Re: [math-fun] Equation mixing integers and reals
I wrote: << Jim Buddenhagen writes: << This polynomial: x^4 - 48*x^3 - 12*x^2 - 33*x + 1613 has a root very near Pi. How can I know if this is unusual for the size of the coefficients? . . .
. . . The difference [between Pi and the root} seems astonishingly low, to 6 places: 1.369107 x 10^(-12). But if we consider all monic integer polynomials up to degree 4 using coefficients of absolute value up to 1613, there are . . . 108,441,586,233,841 polynomials to consider, which makes this sound less astonishing.
Another valid viewpoint might be to consider only monic integer polynomials with sum-of-absolute- coefficients <= 48 + 12 + 33 + 1613 = 1706. For K > 0, the volume V(K) in 4-space of the region where |c_1| + . . . + |c_4| <= K is about the number of lattice points therein. I get V(K) = 2^4 * K^4 / 4! = 2 K^4 / 15, so V(1706) = 1,129,418,361,346 to the nearest integer, or about 1.1 x 10^12, about a trillion. Together with the answer to this question: [QUESTION: What's a good estimate for (maximum root of P) - (minimum root of Q) over all real roots of monic integer polynomials P, Q of degree d whose sum-of-absolute-values- of-coefficients is <= K ???] -- call this difference MM(d;K) -- then we can say the average spacing between such roots is about avspace = MM(4;1706) / 10^12 and we can ask what the expected distance from the nearest root to pi would be; this would be roughly avspace/4. So if MM(4; 1706) is significantly bigger than 4, we can suddenly gasp in surprise over the closeness to pi of the root Jim mentioned. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
Hello, There was a typo error(?) in the posting about this general naive <integer relation> algorithm : Here is the proper code in Maple : ################################################### p:=proc(s) local a, b, c, d, nn, k, i, aa, ss, g, h, m, z; aa := s; nn := nops(aa); k := [seq(frac(abs(aa[i] - aa[i + 1])), i = 1 .. nn - 1), frac(abs(aa[1] - aa[nn]))]; ss := evalf(k); g := sort( [seq(["bobo", convert(1 + ss[j], string), j], j = 1 .. nn)] , lexorder[2]); h := [seq(g[i][3], i = 1 .. nn)]; m := [seq(k[h[z]], z = 1 .. nn)]; return m end: # I remember now the twitch I had to make in order to have # maple to run properly, here <bobo> is just a placeholder # so that the !@#!#! sort algorithm will work. ################################################## ################################################## Now, if I use the vector [1,Pi,Pi^2,Pi^3,Pi^4] and try to find an integer relation with these, and THEN inverting the parameter <pi> to <x> one can find a series of polynomials that have <pi> has a root, interesting isn't ? Results : The following polynomials approximate Pi has a root. ############################################################## What is quite interesting here is the algorithm to find the polynomials, I realize that this could be exploited a lot, since i can use it to gradualy find better and better results maybe it can be refined further to get real and new results with unknown constants!... the polynomial : -1806+41*x^4-x-14*x^2-66*x^3 has 3.1415... has a root but it is NOT as elegant and short as the example you posted. Simon Plouffe -x^2+x^3-21 x^2-x^3+18+x -x^2+2*x^3-49-x x^3-44+x+x^2 -2*x^3+247+x^2-2*x^4 -648-6*x^3+9*x^4-4*x^2-x -7*x^3+350-2*x^4+4*x+5*x^2 19*x^3+310-9*x^4-4*x-x^2 -17*x^2-9*x^3-876-13*x+14*x^4 25*x^2-12*x^3+537+24*x-5*x^4 -42*x^2+x^3-2831-31*x+34*x^4 -1806+41*x^4-x-14*x^2-66*x^3 11185+82*x^3-149*x^4+73*x^2+21*x -13430-50*x^3+161*x^4-68*x^2-10*x 20117-19*x^3-222*x^4+180*x^2+102*x -13422+165*x^3+97*x^4-97*x^2-59*x 16491-194*x^4+372*x+427*x^2-96*x^3 -19616+330*x^4-511*x-601*x^2-161*x^3 -27425+317*x^4+612*x+470*x^2-323*x^3 -56757+591*x^3+485*x^4-718*x^2-549*x 151364+842*x^3-1863*x^4+472*x^2-209*x -302912-3995*x^3+4380*x^4-371*x^2+1207*x 428942+5588*x^3-6431*x^4+2398*x^2+180*x -422515-3724*x^3+6211*x^4-5635*x^2-3632*x 529822+10243*x^3-9324*x^4+5407*x^2+2374*x -23163*x^3-707262+13375*x^4+13221*x+8215*x^2 22236*x^3+619236-10922*x^4-22319*x-17698*x^2 -1033928+7208*x^4-5345*x-8169*x^2+13843*x^3 821*x^3-1178131+17035*x^4-35589*x-40010*x^2 -3919153-90904*x^3+68082*x^4+2333*x^2+26393*x 6426029+50402*x^3-91079*x^4+75105*x^2+45155*x -1923653+4589*x^4+19554*x+15503*x^2+40708*x^3 10983174-296289*x^3-61192*x^4+323671*x^2+308696*x -17024443+339254*x^3+135294*x^4-521139*x^2-487011*x -1099364+87012*x^4-47934*x-71265*x^2-210359*x^3 28342373-438940*x^4+342160*x+478545*x^2+277892*x^3 -66633002+1034179*x^4-254558*x-599254*x^2-883412*x^3 78316338-757810*x^4+1755259*x+1970697*x^2-950229*x^3 -126555223+1509348*x^4-1237895*x-1733570*x^2+17081*x^3 -121525197-9075831*x^3+3508037*x^4+4380350*x^2+5725092*x 326339287+13583887*x^3-6938505*x^4-4883498*x^2-7465445*x -427789612-11832637*x^3+7606202*x^4+3451247*x^2+6270693*x 334313486+3813605*x^3-4335983*x^4-1910253*x^2-3610135*x 14865367*x^2-13000398*x^3+211929969+15293298*x-36923*x^4 -23197499*x^2-1869905*x^3-1038601802-18994787*x+14220486*x^4 33731950*x^2+28904948*x^3+2006015384+23229681*x-33961403*x^4 -13135824*x^2-53346161*x^3-2797025873+2894022*x+46932429*x^4 -331623675+19630148*x^4+41645000*x+34061650*x^2-66036237*x^3 1087241756-47279647*x^4-7812913*x+6800902*x^2+112094996*x^3 -94002246*x^3-8059453944+121167139*x^4-32962934*x-73469122*x^2 84839170-66053718*x^4+182579456*x+193305533*x^2+124747453*x^3 9069388091-24728553*x^4-222886804*x-197355190*x^2-129411379*x^3 -20305728133+221680915*x^4+395942065*x+296788895*x^2-176128402*x^3 38699021825-469299175*x^4-293977305*x-111582676*x^2+291548003*x^3 -49558930834+529678139*x^4-110714671*x-303642443*x^2+42188747*x^3 ...
participants (2)
-
Dan Asimov -
Simon Plouffe