Re: Trinomial decics x^10+ax+b = 0; Help with Mathematica code

*To*: mathgroup at smc.vnet.net*Subject*: [mg93322] Re: [mg93280] Trinomial decics x^10+ax+b = 0; Help with Mathematica code*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Tue, 4 Nov 2008 06:15:20 -0500 (EST)*References*: <200811020657.BAA02645@smc.vnet.net> <BB6FC818-9D69-4DA2-8F6A-0A5B680ECD2C@mimuw.edu.pl> <64fc87850811020944p49edb6a2u38aa53a29b539784@mail.gmail.com> <F85E373D-0B9C-4090-BA7B-AE95575625E1@mimuw.edu.pl>

Sorry, I forgot that one has to Catch whatever might be Thrown ;-) Unfortunately, its unlikely to make any difference :-( Anyway, the correct code is: Catch[NestWhile[({a, b} = RandomInteger[{-10^6, 10^6}, {2}]; If[ck[a] == ck[b] == 0 && (p[a, b] || p[b, a]), Throw[{a, b}], ck[a] = ck[b] = 1]) &, 1, True &, 1, 10^6]];//Timing Andrzej Kozlowski 3 Nov 2008, at 16:08, Andrzej Kozlowski wrote: > I doubt that you will have much luck with this approach. I did: > > f = -a + m^9 - 8 m^7 n + 21 m^5 n^2 - 20 m^3 n^3 + 5 m n^4 ; g = -b > + m^8 n - > 7 m^6 n^2 + 15 m^4 n^3 - 10 m^2 n^4 + n^5; > > p[x_, y_] := > MemberQ[Exponent[FactorList[h /. {a :> x, b :> y}][[All, 1]], m], 5 > | 10] > > (mm = Outer[List, Range[10^3], Range[10^3]]); // Timing > {0.012557, Null} > > (vv = Apply[p, mm, {2}];) // Timing > {7077.94, Null} > > Position[vv, True] > {} > > There is not a single example among the first 10^6. You can try this > approach on larger numbers, though eventually you will find yourself > short of memory to story such large arrays. Or you could try random > searches, with very much larger numbers, e.g. > > ck[_] = 0; > > NestWhile[({a, b} = RandomInteger[{-10^6, 10^6}, {2}]; > If[ck[a] == ck[b] == 0 && (p[a, b] || p[b, a]), Throw[{a, b}], > ck[a] = ck[b] = 1]) &, 1, True &, 1, 10^4]; // Timing > > {0.481902, Null} > > As you see, 10^4 random pairs between -10^6 and 10^6, and no result. > The "flag" ck, by the way, is used to mark the numbers we have > already tried, so when we run this again we do not apply p to them > again. > > Anyway, it looks to me much harder than the proverbial looking for a > needle in a haystack. > > Andrzej Kozlowski > > > > On 3 Nov 2008, at 02:44, Tito Piezas wrote: > >> Hello Andrzej, >> >> The simple trinomials x^6+3x+3 = 0 and x^8+9x+9 = 0 are solvable, >> both factoring over Sqrt[-3]. >> >> However, there seem to be no known decic trinomials, >> >> x^10+ax+b = 0 >> >> such that it factors over a square root (or quintic) extension. >> >> The 45-deg resultant, for some {a,b}, IF it has an irreducible 5th >> or 10th degree factor, will give the {a,b} of such a decic. >> >> To find one, what I did was to manually substitute one variable, >> like "a" (yes, poor me), use the Table[] function for "b" (as well >> as the Factor[] function), and inspect the 45-deg to see if it has >> the necessary 5th or 10th deg factors. Unfortunately, there are >> none for {a,|b|} < 30. >> >> So what I was thinking is to extend the range to {a,|b|} < 1000. >> But that's about 10^6 cases, far more than I could do by hand. >> >> I'm sure there is a code such that we only have to give the upper >> bound for {a,b}, let Mathematica run for a while, and it prints out >> ONLY the {a,b} such that the 45-deg has an irreducible 5th (or >> 10th) factor. I believe there might be a few {a,b} -- but doing it >> by hand is like looking for a needle in a haystack. >> >> But my skills with Mathematica coding is very limited. Please help. >> >> P.S. The 45-deg resultant should be in the variable "m", so pls >> eliminate the variable "n" (not m) between the two original >> equations. >> >> Sincerely, >> >> Tito >> >> >> On Sun, Nov 2, 2008 at 6:14 AM, Andrzej Kozlowski >> <akoz at mimuw.edu.pl> wrote: >> >> On 2 Nov 2008, at 15:57, tpiezas at gmail.com wrote: >> >> Hello guys, >> >> I need some help with Mathematica code. >> >> It is easy to eliminate "n" between the two eqn: >> >> -a + m^9 - 8m^7n + 21m^5n^2 - 20m^3n^3 + 5mn^4 = 0 >> -b + m^8n - 7m^6n^2 + 15m^4n^3 - 10m^2n^4 + n^5 = 0 >> >> using the Resultant[] command to find the rather simple 45-deg >> polynomial in "m", call it R(m). >> >> As Mathematica runs through integral values of {a,b}, if for some >> {a,b} the poly R(m) factors, we are interested in two cases: >> >> Case1: an irreducible decic factor >> Case2: an irreducible quintic factor >> >> What is the Mathematica code that tells us what {a,b} gives Case 1 or >> Case 2? >> >> >> Thanks. :-) >> >> >> Tito >> >> >> >> >> Let f = -a + m^9 - 8 m^7 n + 21 m^5 n^2 - 20 m^3 n^3 + 5 m n^4 ; g >> = -b + m^8 n - >> 7 m^6 n^2 + 15 m^4 n^3 - 10 m^2 n^4 + n^5; >> >> and >> >> h = Resultant[f, g, m]; >> Exponent[h, n] >> 45 >> >> so h is a polynomial of degree 45. Now, let >> >> p[x_, y_] := Exponent[FactorList[h /. {a :> x, b :> y}][[All, 1]], n] >> >> computing p[a,b] gives you the exponents of the irreducible factor >> for the given values of a and b. In most cases you get {0,45} - the >> irreducible case. But, for example, >> p[a, 0] >> {0, 1, 9} >> and >> p[2, 1] >> {0, 9, 36} >> >> So now you can search for the cases you wanted. You did not >> seriously expect Mathematica would do it by itself, I hope? >> >> Andrzej Kozlowski >> >

**Follow-Ups**:**Re: Re: Trinomial decics x^10+ax+b = 0; Help***From:*Artur <grafix@csl.pl>

**References**:**Trinomial decics x^10+ax+b = 0; Help with Mathematica code***From:*tpiezas@gmail.com