Is it true that the equation $27x^2+1=7^3y^2$ has infinitely many solutions in positive integers $x,y$ ?

Question

Is it true that the equation $27x^2+1=7^3y^2$ has infinitely many solutions in positive integers $x,y$ ?

Answer 1

I. Yes.

$$27x^2 + 1 = 7^3y^2\tag1$$

where,

$$x =\color{brown}{1342879} u^2 + 686\cdot\color{green}{376766} u v + 21^3\cdot\color{brown}{1342879} v^2$$

$$y =\color{green}{376766} u^2 + 54\cdot\color{brown}{1342879} u v + 21^3\cdot\color{green}{376766} v^2$$

and $u,v$ satisfy the Pell equation,

$$u^2-21^3v^2=1$$

---

II. In general:

If you have an initial solution $p,q$ to $mp^2-nq^2=d$ for any constant $d$, then an infinite more can be found by solving the Pell equation $u^2-mnv^2=\pm1$ and using the identity,

$$mx^2-ny^2 = d(u^2-mnv^2)^2\tag2$$

where,

$$x = pu^2+2nquv+mnpv^2$$

$$y = qu^2+2mpuv+mnqv^2$$

Thus, $(2)$ explains why we should set $u^2-mnv^2=\pm1$.

---

III. Even more generally:

If you have an initial solution $p,q$ to $ap^2+bpq+cq^2=d$, then an infinite more can be found by solving $u^2-Dv^2=\pm1$ with the familiar $D=b^2-4ac$ and using,

$$ax^2+bxy+cy^2 = d(u^2-Dv^2)^2\tag3$$

where,

$$x = p u^2 - 2(b p + 2c q)u v + D p v^2$$

$$y= q u^2 + 2(2a p + b q) u v + D q v^2$$

and $u,v$ solves $u^2-Dv^2=\pm1$, where $D=b^2-4ac$.

Answer 2

From this site,

$27 x^2 - 343 y^2 +1 = 0$

by Dario Alejandro Alpern

X0 = 1 342879 (7 digits)
Y0 = 376766
If (x,y) is a solution, (-x,-y) is also a solution.

Xn+1 = P Xn + Q Yn
Yn+1 = R Xn + S Yn

P = 97 379496 466615 (14 digits)
Q = 347 082488 429404 (15 digits)
R = 27 321362 062956 (14 digits)
S = 97 379496 466615 (14 digits)

Answer 3

Here is some output from a program that anyone can write, it is Lagrange's method for indefinite forms. First, the cycle for $343s^2 - 27 t^2$ shows that it really does represent $1.$ This is in Dickson (1929) Introduction to the Theory of Numbers. I like the presentation in D. A. Buell, Binary Quadratic Forms. Program in C++, eventually I added in some features for oversize integers using GMP.

You Can Do This Yourself.

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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle 343 0 -27

  0  form            343           0         -27  delta     -3
  1  form            -27         162         100


           0          -1
           1          -3

To Return  
          -3           1
          -1           0

0  form   -27 162 100   delta  1     ambiguous  
1  form   100 38 -89   delta  -1
2  form   -89 140 49   delta  3
3  form   49 154 -68   delta  -2
4  form   -68 118 85   delta  1
5  form   85 52 -101   delta  -1
6  form   -101 150 36   delta  4
7  form   36 138 -125   delta  -1
8  form   -125 112 49   delta  3
9  form   49 182 -20   delta  -9
10  form   -20 178 67   delta  2
11  form   67 90 -108   delta  -1
12  form   -108 126 49   delta  3
13  form   49 168 -45   delta  -4
14  form   -45 192 1   delta  192
15  form   1 192 -45   delta  -4     ambiguous  
16  form   -45 168 49   delta  3
17  form   49 126 -108   delta  -1
18  form   -108 90 67   delta  2
19  form   67 178 -20   delta  -9
20  form   -20 182 49   delta  3
21  form   49 112 -125   delta  -1
22  form   -125 138 36   delta  4
23  form   36 150 -101   delta  -1
24  form   -101 52 85   delta  1
25  form   85 118 -68   delta  -2
26  form   -68 154 49   delta  3
27  form   49 140 -89   delta  -1
28  form   -89 38 100   delta  1
29  form   100 162 -27   delta  -6
30  form   -27 162 100


  form   -27 x^2  + 162 x y  100 y^2 

minimum was   1rep   x = 212581   y = -376766 disc   37044 dSqrt 192.46817919  M_Ratio  50.81481
Automorph, written on right of Gram matrix:  

===============================================

Let's see, as Tito pointed out, we also want to solve $u^2 - 21^3 v^2 = 1,$

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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./Pell 9261


0  form   1 192 -45   delta  -4
1  form   -45 168 49   delta  3
2  form   49 126 -108   delta  -1
3  form   -108 90 67   delta  2
4  form   67 178 -20   delta  -9
5  form   -20 182 49   delta  3
6  form   49 112 -125   delta  -1
7  form   -125 138 36   delta  4
8  form   36 150 -101   delta  -1
9  form   -101 52 85   delta  1
10  form   85 118 -68   delta  -2
11  form   -68 154 49   delta  3
12  form   49 140 -89   delta  -1
13  form   -89 38 100   delta  1
14  form   100 162 -27   delta  -6
15  form   -27 162 100   delta  1
16  form   100 38 -89   delta  -1
17  form   -89 140 49   delta  3
18  form   49 154 -68   delta  -2
19  form   -68 118 85   delta  1
20  form   85 52 -101   delta  -1
21  form   -101 150 36   delta  4
22  form   36 138 -125   delta  -1
23  form   -125 112 49   delta  3
24  form   49 182 -20   delta  -9
25  form   -20 178 67   delta  2
26  form   67 90 -108   delta  -1
27  form   -108 126 49   delta  3
28  form   49 168 -45   delta  -4
29  form   -45 192 1   delta  192
30  form   1 192 -45

 disc   37044
Automorph, written on right of Gram matrix:  
236875798327  45535603438260
1011902298628  194522117134903


 Pell automorph 
97379496466615  9371227187593908
1011902298628  97379496466615

Pell unit 
97379496466615^2 - 9261 * 1011902298628^2 = 1 

=========================================

9261      3^3 * 7^3

===============================================

Name the matrix called "Pell Automorph" with the letter $A.$ Note that the first solution of $u^2 - 9261 v^2 = 1$ is just the left hand column of $A.$ There are infinitely many solutions, they can all be written as the left hand column of the matrix $A^n,$ for integer exponent $n \geq 1.$