A383181 Family of 2-colorings of {1..7824} with no monochromatic Pythagorean triples.

0, 0, 2, 0, 1, 0, 1, 2, 0, 1, 0, 2, 2, 0, 1, 1, 0, 1, 0, 2, 2, 0, 0, 2, 2, 0, 1, 1, 1, 2, 0, 1, 0, 1, 1, 0, 0, 0, 2, 2, 1, 1, 2, 2, 2, 0, 0, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 0, 1, 2, 2, 2, 1, 1, 1, 0, 2, 2, 1, 0, 1, 0, 1, 1, 1, 2, 2, 0, 2, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 0, 2, 2, 2, 2, 1, 2

Offset

1,3

Comments

We use the codes: 1=red, 2=blue, 0=unconstrained (may be red or blue).

Choose any Pythagorean triangle (r,s,t) with t<=7824, then a(r), a(s), a(t) cannot all be the same color (see Examples).

Solution and proof by Heule, Kullmann, and Marek (2016).

Because each of the 2899 numbers for which a(n)=0 can be independently colored red or blue, this sequence represents 2^2899 unique 2-colorings with no monochromatic Pythagorean triples.

There is no 2-coloring of {1..7825} with no monochromatic Pythagorean triples.

Links

David Dewan, Table of n, a(n) for n = 1..7824

David Dewan, Image Decoder Program for Boolean Pythagorean Triples

Marijn Heule, Visualization of a solution of the Pythagorean Triples Problem, 21 May 2016.

Marijn J. H. Heule, Oliver Kullmann, and Victor W. Marek, Solving and Verifying the boolean Pythagorean Triples problem via Cube-and-Conquer, arXiv:1605.00723 [cs.DM], 3 May 2016.

Wikipedia, Boolean Pythagorean triples problem.

Bob Yirka, Computer generated math proof is largest ever at 200 terabytes, Phys.org, 30 May 2016.

Example

The triple (5,12,13) is not monochromatic:
  a(5)= 1 red,
  a(12)=2 blue,
  a(13)=2 blue.
The triple (3,4,5) is not monochromatic whether 4 is red or blue:
  a(3)=2 blue,
  a(4)=0 red or blue,
  a(5)=1 red.

Crossrefs

Cf. A272709, A224921, A156685, A009003.

Sequence in context: A119346 A014586 A122924 * A133450 A029410 A242082

Adjacent sequences: A383178 A383179 A383180 * A383182 A383183 A383184

Keyword

nonn,fini,full

Author

David Dewan, Apr 18 2025

Status

approved