[seqfan] Re: Fwd: A nice (decimal) property of 78

David Wilson davidwwilson at comcast.net
Sat Nov 8 19:12:10 CET 2008

----- Original Message ----- 
From: <f.firoozbakht at sci.ui.ac.ir>
To: "Maximilian Hasler" <maximilian.hasler at gmail.com>
Cc: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>; "Alexander 
Povolotsky" <apovolot at gmail.com>
Sent: Friday, November 07, 2008 12:49 PM
Subject: [seqfan] Re: Fwd: A nice (decimal) property of 78

> 31977, 437388, 773976 & 778998 are four terms of the sequence that are
> concatenation of more than two numbers.
> phi(31977)=phi(31)*phi(97)*phi(7)
> phi(437388)=phi(43)*phi(73)*phi(88)
> phi(773976)=phi(7)*phi(73)*phi(976)
> phi(778998)=phi(77)*phi(89)*phi(98)
> Some primitive terms greater than 10^9:

And why stop there? We should do sigma as well.

> phi(1000396644)=phi(100039)*phi(6644)
> phi(1000857375)=phi(100085)*phi(7375)
> phi(1001637585)=phi(100163)*phi(7585)
> phi(10000279191)=phi(10000270)*phi(9191)

The problem is, in order to prove there are an infinitude of primitive 
solutions, you generally have to exhibit and infinite family of primitive 
solutions. But then only one member of the family is primitive, the others 
become nonprimitive members of the family. 

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