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.

That works, you're right

In[1]:= 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]))

In[46]:= With[{l = RandomInteger[{1, 2^16}, 17]},
   Print["l: ", l];
   And @@ (IntegerQ /@ FullSimplify[Expand[x[#]] & /@ l])
 ]

During evaluation of In[46]:= l: {8309,55053,28081,62999,49730,51523,40250,65370,30882,4898,16978,48880,6337,6133,9837,26546,49300}
Out[46]= True

In[47]:= With[{l = RandomInteger[{1, 2^16}, 17]},
   Print["l: ", l];
   And @@ (IntegerQ /@ FullSimplify[Expand[y[#]] & /@ l])
]

During evaluation of In[47]:= l: {447,23621,25471,49476,26678,6573,60939,63208,56458,51985,24444,34525,59990,12914,10239,64048,34724}
Out[47]= True

thank you.

It has to work because in $x$ the odd powers of $\sqrt{7}$ cancel out as well in $y$ the even powers of $\sqrt{7}$ cancel out and in $y$ a division by $\sqrt{7}$ follows: $x$ and $y$ are both root-free. The factor $\frac{1}{2}$ does also not create a rational number, because the previous step brought a factor 2 in the numerator.