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 A134572 Prime numbers p for which there is exactly one root x of x^3 - x - 1 in F_p and x is a primitive root mod p. 1

%I #15 Jul 08 2023 08:06:52

%S 5,7,11,17,37,67,83,113,199,227,241,251,283,367,373,401,433,457,479,

%T 569,571,613,643,659,701,727,743,757,769,839,919,941,977,1019,1031,

%U 1049,1103,1109,1171,1187,1201,1249,1279,1367,1399,1423,1433,1471,1487,1493,1583,1601

%N Prime numbers p for which there is exactly one root x of x^3 - x - 1 in F_p and x is a primitive root mod p.

%C Gil, Weiner, & Zara prove that there is a unique complete Padovan sequence in F_p for each prime p in this sequence, which is generated by x. - _Charles R Greathouse IV_, Nov 26 2014

%H Juan B. Gil, Michael D. Weiner and Catalin Zara, <a href="http://www.fq.math.ca/Papers1/45-1/quartgil01_2007.pdf">Complete Padovan sequences in finite fields</a>, The Fibonacci Quarterly, Volume 45 Number 1, Feb 2007, pp. 64-75, see p. 71.

%o (PARI) is(n)=if(!isprime(n), return(0)); my(f=factormod('x^3-'x-1,n)[,1]); f=select(t->poldegree(t)==1, f); #f==1 && znorder(-polcoeff(f[1], 0))==n-1 \\ _Charles R Greathouse IV_, Nov 26 2014

%Y Cf. A134573.

%K nonn

%O 1,1

%A _Gary W. Adamson_, Nov 01 2007

%E Corrected and extended by _Charles R Greathouse IV_, Nov 26 2014

%E New name from _Charles R Greathouse IV_, Nov 26 2014

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Last modified July 15 10:49 EDT 2024. Contains 374332 sequences. (Running on oeis4.)