Twin Prime Sandwich

Building up on the lesson from a prior post, "Prime Numbers - a property rediscovered," it appears that all numbers between every pair of twin primes are multiples of 6. 

This isn't too surprising given that all primes are in the form 6n+/-1, but that fact alone doesn't guarantee that the number between a pair of twin primes is of the form 6n.

Here are the steps of the proof.

1. Let a, b, c, d, e represent 5 consecutive integers for which b and d are twin primes.

2. Consider the prime, b. Given the 6n+/-1 theorem, then either a is a multiple of 6 or c is a multiple of 6.

If a is a multiple of 6, then d cannot be prime because d=a+3 making d a multiple of 3. Therefore c must be a multiple of 6.

3. Likewise, consider d. Either d is prime because c is a multiple of 6 or e is a multiple of 6. However, if e is a multiple of 6 then b=e-3 so b would be a multiple of 3 and not prime. Therefore c must be a multiple of 6 if b and d are twin primes.

Prime Numbers - a property rediscovered

I stumbled across an interesting property of prime numbers last year. The numbers 2 and 3 are the only primes that cannot be expressed as a multiple of 6 plus or minus 1. For example, 7 is 6 + 1, 11 is 12 -1, 13 is 12+1 and so on. However, not all numbers of that are a multiple of 6 plus or minus 1 are prime. For example, 25 = 24 + 1 and 25 is not prime. Another example is 35 = 36 - 1 and 35 is not prime.

After a sleepless night, I put together a short proof of all primes greater than 3 being expressed as a multiple of 6 plus 1 or minus 1.

Steps of the proof:

1. Let S represent a multiple of 6. All numbers greater than 3 can be expressed as either S, S+/-1, S+/-2 or S+/-3. (Why not include S+/-4 and S+/-5? An S+4 number such as 10 is already covered by the next S-2. That is, 12-2=10. Likewise for S-4; for example the number 8 or 12-4, is already covered by the previous S, 6 or 8 = 6+2. Again, for the numbers expressed by S+/-5 are covered by next, or previous, S+/-1.)

2. Since all numbers greater than 3 can be expressed as either S, S+/-1, S+/-2 or S+/-3, then so can all prime numbers.

3. Eliminate reasons for S, S+/-2, S+/-3

    Any number that is S (a multiple of 6) is not prime.

    Any number that is S+/-2 is an even number greater than 2 and therefore is not prime.

    S, as a multiple of 6, is also a multiple of 3 so any number that is S+/-3 is also a multiple of 3 and therefore is not prime (note: in the case of 6-3=3, 3 is prime but we've stated this property is for primes greater than 3.)

4. From Step 2 we established all primes can expressed as either S, S+/-1, S+/-2 or S+/-3. From Step 3 we eliminated primes being expressed as S, S+/-2 and S+/-3, so therefore by elimination, all primes greater than 3 must be able to be expressed as S+/-1.

I wondered why I was never taught this before. Was I dozing through this lesson in math? Certainly someone has come across this property before me. I entered "primes as 6n +/-1" and was greeted by the following ego-deflating statement "this is the most re-discovered property of prime numbers." Nearly the same proof is presented by Chris Caldwell, University of Tennessee at Martin. 6n+/-1.

I didn't at all expect my approach to be unique, but it is always very satisfying to prove something on your own and that should always be encouraged.

Update 3-16-2021

One can build up on this property of prime numbers with the fact that the square of all prime numbers above 3 is 1 plus a multiple of 24. For example, 7 square is 49 which equals 1+2x24. Matt Parker proves this additional prime number fact two different ways in this Numberphile YouTube video.

Proofs of the Pythagorean Theorem

There are hundreds of proofs of the Pythagorean Theorem. I've been testing my own limited knowledge of geometry trying to master as many as I can. Many of the proofs use similar techniques (recombining various triangular, square and rectangular shapes in various ways to show the desired result), so sometimes it is difficult to say if a proof is truly unique.

Of the many reconstruction methods, a popular technique is arranging multiple copies of a right-angle tri angle into a square shape then calculating the area of the resultant square in two ways. First, the squaring the length of a side. Second, calculating the areas of the components that make up the square. Setting these two areas equal, produces the desired result, showing:

C² = A² + B²

An example of this approach is given by Bhaskara's First Proof: Bhaskara's First Proof

The most concise proof, in my opinion, is Bhaskara's second proof: 

Start with a right-angle triangle, ABC. Let the lengths of the sides be a, b, and c.

Draw the altitude line, from point C to the opposite side at point D. Line CD is perpendicular to side AB. 

The three triangles, ABC, CBD and CAD are all similar. Let the distance from A to D be x and then from B to D is c - x.

Since the ratio of sides of similar triangles are equal,

 

a/c = (c-x)/a                              b/c = x/b

Simplifying,  

a² = c² – cx                                 b² = cx

Adding these two equations together, results in (note, the term cx and -cx cancel),

 a² + b² = c²

The broken domino proof starts with a domino-shape tile that consists of two equal squares (each representing  c²). Three triangular pieces are broken from one of the squares and rearranged to form the squares representing a² + b². See proof 35 at Gary Zabel's page: Gary Zabel UMB

As a practical use of the broken tile proof, there are actual tiling patterns based on the Pythagorean Proof. Tiling with Pythagoras

A prior post outlines President James Garfield's proof of the Pythagorean Theorem. Garfield Proof

Wikipedia entry: Pythagorean Theorem