Written: Feb. 7, 2024

In their recent claimed computer-free proof of the Four Color Theorem, David Jackson and Bruce Richmond attempted to use sophisticated "asymptotic analysis" to explicitly compute a certain number whose positivity (according to them) implies this famous theorem. While the jury is still out whether their valiant attempt holds water, we prove, in this modest note, that this constant equals exactly 10/27. We also point out that their evaluation of this constant must be erroneous, for two good reasons. Finally, as an encore, we state many similar, but more complicated, results.

Maple package

Jackmond.txt , a Maple package to find explicit expressions, as rational functions of n, of the ratio of the coefficient of x^n in
(Sum(f(i)*x^i,i=1..infinity))^r divided by f(n)
if
Sum(f(i)*x^i,i=1..infinity)
is an algebraic formal power series.

Sample input an output for Jackmond.txt

• If you want to explicit expressions as rational functions of n, of the ratio of the coefficient of x^n of (Sum(2*(4*n+1)!/((n+1)!*(3*n+2)!)*x^n,n=1..infinity)^r divided by 2*(4*n+1)!/((n+1)!*(3*n+2)! for 2 ≤ r ≤ 11
  the input file yields the output file.

• If you want to go all the way to r=18,
  the input file yields the output file.

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger