A134572 - OEIS

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.

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

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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.

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