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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
5, 7, 11, 17, 37, 67, 83, 113, 199, 227, 241, 251, 283, 367, 373, 401, 433, 457, 479, 569, 571, 613, 643, 659, 701, 727, 743, 757, 769, 839, 919, 941, 977, 1019, 1031, 1049, 1103, 1109, 1171, 1187, 1201, 1249, 1279, 1367, 1399, 1423, 1433, 1471, 1487, 1493, 1583, 1601 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..52.

Juan B. Gil, Michael D. Weiner and Catalin Zara, Complete Padovan sequences in finite fields, The Fibonacci Quarterly, Volume 45 Number 1, Feb 2007, pp. 64-75, see p. 71.

PROG

(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

CROSSREFS

Cf. A134573, A134574.

Sequence in context: A046140 A023241 A174357 * A106954 A027755 A260828

Adjacent sequences:  A134569 A134570 A134571 * A134573 A134574 A134575

KEYWORD

nonn

AUTHOR

Gary W. Adamson, Nov 01 2007

EXTENSIONS

Corrected and extended by Charles R Greathouse IV, Nov 26 2014

New name from Charles R Greathouse IV, Nov 26 2014

STATUS

approved

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Last modified April 22 16:32 EDT 2021. Contains 343177 sequences. (Running on oeis4.)