From 19 Aug '05 math-fun: "Apropos the cyclotomic discussion, I assume everybody knows about the 2s in factor(x^105-1). What is the lowest degree polynomial with coeffs in {-1,0,1} with a coeff of 2 in its irreducible factorization? --rwg" From 8 May 09:"rwg>> Puzzle: find a trinomial with maximal |coefficient| < some coefficient of >> a(n irreducible) factor. > Edwin Clark> > x^70-x^35+1 has irreducible factor > > x^48-x^47+x^46+x^43-x^42+2*x^41-x^40+x^39+x^36-x^35+x^34-x^33+x^32 > -x^31-x^28-x^26-x^24-x^22-x^20-x^17+x^16-x^15+x^14-x^13+x^12+x^9 > -x^8+2*x^7-x^6+x^5+x^2-x+1 > > Note the coefficient of x^41 and also of x^7 is 2. Right, equivalently x^70+x^35+1 = (x^(3*5*7)-1)/(x^(5*7)-1). More surprising (to me), your factor/.x->-x, 1+x+x^2-x^5-x^6 -2*x^7 -x^8-x^9+x^12+x^13+x^14+x^15+x^16+x^17-x^20-x^22-x^24-x^26 -x^28+x^31+x^32+x^33+x^34+x^35+x^36-x^39-x^40 -2*x^41 -x^42-x^43+x^46+x^47+x^48, divides x^245 + x^70 + 1. I.e., (x^2+x+1)*(x^5-x^4+x^2-x+1) = x^7+x^2+1. > JamesB> > 2*x^5 - 5*x^2 + 3 = (2*x^3 + 4*x^2 + 6*x + 3)*(x-1)^2 > > 5 < 6. > > Jim Buddenhagen > Wow, I missed that one! --rwg"
My searchware refuses to find Michael Kleber's solution of a related puzzle. Anyway, here's another: Find the height 1 polynomial of least degree d with an irreducible factor of height > d. And >=d? I'll start the bidding with d = 37, height 41. Whose value I'll sequester for now just to make things harder. --rwg