Problems & Puzzles:
Primes p such that R(p) is prime
and p^2 = R(R(p)^2)
means "the reversible of p".
least example is 13, because:
R(p)=31 is prime &
13^2 = 169 = R(31^2)= R(961)=169
rather large of this kind of primes is 3011110001 (Digits =10):
R(p)=1000111103 is prime &
= 9066783438122220001 = R(1000111103^2) = R(1000222218343876609) = 9066783438122220001
property seems to be not entirely trivial, because:
not all primes are such that its reversible are primes
b) not all primes p such that its reversible are primes, are such that p^2
R(p)=1321 is prime but
1231^2=1515361 is not equal to R(1321^2)=R(1745041)=1405471
Find an example of D digits for D = 20, 30, 40,
It seems that all these type
of primes in his decimal
expression are composed of digits less than 4. If
this is true, can
you visualize any other rule about these primes?
Russo has sent (10-11/07/2000) the following primes for question a):
Faride Firoozbakht wrote (March 2004):
I'm sure that if n is a number such that n^2=R(R(n^2)) the following
three statements are true but I don't know how to explain the reasons.
1.All of the digits of n (in decimal expression) are less than 4.
2.If a and b are two arbitrary digits (in decimal expression) of
n then a+b<5.
3.In his decimal expression of n there doesn't exist three
2 beside each other(222 doesn't appear).