Problems & Puzzles:
Puzzle 309. A
property of the prime '5'
Firoozbakht poses the following puzzle:
1) I proved that 5 is the only
prime solution of the equation:
sigma(phi(x)) - phi(sigma(x)) = x [*]
2) Can you find a composite solution for
the equation [*] or can you prove that 5 is the only solution of
3) My question: Can you prove what FF
Contributions for the question 3) came from: Rudolph Knjzek, Joseph L.
Pe, Dan Dima, Johann Wiesenbauer & Salvatore Tringali. The proff given by
all of them is about the same:
You can verify that sigma(phi(p)) - phi(sigma(p)) =\= p, if p = 2 or
p = 3. On the contrary, the property is trivially verified for p = 5,
since sigma(phi(5)) - phi(sigma(5)) =
sigma(2^2) - phi(2*3) = 5. So, in the following, you can suppose p > 5.
Therefore, p is odd and subsequently (p-1)/2 and 2 are distinct divisors
of p-1. Ergo: sigma(phi(p)) - phi(sigma(p)) = sigma(p-1) - phi(p+1) > (1
+ 2 + (p-1)/2 + p-1)) - (p+1)/2 > p. That suffices to prove Faride
No one of them has gotten neither a composite x solution for x values
as large as 22,700,000 (J.W.), 10^9 (D.D.), 21,000,000 (J.L.P.), nor a
proof of its impossibility.