Pythagoras and Fermat

By Michael Griffin MLS

This is the first article in a series about using ancient Greek formulas with Fermat's Last Theorem.

Although it has been solved by modern methods, Fermat's Last Theorem will continue to draw anyone curious as to a method Fermat himself might have used to solve it. To that end,  a few idle investigations are reported here, with some conclusions that by no means make a rigorous proof.

In Fermat's problem:

An + Bn = Cn  has infinitely many solutions for n but the highest whole number solution is 2, as with the sides of a right triangle in the Pythagorean theorem:

A2 + B2 = C2 

And Pythagoras found that you can always generate solutions from

A2 =  B + C where C - B = 1

This is almost a midpoint formula since the length of A2 is divided by a midpoint which is added to length B to make length C. The clear property of this type of Pythagorean midpoint triple is that every single  odd  digit is the beginning of a set of Pythagorean triples. 3, 5, 7, … etc  all fit the same formula that generates triples. We get (3,4,5) (5,12,13) (7,24,25) … etc. The same cannot be said for the second digit B. We cannot say that every even digit B is part of a set of triples, but there is something we can say. There is a clear relation between subsequent B digits proceeding off of an increasing sequence of triples:  the next B digit that fits a triple pattern is 4n more than the previous: 

Bn = Bn--1 + 4n

Where n is the ordinal value of that triple. So after 7 is 9 for the 4th triple, n=4, 24+4(4)=24+16=40 so the next triple is (9,40,41) and 81+1600=1681=412.

 It doesn't take much work to see this pattern. 

Set up a row of the first 6 even members of Pythagorean midpoint triples:

4, 12, 24, 40, 60, 84

Then make a row of the increasing distance between members:

8, 12, 16, 20, 24

The common factor is 4 so divide it out:

2, 3, 4, 5, 6

Which gives the n-ordinal value of that member’s triple.

So each next B value is 4n more than the previous one. 

Again, Bn = Bn--1 + 4n  formula notation, however that is as far as that goes for now.

We shall set it aside and instead make a generalization of the Pythagorean midpoint formula for triple solutions (A,B,C) where n is any positive whole digit: 

An  = Bn--1 + Cn--1 

As with Fermat’s theorem, this is easily true if we make no restrictions on the values of A,B,C, or n.  For the sake of argument and the rules of the game, we make the unfounded assumption that A,B,C, and n are natural numbers only. 

This unfounded notion leads to a result that there are no solutions for this kind of Pythagorean midpoint triple for n  greater than 2:

If  A3 + B3 = C3  then 

A3  = B2 + C2  and substituting into the first equation:

 B2 + C2 + B3 = C3  rearranging:

B2 +  B3 = C3 - C2  and factoring:

B2 (1 +  B) = C2 (C - 1)

For the smallest pair of C - B = 1  (B,C) is (1,2)

12 (1 +  1) = 22 (2 - 1)

1(2) = 4(1) = 4   2 = 4 false

The larger (B,C) gets, the worse the inequality gets

Take the next pair (2,3) 

22 (1 +  2) = 32 (3 - 1)

4(3) = 9(2) = 18   12 = 18 false

The inequality has grown from a difference of 4 - 2 = 2  to 18 - 12 = 6 

And it will just get bigger. 

Also the larger n gets, the worse the inequality gets:

Instead of n = 3 let n = 4

If  A4 + B4 = C4  then 

A4  = B3 + C3  and substituting into the first equation:

 B3 + C3 + B4 = C4  rearranging:

B3 +  B4 = C4 - C3  and factoring:

B3 (1 +  B) = C3 (C - 1)

For the smallest pair of C - B = 1  (B,C) is (1,2)

13 (1 +  1) = 23 (2 - 1)

1(2) = 8(1) = 8   2 = 8   false

 The inequality has grown from a difference of 4 - 2 = 2  to 8 - 2 = 6 

And it will just get bigger. 

So as pairs (B,C) or the power n increases, so does the inequality.

So if  C - B = 1  there is no A,B,C for n > 2 such that 

An  = Bn--1 + Cn--1

∎

QED

This only applies to Pythagorean midpoint triples based on the unfounded: 

An  = Bn--1 + Cn--1 Other articles will address formulas made by Euclid and Plato.

Sources

Fermat's Last Theorem

From Wikipedia, the free encyclopedia

This page was last edited on 31 August 2020

https://en.wikipedia.org/wiki/Fermat's_Last_Theorem

Formulas for generating Pythagorean triples

From Wikipedia, the free encyclopedia

This page was last edited on 24 June 2020

https://en.wikipedia.org/wiki/Formulas_for_generating_Pythagorean_triples

Proof of Fermat's Last Theorem for specific exponents

From Wikipedia, the free encyclopedia

This page was last edited on 19 October 2020

https://en.wikipedia.org/wiki/Proof_of_Fermat%27s_Last_Theorem_for_specific_exponents#Two_cases

    

Pythagorean triple

From Wikipedia, the free encyclopedia

This page was last edited on 3 September 2020

https://en.wikipedia.org/wiki/Pythagorean_triple#Special_cases_and_related_equations