1,1

Condition on odd prime p so that Q(Cp) is not rational over Q: p = 2q+1 where q = -1 (mod 4), q is squarefree, and any of 4p-q and q+1 is not square.

Consider the even integers E > a(1), and constrain distance <= E. The absolute distance from E to one of the sequence terms is conjectured to nearly always be prime (or adjacent). For example, 1488768 = a(40046)-1488623. There are four early exceptions thru E = 1488694, {104, 128, 134, 212}. Where is the next exception? - Bill McEachen, May 31 2025

Shizuo Endo and Takehiko Miyata, Invariants of finite abelian groups, J. Math. Soc. Japan, Volume 25, Number 1 (1973), 1-167 (see Proposition 3.6 (i) p.18).

Akinari Hoshi, On Noether's problem for cyclic groups of prime order, arXiv:1402.3678 [math.NT], 2014 (see Proposition 3.1 (i) p.4 and Table 1 p.19).

Select[Prime[Range[2, 300]], (q = (#-1)/2; SquareFreeQ[q] && Mod[q, 4] == 3 && !IntegerQ[Sqrt[4#-q]] && !IntegerQ[Sqrt[q+1]])&] (* Jean-François Alcover, Sep 22 2018 *)

(PARI) isok(p) = isprime(p) && (q = (p-1)/2) && issquarefree(q) && ((q % 4) == 3) && (!issquare(4*p-q)) && (!issquare(q+1)); \\ corrected by Michel Marcus, May 31 2025

Cf. A240584, A240585.

Sequence in context: A046002 A347695 A132257 * A132255 A142041 A190799

Adjacent sequences: A240580 A240581 A240582 * A240584 A240585 A240586

Michel Marcus, Apr 08 2014

approved