Conference rusure::math

1316.0. "Reply on trinomial factorization in Z[x]" by CADSYS::COOPER (Topher Cooper) Mon Oct 22 1990 17:54

RE: 1315.1

>    Every binomial in Z[x] has an irreducible factor with at most 3 terms.
>    	-- Davenport, Factorisation of Sparse Polynomials, in Proceedings
>    	of Eurocal '83.  Springer-Verlag, 1983.  pp. 214-224.
>    
>    (I believe a binomial in Z[x] is something of the form ax^b+cx^d, where
>    a, b, c, and d are integers and b and d are non-negative and unequal.)
>    
>    A strict limit is not known for trinomials, but it is at least 8; in
>    1981, Brenner showed this with:
>    
>    	x^14+Ax^2-B = P(x)Q(x),
>    
>    where
>    
>    	A = 10307342028165274266525889239823042363346833922876
>    	    075828731741952918006405132760000000,
>    	B = 12841208948559362541055001023860063131661452126861
>    	    6325666077934457967306716740794924967132900000000
>    
>    and where
>    
>    	P(x) = x^7+2pax^6+2pbx^5+2pcx^4+2pdx^3+2pex^2+2pfx+pg
>    and
>    	Q(x) = x^7-2pax^6+2pbx^5-2pcx^4+2pdx^3-2pex^2+2pfx-pg
>    
>    are irreducible with
>    
>    	p = 17
>    	a = 470820
>    	b = 3768415030800
>    	c = 44978423066488340100
>    	d = 478593017683468165593108000
>    	e = 937523138375225928321188658341000
>    	f = 123149310992981992534109963118406585650000
>    	g = 666582695222076790155887095950281112938435190000
>    
>    (Somebody want to check this?)

    I fed the trinomial into MAPLE and asked it to factor it.  I didn't
    bother to check that the resulting coefficients were equal to the ones
    specified above, but the general form is the same (i.e., 8 terms each,
    coef's equal with alternating sign on one of them).

				    Topher