Fermat's Last Theorem case n=4; Proof by Leonardo of Pisa

Fermat's Last Theorem has captured the popular imagination, remaining the prominent unsolved problem for over three centuries. In this webpage we consider only the n=4 case, a subproblem much easier even than the n=3 case, let alone the general case. Yet despite its relative easiness, the proof that a4 + b4 = c4 has no solutions is considered the best evidence of Fermat's talent among his theorems for which his own proofs have been preserved. It is also part of the work which is considered the best demonstration of Leonardo (`Fibonacci') of Pisa's talent. Most historians of mathematics would agree, I think, that Leonardo was the best Western arithmetician between Archimedes and Fermat. (Diophantus would qualify if he actually had proofs for some of the lemmas he claimed.) Yet that Leonardo proved this theorem is largely ignored or disputed. The purpose of this webpage is to pursue a case that Leonardo did indeed prove the n=4 case of FLT.

Please note that all mathematical expressions throughout this page are restricted to positive integers and follow the convention aa denotes a2, aaa denotes a3, etc.

First let us introduce the four theorems we need to discuss.

(I) Pythagorean Triplet Decomposition

      aa + bb = cc
with a odd and relatively prime to b, there exist e, f such that
      a = ee - ff
      b = 2ef
      c = ee + ff

Pythagoras himself is often credited with this discovery. Leonardo applied it with great mastery.

(II) Leonardo's Congruum Theorem

Given three square numbers with no common factor in consecutive arithmetic sequence
      s-t, s, s+t
t is called a "congruum" and these numbers must have the form
      t   = 4xy (x+y) (x-y)
      s-t = (xx-2xy-yy)2
      s   = (xx+yy)2
      s+t = (xx+2xy-yy)2

Since s, t have no common factor and t is even, s is odd and x, y have no common factor. Since s is odd, (xx+yy) is odd so one of x, y is even and the other is odd.

The proof of this is considered Leonardo's pièce de resistance and, as far as I know, its validity is unquestioned. The Congruum Theorem is shown as Proposition XIII in Liber Quadratorum (The Book of Squares), published by Leonardo in 1225. It is said that his discovery of this Congruum Theorem is the reason Leonardo wrote Liber Quadratorum at all.

(III) "Mystery Lemma": Congruums are not square

Given the definition of "congruum," this theorem can be rephrased as:
      Among s-t, s, s+t, t
it is impossible that all four terms are simultaneously square.

This Theorem is shown as Proposition XVI in Liber Quadratorum but it might be called the "Leonardo Mystery Lemma." There is no doubt that it is true and has an easy proof; there is no doubt that it leads directly to a proof for FLT4. The "mystery" is whether Leonardo's proof of this key lemma is valid.

(IV) FLT4: Fermat's Theorem n=4 case

There are no solutions (in positive integers) to
      aaaa + bbbb = cccc
In fact, both Fermat and Leonardo take the more general case cccc - bbbb = aa. In either case, it is sufficient to consider only cases where a and b are relatively prime.


We need not show a proof for the ancient Pythagorean Triplet Decomposition. Nor need we show Leonardo's proof of the Congruum Theorem. (The Pythagorean Triplet Decomposition provides the key help.) Our only purpose in this webpage is to examine the controversial claim that Leonardo proved FLT4 but, although the Congruum Theorem is the key step in proving FLT4, no one disputes that Leonardo's proof of the Congruum Theorem is valid. We will demonstrate that the Mystery Lemma proves FLT4 directly:

Applying the Pythagorean Triplet Decomposition to (aa)^2 + (bb)^2 = (cc)^2
we obtain aa = ee - ff and cc = ee + ff
Now s-t = aa = ee-ff, s = ee, s+t = cc = ee+ff, t = ff are the four squares which the Mystery Lemma tells us are impossible. Since we've stipulated that a, c have no common factor, then neither do e, f, nor s, t. (As mentioned above, Leonardo actually proves the stronger form of FLT4, but let's keep this page as simple as possible.)

So, if Leonardo proved his Proposition XVI (the "Mystery Lemma") we must concede that he proved FLT4.

The remainder of this webapge considers only one question:

Did Leonardo prove his Proposition XVI (that Congruums are not square)?

First let us note that this proposition is an easy consequence of Leonardo's own Proposition XIII (The Congruum Theorem). Suppose t = 4 xy (x+y) (x-y) is the smallest square congruum. Since it is minimal, x and y have no common factors and neither do x+y and x-y. (2 would be a possible factor, but we know one of x,y is odd and the other even, so both of x+y, x-y are odd.) For this reason, any factor of t occurs in only one of the terms x, y, x+y, x-y. If t is square, all its factors are squared, so the four terms are each themselves squares. But if x, y, x+y, x-y are each square, then y would be, by definition, a square congruum, contradicting the assumption that t is the minimal square congruum. (This is straightforward mathematical induction, or what Fermat calls "infinite descent.") Leonardo's own phrasing may be similar to

"When x > y ... then x (x-y) ≠ y (x+y) and from this it may be shown that no square number can be a congruum. For [then] ... the four factors x, y, (x+y), (x-y) must severally be squares which is impossible."

Leonardo does assert that it is "impossible" for x, y, x+y, x-y to all be square. Unfortunately, I do not have a copy of Leonardo's original text, and am not sure how his Latin is phrased. But would we not agree that his proof would be complete if it included simply the phrase "by infinite descent"? Of course that phrase was not yet in existence, nor were proofs expected to be fully rigorous by today's standard. Perhaps Leonardo was remiss in his detailed prose (cf. Laplace's "il est aisé a voir"); but for number theorems like these, the algebraic expressions tell most of the story, and I think it is silly to dispute that Leonardo proved FLT4 just because he assumed, perhaps mistakenly, that someone who'd gotten that far in his book would need to be "led by the hand" through the argument details.

Now perhaps Leonardo doesn't explicitly discuss reasoning by "infinite descent," but surely he applied the idea here. Otherwise ... what? The "most talented mathematician of the Middle Ages" managed to commit the absurdity of assuming Proposition XVI while proving Proposition XVI ? I don't believe it.

Liber Quadratorum

I hope some one can e-mail me a "screendump" from an actual copy of a Liber Quadratorum translation. All I have is Sherlock Holmes in Babylon and other tales of math history, by Marlow Anderson & Robin J. Wilson, of which the key excerpt is this from page 146:


Proposition XII:
      ... 4xy(x+y)(x-y) is divisible by 24.
A number of this form is a "congruum" and furnishes the solution to the problem proposed by Johannes of Palermo.
Proposition XIII.
      (xx + yy)^2 - 4xy(xx - yy) = (yy + 2xy - xx)^2
      (xx + yy)^2 + 4xy(xx - yy) = (xx + 2xy - yy)^2
We show that if
      aa-b, aa, aa+b
are squares in arithmetic progression, then b has the congruum form.
Proposition XVI.
      When x>y, x(x-y) != y(x+y) and "from this," Leonardo writes,
"it may be shown that no square number can be a congruum. For if
      xy(x+y)(x-y) could be a square than either
      x(x-y) = y(x+y)
(impossible by Proposition XVI), or else the four factors must severally be square, which is also impossible. Leonardo to be sure overlooked the necessity of proving this last assertion.

It's the "Leonardo to be sure overlooked the necessity of proving this last assertion" that confuses me. A modern proof would be fashioned to simply state at that point "and by the inductive hypothesis." Did Leonardo understand this?

So. Did Leonardo provide the first proof (even if slightly defective) of FLT n=4, and if so what is the best way to phrase that fact?


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