Problems & Puzzles: Conjectures

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.

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