Mythic Algebra and Other Things: October 2024

Euclid and Fermat yet again

By Michael Griffin, MLS

This is the fifth article about using ancient Greek formulas with Fermat's Last Theorem. It begins with parts of the previous articles about Euclid's formula:

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

Euclid is credited with a formula to generate all possible Pythagorean triples: for any natural numbers k, m and n, m>n, a triplet of (A,B,C) results from k(m2-n2, 2mn, m2+n2).

For example, k(m,n) of 1(2,1) makes (3,4,5), the first such triple, for 9+16=25.

Does Euclid's formula cover all possible solutions of Fermat’s? It may not matter since it probably does not cover all possible triples anyway. Let's at least explore its root properties.

In general, if A,B,C,N,k,m,n are real numbers then m and n can always be chosen such that the Nth root C = N√(m2-n2)N + (2mn)N. While this may not cover all possible, there are infinitely many solutions in real numbers, just not integers or natural numbers. That is not necessary to comply with Euclid's formula. Restricting these to natural numbers results in Pythagorean triples. If N=2 the result for the Nth root C is Euclid’s definition for the C term, m2+n2:

√(m2-n2)2 + (2mn)2 =√ (m4-2m2n2+n4 )+ (4m2n2)

= √(m4+2m2n2+n4 ) = √(m2+n2 )2

If N is beyond 2 we do not get Euclid’s C term and Fermat's Last Theorem means at least one of A,B,C is not an integer. How can we show that?

Consider the nth roots where n is greater than 2. Higher roots would be smaller than the second root which is Euclid's m2+n2. Since they are the C term, higher roots are bigger than the A or B terms. On a number line the terms would arrange as A,B, nth root, Euclid's C. For the A and B terms to be natural numbers so must the m and n terms be.

So we are looking for the nth root C term not to be natural. It's possible domain on the number line goes from the B term to Euclid’s C. Because the nth root is less than Euclid's C we can use some properties of Pythagorean triples. Previous articles discussed the three Greek formulas so we will make a case for each:

If Euclid’s C fits Pythagoras’ formula then C-B=1 and so the domain of the nth root is not large enough after B to make a natural number.

If Euclid’s C fits Plato’s formula then C-B=2 and so the domain of the nth root is not large enough after B to make a natural number except for B+1. In that case C-B=1 and we would have a Pythagorean triple for the second power.

If Euclid’s C fits Euclid’s formula, the domain of the nth root is less than C- B which is m2+n2 - 2mn which factors as (m-n)2. Or if A is the larger term, the domain of the nth root is less than C-A which is m2+n2 - (m2-n2) which is 2n2. For example, for m,n=5,2 then ABC=21,20,29 and the domain of the nth root is less than 8. We have room for natural numbers now, although we can make more restrictions. Since the m and n terms must be natural numbers, so are the domains (m-n)2 and 2n2, so the domain boundaries are only square numbers 1,4,9,16,25, … Higher roots of square numbers are usually not natural numbers. An exception example is like when 2 is the 4th root of 16. Of course the actual C term is below A or B plus 1,4,9,16,25, … and each A or B term is a natural perfect nth root of AN or BN.

There can also be nth roots of combinations of natural numbers like (B+X2). For example, the third root of (2+52) = (2+25) = 27 is 3, a natural number. Euclid's formula is not concluding easily like the other Greek formulas. So this method is not as productive as the prior ways to show Fermat's on Pythagorean triples, using Euclid's formula.

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