how do you show this implication ?

Question

$\forall\; n \geq 3 \;\;\exists (x,y) \in \{2m+1;m\in \mathbb{N}\}^2$ such that $2^n=7x^2 + y^2 $

I proved it by induction and I'm curious how you guys would you show it.

Answer 1

By the generalized Lagrange's identity $$(a^2+7b^2)(c^2+7d^2)=(ac-7bd)^2+7(ad+bc)^2 \tag{1}$$ That can be seen as a consequence of the multiplicativity of the norm over $\mathbb{Z}[\sqrt{-7}]$.
It follows that the set $E_7$ of integers that can be represented as $n^2+7m^2$ is a semigroup.
Obviously $8=2^3=1^2+7\cdot 1^2\in E_7$, $16=3^2+7\cdot 1^2\in E_7$ and $32=5^2+7\cdot 1^2\in E_7$.
Then you may use $(1)$ and induction to prove that every integer of the form $2^n$ with $n\geq 3$ belongs to $E_7$, and it has a representation of the form $x^2+7y^2$ with both $x$ and $y$ being odd numbers.

Answer 2

If $a^2 + 7 b^2 = 2^n$ with odd $$ a \equiv b \pmod 4, $$ then we get $$ \left( \frac{a - 7b}{2} \right)^2 + 7 \left( \frac{a+b}{2} \right)^2 = 2^{n+1}, $$ we have preserved oddness as well as equality $\pmod 4,$ as $$ \left( \frac{a - 7b}{2} \right) \equiv \left( \frac{a+b}{2} \right) \pmod 4 $$ since $$ a - 7 b \equiv a + b \equiv \pm 2 \pmod 8 $$

jagy@phobeusjunior:~$ 
     1     1     2               8
    -3     1     6              16
    -5    -1     2              32
     1    -3     6              64
    11    -1     2             128
     9     5     6             256
   -13     7     2             512
   -31    -3     6            1024
    -5   -17     2            2048
    57   -11     6            4096
    67    23     2            8192
   -47    45     6           16384
  -181    -1     2           32768
   -87   -91     6           65536
   275   -89     2          131072
   449    93     6          262144
  -101   271     2          524288
  -999    85     6         1048576
  -797  -457     2         2097152
  1201  -627     6         4194304
  2795   287     2         8388608
   393  1541     6        16777216
 -5197   967     2        33554432
 -5983 -2115     6        67108864
  4411 -4049     2       134217728
jagy@phobeusjunior:~$