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?
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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