The reciprocal Fermat constant is the sum of the reciprocals of the Fermat numbers,
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(1)
|
 |  |  |
(2)
|
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(3)
|
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(4)
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(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 
, with the right side interpreted by continuity at 
,
where its value is 
(NIST DLMF, eqn. 25.5.3). Applying the inverse Mellin
transform at 
and summing the geometric
series 
gives the contour integral representation
 |
(6)
|
where 
is a real number.
A real number 
is disjunctive in an integer base 
if its base-
expansion contains every finite word over the digits 0, 1, ..., 
,
including words with leading zeros. This is the property
of being a b-dense number. Equivalently,
the sequence of fractional
parts 
,
for 
,
is dense in 
, where 
and 
is the floor function.
Every normal number is disjunctive in the same base, but the converse need not hold (Leśniak 2014, p. 150).
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
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(7)
|
where 
are integers, 
is a nonconstant polynomial
whose coefficients are integers,
and 
is the greatest nonnegative integer 
for which 
divides the positive integer 
.
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).
