Helleseth's three-valued conjecture asserts that certain exponential sums over a finite field cannot take exactly
three distinct values when the extension degree
over the prime field is a power of 2 (Helleseth 1976).
Let 
,
where 
is a prime and 
is a positive integer,
and put 
. Using the field trace,
define
 |
where the field trace is interpreted modulo 
in the exponential function.
For a positive integer 
relatively prime to 
,
the map 
is a permutation
of 
.
The associated exponential sum is
 |
The conjecture states that 
never has exactly three distinct values
if 
for a nonnegative integer 
. The restriction to nonzero parameters
is part of the statement. For example, when 
, 
and 
for 
.
The cases of field characteristic 2 and 3 are known (Katz 2012, 2015). Liu (2026) reported a general proof,
including the remaining range 
and 
. Liu reports GPT-6 Astra and Claude Fable assistance
with mathematical development and argument checking. As of Nov. 3, 2026, independent
specialist verification of the complete claimed proof had
not been reported.
