Problems & Puzzles:
Farideh sent the one more nice puzzle:
For n=5187 we have phi(n-1)=phi(n)=phi(n+1)
Q. Does there exist another such
Farideh herself provided an interesting extension for this same
If we define a(n) the smallest number m such that phi(m-n)=phi(m)=phi(m+n)
then the terms for n=1,2,...,12 are:
5187, 10, ?, 20, 25935, 78, ?, 40, ? , ?, 57057,156.
It seems that for each n, a(6n) exists!
Also we can show that if gcd(n, 34889017254)=1 then a(n) exists.
Q: Can you find some of the numbers a(n) for n=3, 7, 9 & 10 or can
prove that at least one of them doesn't exist.
Giovanni Resta wrote on Feb. 2014
For Puzzle 466, I have computed phi(n) for n < 10^13,
looking for values n such that phi(n-k) = phi(n) = phi(n+k),
in particular for k = 1, 3, 7, 9 and 10, as requested by
you and Farideh.
For n < 10^13, the only new result I obtained is
phi(30057431145 - 7) = phi(30057431145) = phi(30057431145 + 7).