**
Conjecture 54.** **Reversible
primes conjecture**

Suman Das sent the following Conjecture
related to primes:

Conjecture: Between 2 and any
integer n (inclusive), there are more reversible primes in the binary
system than in the decimal system. I have checked this conjecture for
all n up to 700 million. I didn't find any exception.

Definition: A reversible prime in any
number system is a prime whose "reverse" in that number system is also a
prime. For example in the decimal system 73 is a reversible prime because
its reverse 37 is also a prime. However I
exclude palindrome primes from the definition.

**Q1. Can you verify the Suman Das claim?**

**Q2. If verified can
you exhibit an explanation for this Conjecture?**

Let's define f(n)=B-D, where B is the
quantity of reversible primes in binary and D the quantity of reversible
primes in decimal, up to n.

**Q3. Please compute &
plot f(n)**

Contribution came from Frank Rubin on
Feb 9, 08:

There is a simple reason why there
are more reversible primes in a prime number base than in a composite
number base. In a prime base, a prime may end in any digit except 0, and
likewise may begin with any digit except 0. Hence, when the digits are
reversed, the low order digit is always a valid ending digit for a
prime. In a composite base, if the leading digit has a factor in common
with the base, then it cannot be the final digit of a multidigit prime.
Hence the reversal cannot be prime. For example, in base 10, primes
starting with 2, 4, 5, 6, or 8 cannot be reversible.

You can verify that this is the correct explanation by counting
reversible primes in base 6 versus base 10. There should be fewer
reversible primes in any sufficiently large range since 3/5 of the
possible leading digits are disallowed, as compared to 5/9 of the
leading digits in base 10. The number of reversible primes in base 30,
base 210, base 2310, etc., should be progressively smaller.

***