For which even integers $k$ does the equation $$\varphi(n+1)-\varphi(n)=k$$ have a solution ?
$\varphi(n)$ denotes the totient function and $n$ is a positive integer.
For the following $|k|\le 1\ 000$, there is no solution $n$ in the range $[3,10^7]$ :
-958 -926 -910 -898 -892 -846 -834 -814 -790 -730 -682 -610 -594 -582 -570 -550
-514 -490 -462 -442 -422 -370 -354 -326 -310 -226 -202 -114 10 86 126 134 182 22
6 242 266 274 278 286 298 326 370 378 386 446 450 466 470 530 538 574 578 610 62
6 634 638 666 678 706 734 738 758 770 786 790 806 822 826 830 842 866 874 898 91
4 926 932 938 970 986
Almost all those numbers are of the form $\ 4k+2$
In fact, the $|k|\le 1\ 000$ divisible by $4$ having no solution in the range $[3,10^6]$ are $-892$ (solution $10\ 814\ 714$) and $932$ (no solution in the range $[3,10^8]$)
Does a solution exist for $k=932$ ?