
Re:    Properties of coprime integers
Posted:
Jan 6, 2014 3:22 PM


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^(p1) + z^(p2)x + z^(p1)x^2 +   + zx^(p2) + x^(p1) (2)
I assume the third term is supposed to be z^(p3)x^2
> Observation: P is even and Q is odd > > Question: Under what condition if any pQ
Consider the case where z = x + 2*p. If I'm not mistaken, each of the terms x^(pj1)*z^j then is of the form x^(p1)+(p^j)*something (via binomial theorem) so that every term of the sum Q is equivalent to x^(p1) 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 pQ NOT possible > > Any helpful comment will be appreciated.
 jiw

