The reciprocal Fermat constant is the sum of the reciprocals of the Fermat numbers,
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
(OEIS A051158), where 
is the binary digit sum
of 
,
i.e., the number of 1s in its binary expansion,
and 
is the exponent of 2 in the prime factorization
of 
.
The digit-sum identity is Problem E2455 in Moorthy (1974-1975); see also Finch (2019,
p. 247).
Golomb (1963) proved that 
is an irrational number,
Duverney (2001) proved that it is a transcendental
number, and Coons (2013) proved that its irrationality
exponent is 2.
The Mellin transform identity
 |
(5)
|
where 
is the gamma function and 
is the Riemann zeta
function, is valid for 
(NIST DLMF, eqn. 25.5.3). Applying the inverse
Mellin transform at 
and summing the geometric
series 
gives the contour representation
 |
(6)
|
where 
is a real number.
A number is disjunctive in base 
if its base-
expansion contains every finite word over the digits 0, 1,
..., 
.
Equivalently, the sequence of fractional parts 
is dense modulo 1. Every normal number is disjunctive,
but the converse need not hold (Leśniak 2014).
CaptainSude (2026) gives an AI-generated argument claiming that 
is disjunctive but not normal
in base 2, and that every finite binary word occurs with positive lower frequency.
It also treats the family
 |
(7)
|
where 
,
is a nonconstant polynomial whose coefficients
are integers, and 
is the greatest integer 
for which 
divides 
.
The claim states that every such 
is base-
disjunctive and nonnormal. The release includes a Lean 4 check;
independent specialist review had not been reported as of Sep. 9, 2026 (CaptainSude
2026, VibeMathed 2026).
