Physicist: That’s a really cool property!
Every prime number (other than 2 and 3) can be written in the form 6j+1 or 6j+5. For example, 17 = 6(2)+5 and 31 = 6(5)+1.
This is because numbers of the form 6j, 6j+2, 6j+4, are all divisible by 2, and all numbers of the form 6j, 6j+3 are divisible by 3. So that restricts the options for primes to just the “+1” set and the “+5” set. Not all of the numbers in these sets are primes, but all the primes are in these two sets.
6j = 6, 12, 18, 24, 30, … All divisible by 6.
6j+1 = 7, 13, 19, 25, 31, …
6j+2 = 8, 14, 20, 26, 32, … All divisible by 2.
6j+3 = 9, 15, 21, 27, 33, … All divisible by 3.
6j+4 = 4, 10, 16, 22, 28, 34, … All divisible by 2.
6j+5 = 5, 11, 17, 23, 29, 35, …
Call the primes p and q, and notice that 
If p and q are both the same type (+1 or +5), then (p-q) will be a multiple of 6. For example: (+1 case) 31-7 = 24 and (+5 case) 29-11 = 18.
If p and q are opposite types, then (p+q) will be divisible by 6. For example: 23+13 = 36.
In both cases, the other bubble, (p+q) or (p-q), will always be divisible by 2, since the sum and difference of any two odd numbers is always even. So, one bubble is always a multiple of 6 and the other is always a multiple of 2, and together the whole thing is always a multiple of 12.
For example: p=11, q=7. 
This is another example of modular arithmetic. It almost should have been included in the “tricks with 9’s post“.
Also: This trick doesn’t really have much to do with “primes”, so much as it has to do with “numbers that don’t have 2 or 3 as a factor”. That isn’t obvious at first. The first composite (not prime and not 1) number with no 2’s or 3’s is 25.
Cute theorem.
I never heard of one calling a factor a “bubble.” T’is a nice idea.
I thought bubbles refer to the parentheses with stuff inside…
Wonderful! It’s easy to show and I didn’t see the result as obvious (even though I have a BA in math).
Given that 2^10+5^12 is a product of two primes, find the difference between these two primes? How do I solve this? I need to explain this to my 6th grade child.
Amar
http://www.wolframalpha.com/input/?i=FACTOR%282%5E10%2B5%5E12%29 =244141649=14657 X 16657
16657 – 14657 = 2000
As far as I can see the difference between such numbers is always divisible by 24, not just 12, right? Does Physicist’s argument still hold?
The difference between a number raised to the power of any two distinct primes (excluding 2 and 3) also seems to be divisible by 24. For example (42^19) – (42^11). That’s my conjecture anyway. Please check.
Question…
“Every prime number (other than 2 and 3) can be written in the form 6j+1 or 6j+5. For example, 17 = 6(2)+5 and 31 = 6(5)+1.”
Is that an observation or can it be proved to be true?
@RMHP
It’s pretty easy to prove through “proof by exhaustion”.
6j is divisible by 6 (clearly).
6j+2 = 2(3j+1) is divisible by 2
6j+3 = 3(2j+1) is divisible by 3
6j+4 = 2(3j+2) is divisible by 2