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From: WM <wolfgang.mueckenheim@tha.de>
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Subject: Improved version: New proof of dark numbers by means of the thinned
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Date: Mon, 12 May 2025 13:25:30 +0200
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Wolfgang Mückenheim
Technische Hochschule Augsburg
wolfgang.mueckenheim@tha.de


Abstract: It is shown by the intersection of the complements of all 
Kempner series belonging to definable natural numbers that not all 
natural numbers can be defined. We call them dark.


1. Dark Numbers

Not all numbers can be chosen, expressed, and communicated as 
individuals such that the receiver knows what the sender has meant. We 
call those numbers dark numbers. Much evidence has been collected and 
discussed [1]. But in the following we will present the shortest proof 
of their existence. Of course the facilities to express numbers depend 
on the environment and the power of the applied system. But this proof 
shows that, independent of the system, infinitely many natural numbers 
will remain dark forever.

A simple example is provided already by the denominators of the harmonic 
series (1/n). Whatever attempts are made to express denominators m as 
large as possible, the sum from 1/1 to 1/m is finite while the remaining 
part of the series diverges.


2. Kempner Series

The harmonic series diverges. But as Kempner [2] has shown in 1914, 
deleting all terms containing the digit 9 turns it into a converging 
series, the Kempner series, here abbreviated as K(9). That means that 
the complement C(9) of removed terms

C(9) = 1/9 + 1/19 + 1/29 + ...

all containing the digit 9, carries the divergence alone. All other 
terms can be removed. Same is true when all terms containing the digit 8 
are removed. That means that the complement C(8)

C(8) = 1/8 + 1/18 +1/28 + ...

of the Kempner series K(8) carries the divergence alone. Since here 
those terms containing the digits 9 without digits 8 belong to the 
converging series K(8) we can conclude that the divergence is caused by 
the intersection only, i.e., by all terms containing the digits 8 and 9 
simultaneously:

C(8) ∩ C(9) = 1/89 + 1/98 + 1/189 + ...

	
3. Proof

But not all terms containing 8 and 9 are needed. We can continue and 
remove all terms containing 1, 2, 3, 4, 5, 6, 7, or 0 in the denominator 
without changing this result because the ten corresponding Kempner 
series K(0), K(1), ..., K(9) are converging and their complements C(0), 
C(1), ..., C(9) are diverging. But only the intersection of all 
complements carries the divergence. That means that only the terms 
containing all the digits 0 to 9 simultaneously constitute the diverging 
series.

But that is not the end! We can remove any natural number k, like 2025, 
and the remaining Kempner series will converge. For proof use base 2026 
where 2025 is a digit. This extends to every defined number, i.e., every 
number k that can be defined, chosen, and communicated such that the 
receiver knows what the sender has meant. When the terms containing k 
are deleted, then the remaining series converges.


4. Result

The diverging part of the harmonic series is constituted only by 
intersection of all complements C(k) of Kempner series K(k) of defined 
natural numbers k, i.e., by all the terms containing the digit sequences 
of all defined natural numbers. No defined natural number exists which 
must be left out. Terms which, although being larger than every defined 
number, do not contain all defined digit sequences, for instance not 
Ramsey's number, belong to converging Kempner series and not to the 
diverging series of the intersection of all complements. All infinitely 
many terms containing not the digit 1 or not the digit sequence 2025 or 
not the digit sequence of Ramsey's number can be deleted without 
violating the divergence.

All Kempner series K(k) of defined, i.e., finite numbers k split off in 
this way are converging and therefore the sum of their always finite 
sums is finite too although it can be very large [3]. The divergence 
however remains. It is carried only by terms which are dark and greater 
than all digit sequences of all defined numbers --- we can even say 
greater than all digit sequences of all definable numbers because, when 
larger numbers will be defined in future, they will behave in the same 
way. It is impossible to choose a natural number such that the 
intersection of the complements of all Kempner series of larger numbers 
is finite.

This is a proof of the huge set of undefinable or dark numbers.


Literature

[1] W. Mückenheim: "Evidence for Dark Numbers", ELIVA Press, Chisinau, 
2024, pp. 1-36.
[2] A. J. Kempner: "A Curious Convergent Series", American Mathematical 
Monthly 21 (2), 1914, pp. 48–50.
[3] T. Schmelzer, R. Baillie: "Summing a Curious, Slowly Convergent 
Series", American Mathematical Monthly 115 (6), 2008, pp. 525–540.