
Re:    Properties of coprime integers
Posted:
Jan 6, 2014 3:34 PM


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^(p1) + z^(p2)x + z^(p3)x^2 +   + zx^(p2) + x^(p1) (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
Sorry, that should be x^(p1) + 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^(p1) mod p. Since there are p terms in the sum, Q is zero mod p.
 jiw

