On Case 1 of Fermat's Last Theorem

Fermat's Last Theorem (FLT) asserts that the equation x^p + y^p = z^p has no solution in integers x,y,z and prime p. Traditionally this proposition was split into Cases 1 and 2, depending on whether xyz is not or is divisible by p.

Case 1 is by far the simpler, and with the exponent p = 3 turns out to be trivial. Here the assertion is that the equation x³ + y³ = z³ has no solution in integers x,y,z assuming none of them is divisible by 3. This can trivially be ruled out by just examining the equation modulo 9. Similarly, it's trivial to show that x⁵ + y⁵ = z⁵ has no solutions modulo 25 if none of x,y,z is divisible by 5. The same applies to many other primes. This is related to the elementary fact (noted by Gauss) that if the congruence

has no "non-trivial" solutions (meaning that none of x, y, (x+y) are zero modulo p) then Case 1 of FLT is true for the prime exponent p. Unfortunately, if p is of the form 3k+1 there are always non-trivial solutions, so the observation isn't of much value for such primes. However, for the first several primes of the form 3k − 1 it's easy to verify that there are no solutions.

Incidentally, since division by non-zero residue is unique modulo p (and recalling that an algebraic factor of p can be cancelled), we can divide through any solution of the above congruence by x^p, and so setting z = y/x we have the congruence (1+z)^p − z^p − 1 = 0 (mod p²). Thus we need only examine this simpler form to check for possible solutions. For a more extensive discussion of this congruence, see the note On the Density of Some Exceptional Primes.

G.B Mathews (1885) took up this idea and evidently concluded that it settles Case 1 of FLT for ALL prime exponents of the form 3n-1. Dickson seems to agree, saying that the method "leaves in doubt the case p=3n+1". Later, in his review of A. Flechsenhaar's 1909 paper on the Fermat equation modulo p², Dickson flatly claims that the congruence has no solutions for primes of the form p = 6m − 1 in integers coprime to p. However, the claim is false, as shown below.

The approach can be summarized by noting that if x^p + y^p = z^p then we have x + y = z (mod p), which implies that z = kp + (x+y) for some integer k. Thus, we have

Expanding the binomial on the right hand side we see that the first term is (kp)^p and every subsequent term (except the last) contains some positive power of (kp) and has a coefficient divisible by p. Thus, all of those terms drop out (mod p²) and we are left with congruence (1). Since Case 1 of FLT stipulates that none of x,y,z are divisible by p, and recalling that z = x + y (mod p), it's clear that Case I is proven if we can show that congruence (1) has no solutions with each of x, y, and (x+y) not equal to 0 (mod p). To see how this can easily be verified for some small primes, notice that (as Mathews observed in 1886)

where Q(x,y) is a homogeneous integer function of degree p−3. Therefore congruence (1) is equivalent to

Consequently, if we can show that the congruence Q(x,y) = 0 has no integer solutions modulo p with x, y, and x+y coprime to p, it follows that Fermat's equation is insoluble under those conditions. To illustrate, consider the p = 3, for which we have the identity

Here Q(x,y) is a constant 1, so Case 1 of FLT is obviously true for p = 3. For a slightly less trivial example, take the case p = 5, which gives

In this case the congruence Q(x,y) = x² + xy + y² ≡ 0 (mod 5) gives

This congruence can be written as

which is plainly impossible, because −3 is not a square modulo 5. Thus we see immediately that Case 1 of FLT with exponent p = 5 is true.

To illustrate a case where this approach fails, consider the case p = 7, which gives

so we must have x² + xy + y² ≡ 0 (mod 7). However, in this case we see that −3 is a square (mod 7), so the congruence has solutions (such as x=2, y=1), so we can't rule out the exponent p = 7 with this argument.

Proceeding in this way, we find that Case 1 of Fermat's Last Theorem is implied by this simple congruence argument for the primes 3, 5, 11, 17, 23, 29, 41, 47, and 53, which includes all the primes of the form 3n−1 less than 59. This often leads people to jump to the conclusion that it settles Case 1 for all prime exponents of the form 3n−1. However, for p = 59 we find that 1⁵⁹ + 3⁵⁹ = (1+3)⁵⁹ (mod 59²). Looking a little further, we find that the following primes less than 1000 of the form 3k−1 possess non-trivial solutions

There are 17 others less than 2000. Is there a simple way of characterizing these exceptional primes? What is their density? (See On the Density of Some Exceptional Primes.)

A slightly more sophisticated, but still completely elementary, approach to Case 1 of Fermat's Last Theorem for cubes is to notice that if x³ + y³ = z³ then

so z−x is a cube if 3 does not divide y. Similarly we can prove z−y is a cube since 3 doesn't divide x, and x+y is a cube because 3 doesn't divide z. Thus if none of x,y,z is divisible by 3 we have shown that (x+y), (z−x) and (z−y) must all be cubes, which is impossible because we have

and a cube cannot equal 3 times a cube. (For comparison, see the general proof of Fermat's Last Theorem for Cubes, which includes the much more challenging Case 2, where xyz is allowed to be a multiple of 3.)

We can obviously perform the same kind of factorization for other prime powers. If we consider the symmetrical form of the Fermat equation x^p + y^p + z^p = 0 we can make use of the algebraic identities such as

and so on. (For more economical expansions, see the note Sums of Powers in Terms of Symmetric Functions.) Of course, the observation that the quantities (x+y), (x+z), and (y+z) must be pth powers (assuming Case 1) applies to all prime exponents p. This was noted by P. Barlow in 1810, who pointed out that if n is prime and xⁿ + yⁿ + zⁿ = 0 is solvable in coprime integers, then there must be integers r,s,t such that one of the following four sets of conditions is satisfied:

Barlow certainly wasn't the first person to notice this, nor was he the last, as amateurs frequently re-discover it. Unfortunately, the factorization on the right hand side of the equivalent of equation (2) for primes greater than 3 includes more than just those three factors, so the simple divisibility argument used in Case 1 for p=3 doesn't immediately apply. This is, however, the essential idea behind what is perhaps the most important elementary result concerning Case 1 of Fermat's Last Theorem, namely, Sophie Germain's Theorem.

Sophie Germain showed that if p is an odd prime and m is a prime such that the congruence x^p + y^p + z^p = 0 (mod m) can only be satisfied if either x, y, or z is congruent to 0 (mod m), and if there is no integer x such that x^p = p (mod m), then x^p + y^p + z^p cannot equal zero in integers x,y,z co-prime to p (nor in integers coprime to m). She found at least one such "auxiliary prime" m for each p less than 100.

The proof of this can be illustrated by the case p = 5 with the auxiliary prime m = 11. First, notice that we can write

and as we discussed above the two factors on the right are mutually coprime (on the assumption that x,y,z are pairwise coprime and none of them is divisible by 5). Therefore, both factors must individually be 5th powers (by unique factorization), and the same argument applies to the factorizations of −y⁵ and −z⁵, so we have integers A,B,C and a,b,c such that

and of course Aa = −x, Bb = −y, and Cc = −z.

Now, let's look at the 5th powers of integers modulo 11. (The reason for choosing the modulus 11 will be clear shortly). For x = 0, 1, 2, ..., 10 we find that the fifth powers (modulo 11) are 0, 1, -1, 1, 1, 1, -1, -1, -1, 1, -1 respectively. So, fortuitously, a fifth power of an integer must be congruent to 0, +1, or −1 modulo 11. It immediately follows that we can only have a solution of x⁵ + y⁵ + z⁵ = 0 if one of the numbers x,y,z is a multiple of 11, because otherwise all three of these 5th powers is either +1 or −1 (mod 11), and there is no way of adding three such values to get zero (mod 11). Therefore, exactly one of the numbers x,y,z must be a multiple of 11. Since the three variables are symmetrical, we can assume x = 0 (mod 11), but notice that

so again we have three 5th powers summing to 0 (mod 11), and by the same reasoning as above, this implies that one of A,B,C must be a multiple of 11. However, since x is already divisible by 11, it's clear than neither B nor C can be divisible by 11 because, because that would imply that Z or y, respectively, was divisible by 11, contradicting the fact that x,y,z are pairwise coprime.

This seems to force us to conclude that A must be a multiple of 11, but this too is in conflict with our assumptions, because if A = y + z = 0 (mod 11) then we have z = −y (mod 11), and plugging this into the above equation for a⁵, evaluated modulo 11, gives the congruence a⁵ = 5y⁴ (mod 11), and substituting x = 0 (mod 11) into the equation for c⁵ gives c⁵ = y⁴ (mod 11). Combining these last two congruences gives a⁵ = 5c⁵ (mod 11), which is obviously impossible, because we know the 5th powers (mod 11) are all 0, +1, or −1, and we can rule out a = c = 0 (mod 11) because z = −Cc and we know z is not divisible by 11. This completes the proof of Sophie Germain's Theorem for the special case p = 5, using the auxiliary prime m = 11. The proofs for many other prime exponents are similar.

This theorem raises some interesting questions. One such question is whether, given a prime p, we can characterize the primes q such that the congruence

implies that either x,y, or z must be divisible by q. For example, if p = 3 we have q = 2, 7, or 13. (It also appears that setting q = p² satisfies the requirements for some, but not all, primes p.) Here's a brief table of some q values for small primes

This table doesn't necessarily include all the q-primes for any given p; it extends only as far as I searched. However, it's clear that the list of q-values for any given p is finite.

An interesting application of this approach can be illustrated by the case of exponent 14. (Historically this case was the next to be resolved - by Dirichlet - following Legendre's proof for exponent 5; it was another seven years before Lame' finally settled the case of exponent 7.) We can prove very simply that if there are integers x,y,z such that x¹⁴ + y¹⁴ = z¹⁴ then xyz must be divisible by 71. This follows because we have 71 = (5)(14) + 1 and therefore

If we assume xyz is coprime to 71 then this equation implies the congruence

It's easy to determine that the 5th roots of 1 (mod 71) are the residues 1, 5, 25, 54, and 57, and none of these can be expressed as a sum of two values in this set (even allowing duplicates). Thus the equation is impossible under the assumption that xyz is coprime to 71.

In general, if kn + 1 = p for some prime p, and if the kth roots of 1 (mod p) are "sum free" in the sense described above, then the Fermat equation with exponent n is impossible in integers x,y,z with xyz coprime to p. (Note that if n = 1 or 2 the kth roots are never sum-free.)

So, to further restrict the possible solutions for exponent 14 (for example), we could note that (3)(14) + 1 = 43, and the cube roots of 1 (mod 43) are 1, 6, and 36. Since these roots are sum-free, we've shown that xyz must be divisible by 43 (as well as 71). Similarly from (2)(14) + 1 = 29 and the square roots of 1 (mod 29) being 1 and 28, we can say that xyz must be divisible by 29.

Let's call the prime p a "Germain Prime" relative to the positive integer n if there exists an integer k such that p = kn + 1 and the equation a + b = c (mod p) has no solutions with a,b,c each a kth root of 1 (mod p). Then we know that if xⁿ + yⁿ = zⁿ then xyz is divisible by every Germain Prime relative to n. For example, relative to n = 2 the only Germain Primes are 3 and 5, so it follows that for every Pythagorean triple x,y,z we have xyz divisible by 3 and 5.

For another example, the Germain Primes relative to n = 7 are 29, 71, 113, and 491, which implies that if x⁷ + y⁷ = z⁷ then xyz is divisible by (29)(71)(113)(491). Of course, we know the Fermat equation with n = 7 has no integer solutions, so this consequence is not terribly important. Nevertheless, the Germain Primes seem interesting in their own right. Following is a table of all the Germain Primes with k < 400 for integers n (not necessarily primes as in the above table) from 1 to 13.

It's known that there are only finitely many Germain Primes relative to any given n, so this approach cannot prove FLT. However, it does give an elementary way of proving that any solutions of the Fermat equation for large n must be extremely rare (if they existed). For example, we can show that if x²¹ + y²¹ = z²¹ then xyz must be divisible by the product of all the Germain Primes relative to n = 21:

This obviously implies that any solution would have to consist of extremely large numbers.

This approach to FLT is actually quite old. G. Libri wrote about Germain Primes (although he didn't name them) in 1832. They were subsequently discussed by (among others) Pepin (1880), Pellet (1886), Mathews (1895), Dickson (1909), Cornacchia (1909), and Mantel (1916). Several of these writers stated (although none ever published a proof) that the number of Germain Primes relative to any given n is finite. As Libri said back in 1832, "hence it is futile to try to prove uⁿ + vⁿ = zⁿ impossible by trying to show that one of the unknowns must be divisible by an infinitude of primes".

One of the more charming incarnations of this approach was that of E. Dubouis. Writing in 1910 he defined a "sophien of n" to be a prime p of the form kn+1 for which xⁿ + yⁿ = 1 (mod p) is impossible.

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