A293201 Primes p with a primitive root g such that g^3 = g^2 + g + 1 mod p.
7, 11, 13, 17, 19, 41, 47, 53, 73, 107, 131, 139, 149, 163, 167, 199, 227, 257, 263, 269, 271, 293, 311, 347, 349, 359, 373, 401, 419, 421, 431, 479, 523, 557, 599, 617, 683, 701, 757, 761, 769, 809, 827, 863, 877, 907, 911, 929, 937, 953, 991, 1031, 1033
Offset: 1
Keywords
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
- D. Shanks, Fibonacci primitive roots, end of article, Fib. Quart., 10 (1972), 163-168, 181.
Crossrefs
Programs
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Maple
filter:= proc(p) local x,r; if not isprime(p) then return false fi; for r in map(t -> rhs(op(t)), [msolve(x^3-x^2-x-1,p)]) do if numtheory:-order(r,p) = p-1 then return true fi od; false end proc: select(filter, [seq(i,i=3..2000,2)]); # Robert Israel, Oct 02 2017 -
Mathematica
selQ[p_] := AnyTrue[PrimitiveRootList[p], Mod[#^3 - #^2 - # - 1, p] == 0&]; Select[Prime[Range[2, 200]], selQ] (* Jean-François Alcover, Jul 29 2020 *)
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PARI
Z(r, p)=znorder(Mod(r, p))==p-1; \\ whether r is a primitive root mod p Y(p)=for(r=2,p-2,if( Z(r,p) && Mod(r^3-r^2-r-1,p)==0 , return(1))); 0; \\ test p forprime(p=2,10^3,if(Y(p),print1(p,", ")) );
Comments