An almost linear time algorithm testing whether the Markoff graph modulo p is connected

Abstract The Markoff graphs modulo p were proven by Chen (Ann Math 199(1), 2024) to be connected for all but finitely many primes, and Baragar (The Markoff equation and equations of Hurwitz. Brown University, 1991) conjectured that they are connected for all primes, equivalently that every solution to the Markoff equation modulo p lifts to a solution over $$\mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> . In this paper, we provide an algorithmic realization of the process introduced by Bourgain et al. [ arXiv:1607.01530 ] to test whether the Markoff graph modulo p is connected for arbitrary primes. Our algorithm runs in $$o(p^{1 + \epsilon })$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ϵ</mml:mi> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> time for every $$\epsilon &gt; 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . We demonstrate this algorithm by confirming that the Markoff graph modulo p is connected for all primes less than one million.