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A294676 Ramsey-Comer numbers: a(n) is the smallest prime p congruent to 1 mod 2n such that for every prime q >= p (also congruent to 1 mod 2n), the multiplicative subgroup H of (Z/qZ)* of index n contains a solution to x+y = z. 1
3, 13, 19, 73, 131, 313, 547, 193, 613, 1201, 1453, 1249, 547, 2857, 2971, 1601, 4217, 3169, 2243, 4441, 9661, 10957, 7039, 7873, 8951, 11701, 14419, 18257, 11311, 29641 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

a(n) <= n^4 + 5 (cf. Alm, 2017).

The subgroup H, along with its n-1 cosets, induces a cyclic coloring on K_q. Labeling the vertices 0 through q-1, color the edge uv by the color corresponding to the coset containing u-v (mod q). Thus if q >= a(n), the coloring induced by H and its cosets must contain a monochromatic triangle. In fact, it contains many monochromatic triangles in each color class.

The data gathered thus far suggest that the bound n^4 + 5 can be replaced by cn^3 for some c > 1, but there is no proof.

a(n) > A263308(n). The reason A263308(8) is zero can be taken to be that a(8) is exceptionally small; similarly, a(13) is small, so A263308(13)=0.

LINKS

Table of n, a(n) for n=1..30.

Jeremy F. Alm, 401 and beyond: improved bounds and algorithms for the Ramsey algebra search, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.4. (Also here: arXiv:1609.01817  [math.NT], 2016.)

CROSSREFS

Cf. A263308.

Sequence in context: A055202 A158016 A281998 * A293465 A271924 A178712

Adjacent sequences:  A294673 A294674 A294675 * A294677 A294678 A294679

KEYWORD

nonn,more

AUTHOR

Jeremy F. Alm, Nov 06 2017

STATUS

approved

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Last modified April 16 05:26 EDT 2021. Contains 343030 sequences. (Running on oeis4.)