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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 3:34 PM
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On Mon, 06 Jan 2014 20:22:31 +0000, James Waldby wrote:
> 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-3)x^2 + --- --- + zx^(p-2) + x^(p-1) (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


Sorry, that should be x^(p-1) + p * something. For example, if
p=7 and z=x+2*p, then z^4*x^2 = x^6 + 8*p*x^5 + 24*p^2*x^4 +
32*p^3*x^3 + 16*p^4*x^2.

Also, the factor 2 as in z = x+2*p is superfluous; ie z = x+p works.

> (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.


--
jiw



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