Problems & Puzzles: Puzzles

Puzzle 1046. Reversed binary are decimal twin primes JM Bergot sent the following very nice puzzle.   1011=11;  reversed 1101=13 Q. Can you find larger twin primes like this?

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During the week 18-23, July 2021, contributions came from Giorgos Kalogeropoulos, Emmanuel Vantiegehm, Simon Cavegn

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Giorgos wrote:

I believe that the example with 11 is unique.

If we consider the more relaxed case of integers and calculate the difference of the binary and its reverse we will see that 11 is the only case where the difference equals 2.

I also noticed a rule for which I don't have proof: 

The minimum difference between a binary and its reverse of an integer k with  2^n < k < 2^(n+2)  (if we exclude zero which is the case of binary palindromes  A006995) is 2^(n/2).

This means that the minimum difference >0 for numbers greater than 2^4 is 2^2=4. So if we are searching for numbers whose difference is 2 we should only look from 2^2 up to 2^4=16.

If we do this we will find that the only number with this property is 11.,

Another example to make this rule clearer is that for integers k with 2^10 < k < 2^12  the minimum difference >0 of the binary and its reverse is 2^5 = 32 

The only numbers with difference 32 are 2081, 2225, 2345, 2489, 2597, 2741, 2861, 3005, 3107, 3251, 3371, 3515, 3623, 3767, 3887, 4031 and they are smaller than 2^12=4096

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Emmanuel wrote;

The  answer to Puzzle 1046 is : there are no further examples.

There are even no more odd numbers whose binary form reverses when 2 is added.

Let's work binary.
Then we state :
The only number that reverses by adding 10 is  1011.

Proof.
It is clear that we must restrict to odd digits.
First of all, the number 1011 after adding 10 becomes 1101, the reverse of 1011.
We see immediately (by exhausting all possibilities) that there is no other four- (or less) digit number with that property.
Take a five( or more) digit number  m.  Then there are four possibilities :
   m = 10...01.  Then : m+10 = 10...11, not the reverse of  m
   m = 11...01 : it's reverse is  10...11, smaller than  m, thus certainly not equal to m+10
   m = 11...11.  Adding  10  ends in  01, certainly not the reverse of  m.  
   m = 10...11.  Then : m+10  must be  11...01.  which can be obtained only if the intermediate digits of  m  are 1.
       But adding  10  will turn them into zeros, which is not giving the reverse !

 

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Simon wrote:

1011 = 11; reversed 1101 = 13 (twin primes)
10111 = 23; reversed 11101 = 29 (consecutive primes)
100101 = 37; reversed 101001 = 41 (consecutive primes)

Checked up to 10^12.
I think there are no other twin/consecutive prime solutions because the smallest possible difference of binary reversed numbers becomes way bigger than the largest prime gap.

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