But you are not comparing like with like --- the LLL products do have more than single digit coefficients; 43267971744 is (quite) large; and output from LLL can hardly be considered uniformly random anyway! On 7/22/18, Bill Gosper <billgosper@gmail.com> wrote:
FindIntegerNullVector constructed 47 products of linear combinations of √2, √5, and √10, all = 1. The very first thing it found was
(3 + 3 √2 - 2 √5 - √10) (3 - 3 √2 - 2 √5 + √10).
Is this even obviously rational? It's 1 iff (3 - 3 √2 + 2 √5 - √10) (3 + 3 √2 + 2 √5 + √10) is.
Things got rapidly worse:
(√2 - 1) (3 + √2 - √10) (43 + 26 √2 + 10 √5 + 11 √10)/71,
until bewilderment: replaced doubt:
(51 + 38 √2 - 15 √5 - 16 √10) (1 + 3 √2 - √10) (-1 + 2 √5 - √10) (1 - √2 + 2 √5 - √10) (-1 + 3 √5 - √10)^3 (4 + 3 √2 + √10) (3 + √2 - 3 √5 + √10) (3 + 7√2/3 + 4√5/3 + √10) (17 + 6 √2 + 8 √5 + √10)^3 (-29 - 20 √2 + 18 √5 + 11 √10)/43267971744
In:= RootReduce@% Out= 1
Such things ought to be trivial to construct: Just multiply a random pile of surds by their reciprocal, with denominator rationalized. But I'd expect these reciprocals to have coefficients a lot bigger than 51:
In[707]:= RandomInteger[{-9, 9}, {4, 4, 4}].Sqrt[{1, 2, 5, 10}]
Out[707]= {{6 + 5 Sqrt[2] + 8 Sqrt[10], 6 + 7 Sqrt[2] + 8 Sqrt[5] - 4 Sqrt[10], -8 + 5 Sqrt[2] - 9 Sqrt[5] + 2 Sqrt[10], 5 + 3 Sqrt[2] - 8 Sqrt[5] + 5 Sqrt[10]}, {2 + 4 Sqrt[2] + 6 Sqrt[5] + 9 Sqrt[10], 1 - 5 Sqrt[2] + 8 Sqrt[5] - Sqrt[10], 1 + Sqrt[2] - 6 Sqrt[5] + 9 Sqrt[10], -5 + 8 Sqrt[2] + 5 Sqrt[5] + 3 Sqrt[10]}, {-2 + 3 Sqrt[2] - 2 Sqrt[5] + 7 Sqrt[10], 5 - 6 Sqrt[2] - 6 Sqrt[5] - 9 Sqrt[10], -3 + 4 Sqrt[2] - 4 Sqrt[5] + 5 Sqrt[10], -1 + 7 Sqrt[2] - Sqrt[5] - 6 Sqrt[10]}, {1, 5 + 4 Sqrt[2] + 9 Sqrt[5] + 9 Sqrt[10], 4 - 5 Sqrt[2] + 9 Sqrt[5] - 9 Sqrt[10], 3 + 4 Sqrt[2] + 6 Sqrt[5] + 9 Sqrt[10]}}
In[708]:= Times @@ # & /@ %
Out[708]= {(6 + 7 Sqrt[2] + 8 Sqrt[5] - 4 Sqrt[10]) (-8 + 5 Sqrt[2] - 9 Sqrt[5] + 2 Sqrt[10]) (5 + 3 Sqrt[2] - 8 Sqrt[5] + 5 Sqrt[10]) (6 + 5 Sqrt[2] + 8 Sqrt[10]), (1 - 5 Sqrt[2] + 8 Sqrt[5] - Sqrt[10]) (-5 + 8 Sqrt[2] + 5 Sqrt[5] + 3 Sqrt[10]) (1 + Sqrt[2] - 6 Sqrt[5] + 9 Sqrt[10]) (2 + 4 Sqrt[2] + 6 Sqrt[5] + 9 Sqrt[10]), (5 - 6 Sqrt[2] - 6 Sqrt[5] - 9 Sqrt[10]) (-1 + 7 Sqrt[2] - Sqrt[5] - 6 Sqrt[10]) (-3 + 4 Sqrt[2] - 4 Sqrt[5] + 5 Sqrt[10]) (-2 + 3 Sqrt[2] - 2 Sqrt[5] + 7 Sqrt[10]), (4 - 5 Sqrt[2] + 9 Sqrt[5] - 9 Sqrt[10]) (3 + 4 Sqrt[2] + 6 Sqrt[5] + 9 Sqrt[10]) (5 + 4 Sqrt[2] + 9 Sqrt[5] + 9 Sqrt[10])}
made four quadruples with single digit coefficients. Reciprocating, In[709]:= RationalizeDenominator /@ (1/%)
Out[709]= {(-8740961902019984 - 5440587070445184 Sqrt[2] + 2455653039709912 Sqrt[5] + 1573532047536568 Sqrt[10])/ 510888132844109317136, (4549175633470544 + 1131438893999820 Sqrt[2] - 1605347641058392 Sqrt[5] + 706649915148436 Sqrt[10])/ 968178139395832169104, (508007526470048 - 194043325345184 Sqrt[2] - 158757589376544 Sqrt[5] + 99676189957856 Sqrt[10])/ 20517561741702420224, (-102601486842 + 402200749082 Sqrt[2] + 176375285826 Sqrt[5] - 343223324388 Sqrt[10])/7676200432663996} --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun