Expert: Scott A Wilson - 7/31/2009

Question
hi, I have question in number theory but since all experts are on vacation in that field I thought you may help me in this :

Twin primes 5 and 7 are such that half of  their sum is a perfect number, are there other twin primes withthis property.
Hint: given the twin primes p and p+2,with p>5, 1/2(p+p+2_=6k for some k>1
Thank you so much for the help :)

Answer
The rule for finding perfect numbers is by testing 2^n-1.  
If that is prime, then (2^n - 1)2^(n-1) is a perfect number.

2-1 = 1, and that is not prime.  All prime numbers are numbers with two distinct factors - 1 and themselves.  1 does not have two distinct factors, for they are both the same number.

2^2 - 1 = 4-1 = 3, which is prime.  This means that (2^2 - 1)(2^(2-1)) = (4-1)(2^1) =
3*2 = 6 is a perfect number.  Yes, 5 and 7 are prime.

2^3 - 1 = 8 - 1 = 7, which is prime.  This means that (2^3 - 1)(2^(3-1)) = 7*4 = 28 is a perfect number.  The factors that are less than that number are 1, 2, 4, 7, and 14.  They add to 28.
The number n+1 is 29, which is prime.  However, if I get this correctly, we look at 28-1=27, and that is not prime for 27 = 3*3*3.

2^4 - 1 = 16 - 1 = 15, which is not prime.

2^5 - 1 = 31, which is prime.  Using the formula, we get 31*16 = 496, which is the next perfect number.  496+1=497, which is known to be 7*71.  496-1=495, which is known to be 3*3*5*11, so it is not prime either.

If there is something I'm missing, try it on the following perfect numbers and get back to me.
They are 6, 28, 496, 8,128, 33,550,336, and 8,589,869,056.  Those are the first six perfect numbers.  They are 2^(n-1)(2^n - 1) where n is 2, 3, 5, 7, 13, and 17.

The factors of 28 are 1, 2, 4, 7, and 14.

The factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, and 248.

The factors of 8,128 are 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, and 4064.

The factors of 33,550,336 are 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048,  4,096,
8,191, 16,382, 32,764, 65,528, 131,056, 262,112, 524,224, 1,048,448, 2,096,896, 4,193,792,
8,387,584, 16,775,168.

I won't even try to factor the last one, though.


In case this helps at all, the following don't lead to perfect numbers.  They are
2^4 - 1 = 15 = 3*5
2^6 - 1 = 64 - 1 = 63 = 3*3*7
2^8 - 1 = 255 = 3*5*17
2^9 - 1 = 511 = 7*73
2^10 - 1 = 1,023 = 3*11*13
2^11 - 1 = 2,047 = 23*89
2^12 - 1 = 4,095 = 3*3*5*7*13
2^14 - 1 = 16,383 = 3*43*127
2^15 - 1 = 32,767 = 7*31*151
2^16 - 1 = 65,535 = 5*17*771
2^18 - 1 = 262,143 = 3*3*3*7*19*73


The next number was too large to even worry about when I attended college.
2^19 - 1 is 524,287, and that is prime.
However, (2^19 - 1)(2^18) can now actually be found.
It is the next perfect number: 137,438,691,328.


And that's all I've got to say about that...