A306787 Prime numbers p such that there exists an integer k such that p-1 does not divide k-1 and x -> x + x^k is a bijection from Z/pZ to Z/pZ.

31, 43, 109, 127, 157, 223, 229, 277, 283, 307, 397, 433, 439, 457, 499, 601, 643, 691, 727, 733, 739, 811, 919, 997, 1021, 1051, 1069, 1093, 1327, 1399, 1423, 1459, 1471, 1579, 1597, 1627, 1657, 1699, 1723, 1753, 1777, 1789, 1801, 1831, 1933, 1999, 2017

Offset

1,1

Comments

If x -> x + x^k is a bijection from Z/pZ to Z/pZ then the following facts hold:

-v_2(k-1) >= v_2(p-1)

-gcd(k+1,p-1) = 2

-2^(k-1) = 1 (mod p).

The third fact is very important as it shows that for a given k there are a finite number of solutions p.

If p = 1 (mod 3) and 2^((p-1)/3) = 1 then either k = (p-1)/3+1 or k = 2*(p-1)/3+1 has the wanted property (see sequence A014752 for more information when this happens). It is a sufficient but not necessary condition since 3251 also appears in this sequence but 3 does not divide 3250.

Links

Elias Caeiro, Table of n, a(n) for n = 1...212

Problèmes du 9ème Tournoi Français des Jeunes Mathématiciennes et Mathématiciens, Problem 7 question 7, 2019 (in French).

Example

For p = 31 and k = 21, x -> x + x^k is a bijection.

Crossrefs

Cf. A014752.

Sequence in context: A059898 A016108 A014752 * A227622 A020348 A033905

Adjacent sequences: A306784 A306785 A306786 * A306788 A306789 A306790

Keyword

nonn

Author

Elias Caeiro, Apr 16 2019

Status

approved