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

Author

David Dewan, Apr 18 2025

Keywords

• nonn
• fini
• full

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.

Examples

			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.
		

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.

Crossrefs

Cf. A272709, A224921, A156685, A009003.