The achromatic number 
of a finite simple graph 
is the greatest number of colors in a proper vertex
coloring such that every pair of distinct colors occurs at the ends of some graph edge. A proper vertex
coloring alone would allow a different color on every graph
vertex. Consequently 
, where 
is the chromatic number.
Timmons (2026) proved that the Peisert graph of order 
,
where 
is a prime number with 
, has 
for all sufficiently large admissible 
. For a fixed integer 
and sufficiently large prime
power 
,
the Cayley graph on the finite
field 
joining pairs whose difference is a nonzero 
th power satisfies 
. Timmons (2026) credits ChatGPT with mathematical
proof assistance.
