Mohanty's conjecture

Question

Is there any proof or counter-proof of Mohanty's conjecture (1988) in the litterature:

The numbers n, n + 6, and n + 12 cannot be expressed simultaneously as sum of two squares.

Answer 1

For the system of equations.

$$\left\{\begin{aligned}&N=c^2+q^2\\&N+T=a^2+b^2\\&N+2T=x^2+y^2\end{aligned}\right.$$

Lay on multipliers. $T=2ps$ Solutions written in this form.

$$c=T+k^2+k(p+s-2)-p-s+\frac{1}{2}$$

$$q=T+k^2+k(p+s)-\frac{1}{2}$$

$$a=T+k^2+k(p+s-1)-p+\frac{1}{2}$$

$$b=T+k^2+k(p+s-1)-s+\frac{1}{2}$$

$$x=T+k^2+k(p+s-1)-p-s+\frac{1}{2}$$

$$y=T+k^2+k(p+s-1)+\frac{1}{2}$$

$k - $ Any whole number. It is seen that solutions in integers there is not only for $T=6$ but for any other integer.

This formula will be better ....

Decompose the number $T$ in two different ways. $T=2ps=kt$

$$c=kn^2+(2k+s-p)n+s-p+\frac{t+k}{2}$$

$$q=kn^2+(s-p)n+\frac{t-k}{2}$$

$$a=kn^2+(k+s-p)n+s-p+\frac{t+k}{2}$$

$$b=kn^2+(k+s-p)n+\frac{t+k}{2}$$

$$x=kn^2+(k+s-p)n+s+\frac{t+k}{2}$$

$$y=kn^2+(k+s-p)n-p+\frac{t+k}{2}$$