A Note on Fermat's Last Theorem—Was It a Right Question?

          Pierre de Fermat asked whether the Babylonian result can hold for n 3. That is, he asked:

I would argue that Fermat's Last Theorem was an answer to possibly a wrong question, if the intent was to extend the Babylonian triples to higher dimensions.

A Generalized Babylonian Result

          When there are three circles with radius x, y and z, and if the sum of the areas of the first two circles with radiuses, x and y, equals that with radius z, Pythagoras's Theorem tells the exact geometric procedure to get z: use the hypotenuse derived from a right triangle with the two sides, x and y. However, in the three dimensional case, a geometric means or procedure to get the radius of the ball whose volume is equal to the sum of the other three balls has yet to be found.

3. What Is a Right Question?

          Consider a Diophantine equation which consists of finding k n^th powers equal to an n^th power. Such an equation might be called an "n × k equation." (This definition is somewhat different from the usual "m-n equation," and our problem is a special case, i.e., n-1 Diophantine equation). Fermat's Last Theorem simply states that no solution exists for n × 2, if n > 2. Our interest here is limited to the case of n × n Diophantine equation. Solutions are known to exist for 2 × 2 (the Babylonian triples), 3 × 3, 4 × 4, and 5 × 5. However, no known solutions exist for n × n for n 6.

          Cursory reading of a few books on the subject has not revealed any conjecture or a theorem on the "even" or "balanced" case, where the number of summands is equal to the power of z. Specifically, I am interested in the following existence question:

Are there positive integers x₁, x₂, ..., x_n, z, and n such that

When n = 1, the above is trivial, and hence an alternative representation of the existence question can be written as:

          The Babylonian triples suggest that an integer square can be expressed as the sum of two integer squares. Intuitively, in the spirit of Fermat, the theorem, if true, states that n^th power of a positive integer could be expressed as the sum of n (rather than 2) n^th powers of some other smaller integers.

          Davis Wilson has compiled a long list of the smallest n^th powers which are the sums of distinct smaller n^th powers. For instance, 15⁴ = 4⁴ + 6⁴ + 8⁴ + 9⁴ + 14⁴, and 12⁵ = 4⁵ + 5⁵ + 6⁵ + 7⁵ + 9⁵ + 11⁵. As the powers increase, however, the number of terms of n^th powers on the right hand side seems to increase much faster when the smallest n^th powers are used. If integers are not limited to only the smallest n^th powers or nondistinct integers are allowed, the number of terms of n^th powers might be much smaller. Nevertheless, this observation suggests that as the power increases, the chance for the existence theorem to hold might decrease.

          Is the existence theorem (or conjecture) in equation (1) true? If so, how do we prove it? If not, why not? Answering this existence or nonexistence question might be a real challenge.

          Babylonians had many examples of the triples when n = 2. John Conway also gives another example of the quadruple, (1, 6, 8, 9) when n = 3, and I am sure there are many others. Noam Elkies also gave a monstrous example with nonnegative integers when n = 4, namely the quintuple (0, 2682440, 15365639, 18796760, 20615673), and Roger Frye found a smaller quintuple (0, 95800, 217519, 414560, 422481).[See Lopez-Ortiz and "Several Years," Math. Comp. 51 (1988), 825-835)]. Although interesting, these examples involve zero as a member of the quintuple.

          Note that in the Babylonian example, you can add more integer variables. For instance, from 3² + 4² = 5², 5² + 12² = 13², and 13² + 84² = 85², ..., we also get

or

Thus, Pythagorean N-tuple states that an odd integer square can be broken into (N - 2) even integer squares and one odd square (Oliverio, 1996).

Moreover, by Fermat's Last Theorem, even though there exist no positive integers, x, y, and z for which

The quadruples (3,4,5,6) and (1,6,8,9) demonstrate that there exist positive integers, x, y, w, and z for which

I suppose, for given n, adding more integer variables increases "freedom," and makes it easier to find the modified theorem to hold. Thus, we should be able to find somewhat simpler quintuples with all positive integers than the monstrous ones Elkies and Frye found for n = 4.

Thus, another interesting conjecture is:

Note that when k = 1 and n = 3, the trivial Fermat equation, x₁³ = z³ permits an easy solution, x₁ = z. In this case, adding another variable, x₂, yields:

for which an integer solution does not exist by FLT. This explains the restriction, 1 < k < n, in Conjecture 2. But why is this restriction necessary? When k = 2 and n = 3, we already know that adding another variable suddenly permits a solution to Fermat's equation. Is there periodicity here? Probably not. We have already seen that three fourth powers of positive integers can sum to another. Moreover, it is known that four fifth powers of positive integers can sum to another. Thus, periodicity is not likely to be present here, and that is the reason for the above restriction, k > 1.

          Shall we follow the footsteps of other mathematicians and prove the existence for each n? or will there emerge another Andrew Wiles who will prove the general existence theorem or non-existence theorem beyond some k? In any case, proving the existence conjecture for each n is probably too time-consuming, and computers might help us prove existence up to a very large number. However, a general proof might be needed, and the mathematical machinery that has been developed so far might just be sufficient.

          We should all congratulate Andrew Wiles wholeheartedly, and many others who have contributed to various steps that led to the proof of the 350 year old puzzle. However, I often wonder how many pages God would have needed to prove FLT (with apologies to God for presumptuously comparing divinity and humanity). Mathematicians in other inhabited planets might have proved it in fewer lines. It is said "the argumentative defense of any proposition is inversely proportional to the truth contained." This adage seems to suggest a 200 page-long proof of Fermat's Last Theorem might be a Pyrrhic victory. The two conjectures in this note may offer a new challenge.

Acknowledgement
I am grateful to David Rusin and S Butcher for helpful comments on an earlier version.
Timeline of Fermat's Last Theorem