[seqfan] About number theory and combinatorics (Invitation)

R J Cano slackgorithmus at gmail.com
Fri Jan 18 02:48:50 CET 2013


Subject: On a proof related to those sequences similar to A215940 and the
consequences of the "abc" conjecture.

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Greetings.

Dear SeqFans,

Please look this*. It is the sixth page from a Google book about elements
on number theory. There is explained the importance of the so called "abc"
conjecture due the ease to certain proofs if it were true. Open it and go
to the end inside the linked page.

Can you spot that there is present in the context of another problem, the
same kind of polynomial which is a factor inside the inductive proof given
in the LINKS section of A215940?. Then:

Would it exist in fact a connection between these two problems???. Well,
what It mean is: Is it there plausible an equivalence between these two
problems ??? (respectively (i) and (ii) as follows):

(i) Two distinct permutations for the same digits in an arbitrary base r are
congruent modulo (r--1) **

(ii) All the polynomials of the form (x^n-1)/(x-1) have infinite
square-free values.

My rough idea there would be for now: Take advantage of having (i) already
proved and use it looking for an acceptable interpretation*** of (ii) when
it is assumed true.

Regards,

R. J. Cano, <slackgorithmus at gmail.com>

P.S.: Note(s):

* Full URL:
http://books.google.co.ve/books?id=1MTcYrbTdsUC&lpg=PR3&ots=9F82GiXs14&dq=generalization%20of%20pandigital%20numbers&lr&hl=es&pg=PA6#v=onepage&q&f=false

** Minus sign duplicated for readability.

*** In terms of permutations and their differences.

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