H o w Probable Is Fermat's Last Theorem?

Manfred Schroeder

The distance between z n, the nth power of z E [q, and (z + 1) n is approximately nz ~-1. Thus, the probability that a randomly chosen number s E N near s = z n equals the nth power of an integer is approximately 1 /nz ~- 1 =

n -1 s -I+U~. I shall call the latter expression the "densi ty" of the nth powers at s.

Similarly, the density of s = x n + yn (x, y E I~, y > x) is given by the convolution integral

1 fsn dt dn(s) = ~-$ J0 [(s/2 + t )(s/2 - t ) ] 1 - 1 / n

1 ( ! ) 1 - 2 / n ~01 d u

= n-2 (1 -- U2) 1-1/n

= 1 s - l + 2 / n B ( 1 , 1 ) 2n 2

(1)

Here B(a, b) is the complete beta function (Euler's inte- gral of the first kind):

B(a, b) - r (a)r (b) F(a + b)

For a = b = 1/n, one obtains

( 1 1 ) p2 (1 /n ) B , = F(2/n) "

For the exponent n = 2, we have B (�89 �89 = 7r. Thus, with Eq. (1), the density becomes d2(s) = 7r/8 ~ 11/28, i.e., the density of the sum of two squares is constant: Approxi- mately 11 out of every 28 randomly chosen integers are, on average, equal to a sum of two squares.

This result also follows from the observation that the number of lattice points of the integer lattice ~2 within a circle of radius v ~ is approximately 7rs. Thus, given

a random integer z, the approximate probability that its square z 2 is equal to the sum of two squares, z 2 = x 2 q- y2, is 7r/8, provided the pure squares and the sums of two squares are distributed independently. Numerical evidence, as well as the density of Pythagorean triplets, suggests that this may indeed be the case.

For n = 3, the density according to Eq. (1), with B (�89 �89 = 5.2999. . . , is about 0.294s -1/3. Thus, the prob- ability that the cube z 3 of a randomly chosen integer z is equal to the sum s of two cubes, s = z3, is approximately equal to 0.294/z. (Numerical counting gives 0.295/z.)

THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4 (~)1994 Springer-Verlag New York 19

Summing over z from 1 to Z, tells us that Fermat 's Last Theorem (FLT) for exponent n = 3 would be violated in about 0.294 log g cases for z < Z. The first such viola- tion would be expected to occur below g ~ e 1/0"294 ~ 30 ,

corresponding to s ~ 27, 000. We know, of course, from the proofs of FLT for n = 3 by Euler, Legendre, Sophie Germain, and others, that there are, in fact, no such coun- terexamples.

For n = 4, we have B(�88 �88 = 2rc/U = 7.4163 . . . . where U = 0.847213... is the so-called "ubiquitous constant," i.e., the common (arithmetic-geometric) mean of 1 and 1/v~. Thus, the density according to Eq. (1) is about 0.23176s-U2 = 0.23176/z 2. (Numerical result: 0.23175/z2.) By summing over z from 1 to infinity, we obtain the probabilistic estimate of the number of viola- tions of FLT for n = 4. With the infinite sum over 1/z 2 equal to rr2/6, we see that the probability of such a vio- lation is approximately 0.38 or less than 0.50.

For n = 5, the density according to Eq. (1) is about 0.1900/z 3. (Numerical counting gives 0.1896/z 3. ) By sum- ming over all z, one obtains, with ~(3) ~ 1.202, a proba- bility of 0.23 for a violation of FLT for n = 5.

For large values of n, we use the approximation

B ( 1 , 1 ) = 2 n + O ( 1 )

and obtain with Eq. (1)

dn(8) = n - 1 8 - l+2/n = n - l z 2-n. (2)

The density summed over all values of z from I to infin- ity equals approximately 1/n. Thus, whereas violations of FLT become increasingly unlikely with increasing n, the grand total of expected violations for all values of n is infinite. In other words, assuming x" + pn and z n are independently distributed, the truth of FLT is highly un- likely. Only a completed proof could lay these doubts to rest.

For what exponents n would FLT be expected to fail, if it does fail? Suppose FLT was proved for all exponents below some large integer N. Given the fact that FLT has also been proved for all regular primes (Kummer, 1850) [1], the number of cases for which FLT would be violated between N and M (disregarding all other proven cases) is given by

However, so far we have ignored an important re- striction. For prime n, x n = x mod n, and x ~ + yn = z,~ would, therefore, imply x + p ~ z mod n. Assuming z < n, the congruence can be writ ten as the equation x + p = z, implying x n + y'~ < z ~. Thus, for x n + y'~ = z '~ it is necessary that z > n. Summing Eq. (2) accordingly yields approximately n 1-~, which, summed over n > N = 106, gives a negligible probability (< 10 -6~176176 for FLT to fail.

Probabilistic analysis has shown its value in estimat- ing the densi ty of Mersenne primes. Although there is no proof of the existence of infinitely many Mersenne primes, such methods have given good predictions of undiscovered Mersenne primes [3]. How will it be for FLT?

References

1. P. Bachmann, Das Fermatproblem in seiner bisherigen En- twickelung, deGruyter, Berlin & Leipzig, 1919. Reprinted by Springer-Verlag, Berlin, 1976.

2. C. L. Siegel, "Zu zwei Bemerkungen Kummers," Nachr. Akad. d. Wissen. Gf~'ttingen, Math. Phys. Kl., II (1964), 51-62. Reprinted in Gesammelte Abhandlungen (edited by K. Chan- drasekharan & H. Maat~), Vol. III, 436-442. Springer-Verlag, Berlin, 1966.

3. M. R. Schroeder, Number Theory in Science and Communica- tion, 2nd ed., Springer-Verlag, New York, 1990.

Drittes Physikalisches Institut Universita't G6ttingen D-37073 GiJttingen, Germany

M 1 F = ~ ~ ~ (1 - e-1/2)(log log M - log log N), (3)

N

where the factor I - e -1/2 ~ 0.39 is the asymptotic frac- tion of irregular primes/5 (Siegel, 1964) [2]. The "fail- ure estimate" F exceeds unity for M > M0, where log M0 = 12.7 log N. For N = 106, say, the "crossover exponent" M0 equals about 1076. The next failure would be expected around 10152 (give or take a few dozen orders of magnitude).

20 THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 4,1994