Relatively Prime Pythagorean Triples

Date: 09/13/97 at 18:23:22
From: Ben Love
Subject: Relatively Prime Pythagorean Triplets

A relatively prime Pythagorean triplet (RPPT) (in case you know it 
by other names) is a triplet such as 3,4,5 that cannot be reduced.
Ex. 3,4,5 cannot be reduced.
Ex. 6,8,10 CAN be reduced (by 2)

Anyway, I found using a computer that whenever A is a positive prime 
number, then B is even and C = B + 1! Another interesting fact is that 
so far I have only been able to find exactly one RPPT for each prime 
from 1 to 100 except 2. 2, as far as I can see, doesn't work at all.

I can't say I've proven that there is no other triplet with 3, but 
I have found that 4 is the ONLY number from 1 to 10,000 that works 
with 3. The third thing I've found is that if you take each prime 
from 3 to 100 and then find the difference between the "B"s (ex: 
3 _4_ 5 and 5 _12_ 13 -- 12 - 4 = 8) all of the differences except 
the first and second are divisible by twelve. This brings me to my 
first question: Why is that? Or is that just a coincidence?

Second, if that is true, how can I prove that C - B = 1 (when A is a 
pos. prime and B and C are pos. whole numbers)? I have been working 
on this and cannot figure it out.

Here is a chart that I have printed up:

The first three columns represent A, B, and C in that order.
The fourth is the difference between each prime (A). The fifth is the
difference between the other numbers (B). The last is what you get 
when you divide B by 12.

------ Copied from file rppt.prf --------

RPPT = Relatively Prime Pythagorean Triplet

A triplet is relatively prime if the triplet cannot be reduced.
Ex: 3,4,5 cannot be reduced.
Ex: 6,8,10 CAN be reduced (by 2) so it is not.

These are all RPPTs.

               A    B    C      CC          Diff. A   B   B / 12      
              ---  ---  ---   ------             --- --- ---
Here is one:    3    4    5       25 Difference:  3   4   0       
Here is one:    5   12   13      169 Difference:  2   8   0
Here is one:    7   24   25      625 Difference:  2  12   1
Here is one:   11   60   61     3721 Difference:  4  36   3 ----------
Here is one:   13   84   85     7225 Difference:  2  24   2
Here is one:   17  144  145    21025 Difference:  4  60   5  10 primes 
Here is one:   19  180  181    32761 Difference:  2  36   3
Here is one:   23  264  265    70225 Difference:  4  84   7  in a row?
Here is one:   29  420  421   177241 Difference:  6 156  13
Here is one:   31  480  481   231361 Difference:  2  60   5  Why?
Here is one:   37  684  685   469225 Difference:  6 204  17
Here is one:   41  840  841   707281 Difference:  4 156  13
Here is one:   43  924  925   855625 Difference:  2  84   7 ----------
Here is one:   47 1104 1105  1221025 Difference:  4 180  15
Here is one:   53 1404 1405  1974025 Difference:  6 300  25
Here is one:   59 1740 1741  3031081 Difference:  6 336  28
Here is one:   61 1860 1861  3463321 Difference:  2 120  10
Here is one:   67 2244 2245  5040025 Difference:  6 384  32
Here is one:   71 2520 2521  6355441 Difference:  4 276  23
Here is one:   73 2664 2665  7102225 Difference:  2 144  12
Here is one:   79 3120 3121  9740641 Difference:  6 456  38
Here is one:   83 3444 3445 11868025 Difference:  4 324  27
Here is one:   89 3960 3961 15689521 Difference:  6 516  43
Here is one:   97 4704 4705 22137025 Difference:  8 744  62

Prove: When A is a pos. prime and B,C are pos. whole numbers, C-B = 1
    
B = Sqrt((CC)-(AA))
     
C = Sqrt((AA)+(BB))
         
C - B = Sqrt((AA)+(BB)) - Sqrt((CC)-(AA))

If C - B = 1 then B = (AA)-1  AND  (AA) = 2B + 1
                                      
    A    B       A     B              C      45 degrees
  +----+---+   +----+---+             |    /
A | AA |  /|  B|  /  \  |             |  /
  |    |/  |   +/  CC  \|A            |/
  +----+---+   |\     / +       -------------B
B |    | BB|  A|  \ /   |B           /|
  +----+---+   +---+----+          /  |
                 B    A          /    |

------- Copied from file rppt.prf -------

Sqrt = Sq. Root
AA, BB, CC, etc. = A squared, B squared, C squared, etc.

The first two diagrams show the Pythagorean Proof.

The third diagram is missing a curve.
The curve is a hyperbola that touches the 45 degree line exactly once.
It is a representation of a number A where it is a prime. The line 
represents the points where C = B + 1. The only point where A touches 
the line is the RPPT. The other places are where the triplet is not 
all whole numbers. The equation for the hyperbola is C = sqrt(BB+AA).

This is what my dad and I have been able to come up with.
Please help us prove C - B = 1.

Thanks,
Ben Love

Date: 09/13/97 at 19:33:51
From: Doctor Anthony
Subject: Re: Relatively Prime Pythagorean Triplets

The fact that C-B = 1 can be proved quite quickly if you look at the 
formula for generating Pythagorean triples.

If we require integer values of x, y, z for which   x^2 + y^2 = z^2  
then we can use the following generators with a, b taking any integer 
values whatsoever provided a > b.

    x = a^2 - b^2

    y = 2ab

    z = a^2 + b^2

since  x^2 + y^2 =  a^4 - 2a^2.b^2 + b^4 + 4a^2.b^2
                 =  a^4 + 2a^2.b^2 + b^4
                 =  (a^2 + b^2)^2
                 =  z^2 

For example if a=2, b=1 produces   x = 4 - 1     = 3
                                   y = 2 x 2 x 1 = 4
                                   z = 4 + 1     = 5


A prime number has no factors other than 1 and itself.

Now note that x = (a-b)(a+b) so if x is prime the factor (a-b) 
MUST be 1.

Also  z-y = a^2 + b^2 - 2ab = (a-b)^2 = 1^2  =  1.

So if x is prime then z and y MUST differ by 1.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/