Have you seen this, David?
In[1]:= Reduce[x^2 - 7 y^2 == 1 && x > 0 && y > 0, {x, y}, Integers]
Out[1]= C[1] \[Element] Integers && C[1] >= 1 &&
x == 1/2 ((8 - 3 Sqrt[7])^C[1] + (8 + 3 Sqrt[7])^C[1]) &&
y == -(((8 - 3 Sqrt[7])^C[1] - (8 + 3 Sqrt[7])^C[1])/(2 Sqrt[7]))
let's define x
and y
as functions of the exponent
In[2]:= Clear[x, y]
x[n_] := 1/2 ((8 - 3 Sqrt[7])^n + (8 + 3 Sqrt[7])^n)
y[n_] := -(((8 - 3 Sqrt[7])^n - (8 + 3 Sqrt[7])^n)/(2 Sqrt[7]))
and convince ourself that they are integers:
In[5]:= RootApproximant[x[#]] & /@ RandomInteger[{1, 30}, 17]
Out[5]= {32257, 514088, 9156316745224513962640064643072, \
2199950293420883836928, 2081028097, 8193151, 35061166466652774072320, \
33165873224, 130576328, 138038228081368154112, 543466014742176320, \
8193151, 8424001222568, 33165873224, 8, 138038228081368154112, \
574522862194843517270624305152}
In[6]:= RootApproximant[y[#]] & /@ RandomInteger[{1, 30}, 17]
Out[6]= {217149230841226888732894822400, 3096720, 50743789129440, \
13625260145284696589750763520, 12535521795, 49353213, \
223267627569198170583290140455600128, 12535521795, 12535521795, \
217149230841226888732894822400, 12888722657083742, \
53643587967663564981796864, 53643587967663564981796864,
1/4 (1617433305776647 + 13 Sqrt[15479825435713475684361539193]), 765, \
854931483328272233719136256, 223267627569198170583290140455600128}
In[7]:= FactorInteger[15479825435713475684361539193]
Out[7]= {{3, 1}, {7, 1}, {17, 1}, {173, 1}, {1160503, 1}, {215975989374144271, 1}}
they are not: the
$x$ pass this self-test, the
$y$ do not. This is under Mathematica 10.1. Mathematica is a great product, but as Heifetz put it:
There is no top. There are always further hights to reach.
Best possible support for Wolfram Inc. is pointing out the next bug.