## Is there a composite number n; σ(n+2) = σ(n)+2 and φ(n+2) = φ(n)+2?

Prove that if each of the numbers   $n$   and   $n+2$   is prime, then
$\phi \, (n+2) \; = \; \phi \, (n) \; + \; 2$

The equation is satisfied also by composite numbers, for example,

$\phi \, (14) \; = \; \phi \, (12) \; + \; 2 \; = \; 6$
$\phi \, (16) \; = \; \phi \, (14) \; + \; 2 \; = \; 8$
$\phi \, (22) \; = \; \phi \, (20) \; + \; 2 \; = \; 10$
$\phi \, (46) \; = \; \phi \, (44) \; + \; 2 \; = \; 22$
$\phi \, (64) \; = \; \phi \, (62) \; + \; 2 \; = \; 32$
$\phi \, (94) \; = \; \phi \, (92) \; + \; 2 \; = \; 46$

There are no composite odd numbers   $n \; < \; 10^4$   that satisfy the equation.

We may suggest a conjecture that there no odd numbers   $n$,   except for the pairs of twin primes   $(n, n+2)$   for which the equation holds:

$\phi \, (n+2) \; = \; \phi \, (n) \; + \; 2$

Does there exist a composite number   $n$   for which the equations

$\phi \, (n+2) \; = \; \phi \, (n) \; + \; 2$
$\sigma \, (n+2) \; = \; \sigma \, (n) \; + \; 2$

hold simultaneously?