A Surprising Fact about Prime Numbers

Yours truly on a prime hike to Roundtop Lake, in California's Northern Sierra Nevada Range.
Yours truly on a prime hike to Roundtop Lake, in California's Northern Sierra Nevada Range. | Source

Background

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.


Why?

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.

Why?

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.


Mamikon Mnatsakanian
Mamikon Mnatsakanian | Source

Hat-tip

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

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Comments 14 comments

Katherine Katie profile image

Katherine Katie 4 years ago from UK

Very Nice one


Larry Fields profile image

Larry Fields 4 years ago from Northern California Author

Hi Katherine. Thanks for reading.


LewSethics profile image

LewSethics 4 years ago

Hmmm. It took me a minute to see the whole thing. Very interesting.


Larry Fields profile image

Larry Fields 4 years ago from Northern California Author

Hi Lew. Thanks for stopping by.


Cacey Taylor 4 years ago

Very interesting. Is there a reason why it is always divisible by 6?


Larry Fields profile image

Larry Fields 4 years ago from Northern California Author

Hi Cacey. Thanks for reading.

To answer your question. The second section of this hub explains why, but not in a way that will leave everyone saying, "Aha!"

Being a maths geek, I can visualize the whole thing, but it's not reasonable to expect my readers to see it in the same way. The best that I can do is to explain it in several small steps.

However that approach is not as satisfying as a clear, one-line explanation would be--assuming that such an explanation existed in the first place.


Nell Rose profile image

Nell Rose 4 years ago from England

Clear as mud Larry! lol! on a dark day in a tunnel full of mush! but thats not a reflection on your math, its a reflection on my mind and maths! me and numbers just do not get on, fascinating though! lol!


Cacey Taylor 4 years ago

Thanks Larry


Larry Fields profile image

Larry Fields 4 years ago from Northern California Author

Hi Nell. Nice to hear from you, as always.

About you and maths. A friend, Mamikon Mnatsakanian, created a maths enrichment program for a local Montessori school. All of the games and puzzles were his own. The kids loved it. They called him Magic Mamikon. I have a feeling that if you had had this kind of maths education, that you would feel more comfortable with the subject matter.


Larry Fields profile image

Larry Fields 4 years ago from Northern California Author

Hi Cacey. Thanks for stopping by again.


tabrezrocks profile image

tabrezrocks 4 years ago from Calcutta, India

good knowledge based article.


Larry Fields profile image

Larry Fields 4 years ago from Northern California Author

Hi tabrezrocks. Thanks for reading.


Suhail and my dog profile image

Suhail and my dog 3 years ago from Mississauga, ON

I liked your mathematics behind prime numbers. It is very interesting indeed.

At one time, as an engineering student, I was very much interested in secrets of mathematical numbers, but then after graduating, I never practiced engineering. However, I do try to keep connected with mathematics in one way or the other.

I will shortly be reading a book titled 'Mathematical Nature Walk'. It seems to show mathematics behind many of the nature's phenomenon. Mathematics still intrigues and, in a way, intimidates me.


Larry Fields profile image

Larry Fields 3 years ago from Northern California Author

Hi Suhail,

Thanks for stopping by. Your comment reminded me of something that I've noticed. Mathematicians and physical scientists tend to be Nature buffs.

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