**A Prime Number Conjecture in Binary**

Take any prime number p and write it in binary format.

From this prime now generate a series of numbers using this simple algorithm:

You can insert a single digit (0 or 1) anywhere you wish. (But you can't insert a 0 at the beginning because that just gives p unchanged.)

If p is m digits long in binary you will generate 2*m+1 new binary numbers. (btw: some may are duplicates.)

CONJECTURE: At least one of these new numbers is prime.

Let me illustrate the process with a simple example.

Take the prime 5, which in binary is 101. It has 3 digits, so we know the algorithm will generate 2*3+1=7 numbers. Here they are (notice the duplicates)..

1010=10

1011=11

1011=11

1001=9

1101=13

1001=9

1101=13

So in this case we've generated two new prime numbers, 11 and 13.

A note on duplicates:

I think you will always get m duplicates. So if these are removed the algorithm generates just m+1 numbers and the conjecture says at least one of these is prime.

Content written and posted by Ken Abbott abbottsystems@gmail.com

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