Problems & Puzzles:
Puzzle 39.- The Mirrorable Numbers (By Mike
We call a number N a Dihedral Calculator Prime (or
just a Dihedral Prime) if it has the following property:
if we display N on a calculator, then the four numbers
(b) N upside down
(c) N in a mirror, and
(d) N upside down and in a mirror
are all primes. Note that (a), (b), (c), (d) need not
be all different; the only requirement is that they all
It's easy to see that the digits of N must be
0, 1, 2, 5, and 8.
Here are all Dihedral Primes less than 10^6:
108881 110881 118081 120121 121021 121151 150151
151051 151121 180181 180811 181081 188011 188801
which means the number of such primes with 1,2,3,...
(**) 1, 1, 2, 2, 1, 14, ...
Now, the puzzles:
(i) How far can you extend the sequence (**)
(ii) 120121 is the smallest such prime in which all
numbers (a, b, c, d) are distinct. How many primes like
there for 6,7,8,... digits?
(iii) This idea can be extended so that there are even
more than four numbers to look at. For example, place the
calculator on a table and stand a mirror on end touching
the left edge. Look into this mirror while seated at the
bottom of the calculator, and you will see a 2N-digit
number consisting of number (c), N-in-a-mirror followed
by (a), the digits of N. You can also place a mirror on
the right edge of the calculator and do the same (which
gives (a)(c)). You can also turn the calculator upside
down and look in the same two mirrors, which gives two
Thus the final, most challenging puzzle: can you find
a number N such that all eight of these numbers (a, b, c,
d, plus these four 2N-digit numbers) are primes? It is
known that there is no such N less than 10^11, but
heuristic arguments suggest that eventually there will be one.
Felice Russo has solved (June 1999)
the questions (i) and (ii) of this puzzle:
(i): "The sequence of number of Dihedral
primes... is:1, 1, 2, 2,
1, 14, 40,
52, 228, 482"
(ii) "The smallest primes for 7,8,9 and 10
digits in which all four numbers (a,b,c,d) are distinct
are ...: 1028011,
10128011, 100252511, 1000202221"
Russo also obtained other results
related with this kind of numbers:
"the number of Dihedral primes with four numbers
a,b,c,d distinct for 6,7,8,9 and 10 digits is: 4,12,16,132,308"
" Moreover I obtained a more accurate result for
the series SUM(1/Pd) for Pd <= 9*10^9. It is: 0.607924
(we could call it the Keith's constant or the Dihedral
primes constant, K)
Also for the number (#Pd) of Dihedral primes vs the
number of digits d I found a better fit:
#P(d)~2/5*K*exp(5/4*K*d), where with "k" I
indicated the "Keith's constant".
Russo shows that "If the above
approximation for #P(d) is correct, then SUM(1/Pd)
converges... to <~ 0.661", and adds: "Here
below a heuristic to justify the exponential growth
observed for #P(d). Since the Diheadral numbers can
contain only the digits 0,1,2,5 and 8 we can have
4*5^(d-1) of these numbers with d digits (excluding zero
as first digit). According to the Prime Number Theorem
the probability that a random number with d digit is
prime is given by: 1/(d*ln(10)). Anyway this probability
must be increased because the prime numbers are not
divisible by 2 and 5. So it became: 5/(2*d(ln(10)).
The number of Dihedral primes with d digits then is given
For large d the number of Dihedral primes became roughly
Patrick Capelle wrote (Feb. 9, 08):
I would like to mention that the prime
number 5 is missing in the list of dihedral primes less than 10^6.
Hence, the Sloane's sequences A038136, A048660, A048662 should be