implies other) modulo p.

61, 67, 73, 103, 151, 193, 271, 307, 367, 439, 499, 523, 547, 577, 613, 619, 643, 661, 727, 757, 787, 853, 919, 967, 991, 997, 1009, 1021, 1093, 1117, 1249, 1303, 1321, 1399, 1531, 1543, 1549, 1597, 1609, 1621, 1669, 1759, 1783, 1861, 1867

Offset: 1

Author

Warren D. Smith

Keywords

• nonn
• easy

Comments

Primes of the form x^2+xy+61y^2, whose discriminant is -243. - T. D. Noe, May 17 2005

Primes of the form (x^2 + 243*y^2)/4. - Arkadiusz Wesolowski, May 30 2015

References

• K. Ireland and M. Rosen, A classical introduction to modern number theory, Vol. 84, Graduate Texts in Mathematics, Springer-Verlag. Exercise 23, p. 135.

Links

• Bruno Berselli, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A014752, A014755.

Programs

Mathematica

p6 = Select[6*Range[0, 400]+1, PrimeQ]; Select[p6, (Reduce[3 == k^3+m*#, {k, m}, Integers] =!= False)&] (* Jean-François Alcover, Feb 20 2014 *)

PARI

forprime(p=1, 9999, p%6==1&&ispower(Mod(3, p), 3)&&print1(p", ")) \\ M. F. Hasler, Feb 18 2014

PARI

is_A014753(p)={p%6==1&&ispower(Mod(3, p), 3)&&isprime(p)} \\ M. F. Hasler, Feb 18 2014

Extensions

Offset changed from 0 to 1 by Bruno Berselli, Feb 20 2014