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A Surprising Fact about Prime Numbers
Basically, a prime number is an integer that's divisible only by itself and by 1. The second part of the definition is the convention that 1 is not counted as a prime number. Apparently this is a useful convention.
The Larry Primes are prime numbers that are greater than 3. Here are the Larry Primes that are less than 100: 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.
Let m and n be two Larry Primes. And let's suppose that n is greater than m.
Larry's Prime Theorem (LPT): The difference between the squares of any two Larry Primes (written in mathematical shorthand as n^2 - m^2) is always divisible by 6.
The 'hat' notation means "to the power of. Thus
2^3 = 2*2*2 = 8
As an example of LPT, let's consider the primes 5 and 13.
13^2 - 5^2 = 169 - 25 = 144 which is divisible by 6. (144/6 = 24.)
Whadayaknow, I'm right so far! If you're still skeptical, please feel free to test-drive some other prime pairs on this list, before moving on to the next section.
By definition, Larry Primes greater than 3 are not divisible by 2, or by 3.
6 is divisible by both 2 and 3.
When we divide a Larry Prime p by 6, the only possible remainders are 1 and 5.
By definition, zero cannot be a remainder.
2 cannot be a remainder, because that would make p an even number.
Ditto for 4 as a remainder.
3 cannot be a remainder, because that would make p divisible by 3.
Thus any Larry Prime can be written as either 6a + 1 or 6b + 5
where a and b are both integers.
Now let's square both expressions.
Equation 1: (6a + 1)^2 = 36a^2 + 12a + 1 = 6j + 1
where j is an integer.
On the other hand,
Equation 2: (6b + 5)^2 = 36b^2 + 60b + 25 = 6k + 1
where k is an integer.
Note that the right hand sides of Equations 1 and 2 have the same form.
Finding the difference between the two right hand sides,
(6j + 1) - (6k + 1) = 6(j - k)
which is divisible by 6.
A second surprising fact
Let a, b, c, k, m, and n be any six Larry Primes. Then
a^2 + b^2 + c^2 + k^2 + m^2 + n^2 is divisible by 6.
In other words, the sum of the squares of any 6 Larry Primes is always divisible by 6.
To see why, use an approach similar to that of the previous section.
The inspiration for this hub is a conversation from many years ago, with mathematician extraordinaire, Mamikon Mnatsakanian. Mamikon showed me an educational graphic of his design. Spider Math should help students visualize the distribution of prime numbers. However I have not been able to find it online yet.
Here's a LINK to Mamikon's website.
Copyright 2012 by Larry Fields