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Topic: ----- ----- ----- Properties of co-prime integers
Replies: 5   Last Post: Jan 6, 2014 3:34 PM

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James Waldby

Posts: 308
Registered: 1/27/11
Re: ----- ----- ----- Properties of co-prime integers
Posted: Jan 6, 2014 3:22 PM
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On Mon, 06 Jan 2014 10:40:07 -0800, Deep wrote:

> Consider the following three integers x, z, p with the given conditions.
> Conditions: x, z are odd integers each > 1 and z > x, prime p > 3 and
>
> Let P, Q be two integers each > 1 and further let P and Q be as follows:
>
> P = z - x (1)
>
> Q = z^(p-1) + z^(p-2)x + z^(p-1)x^2 + --- --- + zx^(p-2) + x^(p-1) (2)


I assume the third term is supposed to be z^(p-3)x^2

> Observation: P is even and Q is odd
>
> Question: Under what condition if any p|Q


Consider the case where z = x + 2*p. If I'm not mistaken, each
of the terms x^(p-j-1)*z^j then is of the form x^(p-1)+(p^j)*something
(via binomial theorem) so that every term of the sum Q is equivalent
to x^(p-1) mod p. Since there are p terms in the sum, Q is zero mod p.

Examples:
(p,x,z,P,Q) = (5, 7, 17, 10, 140305),
(p,x,z,P,Q) = (5, 7, 27, 20, 716605),
(p,x,z,P,Q) = (5, 7, 37, 30, 2310905),
(p,x,z,P,Q) = (11, 13, 79, 66, 11333239526293341437),
(p,x,z,P,Q) = (11, 13, 101, 88, 126780493918875058603),
(p,x,z,P,Q) = (43, 17, 103, 86, 4144786942253223879718720890134858071927025526479146737326617519545006958907412864649),
etc

> My conjecture is that p, P, Q are co prime integers so p|Q NOT possible
>
> Any helpful comment will be appreciated.






--
jiw



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