Path: ...!3.eu.feeder.erje.net!feeder.erje.net!weretis.net!feeder8.news.weretis.net!reader5.news.weretis.net!news.solani.org!.POSTED!not-for-mail
From: WM <invalid@no.org>
Newsgroups: sci.math
Subject: The existence of dark numbers proven by the thinned out harmonic
 series
Date: Wed, 12 Mar 2025 08:08:28 +0100
Message-ID: <vqrbtd$1chb7$2@solani.org>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 7bit
Injection-Date: Wed, 12 Mar 2025 07:08:29 -0000 (UTC)
Injection-Info: solani.org;
	logging-data="1459559"; mail-complaints-to="abuse@news.solani.org"
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:QNE6nNgnvB/ONS3B0X3CcXA8uSY=
X-User-ID: eJwFwQkBACAIBLBK/GAc0KN/BDfX4Lhp4WG+vme8tcDXp1v6jVdPSi7KkIS0PmL3cdROJ6AoPaWk9EgEH3RAFfA=
Bytes: 2319
Lines: 30

The harmonic series diverges. Kempner has shown in 1914 that when all 
terms containing the digit 9 are removed, the series converges.

That means that the terms containing 9 diverge. Same is true when all 
terms containing 8 are removed. That means all terms containing 8 and 9 
simultaneously diverge.

We can continue and remove all terms containing 1, 2, 3, 4, 5, 6, 7, 8, 
9, 0  in the denominator without changing this. That means that only the 
terms containing all these digits together constitute the diverging 
series. (*)

But that's not the end! We can remove any number, like 2025, and the 
remaining series will converge. For proof use base 2026. This extends to 
every definable number. Therefore the diverging part of the harmonic 
series is constituted only by terms containing a digit sequence of all 
definable numbers.

Note that here not only the first terms are cut off but that many 
following terms are excluded from the diverging remainder.

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

(*) At this point the diverging series starts with the smallest term 
1023456789 and contains further terms like 1203456789 or 1234567891010 
or 123456789111 or 1234567891011. Only those containing the digit 
sequence 10 will survive the next step, and only those containing the 
digit sequence 1234567891011 (where the order of the first nine digits 
is irrelevant) will survive the next step.

Regards, WM