Fermat great theorem

Fermat's famous theorem, Fermat's big theorem, Fermat's last theorem

The assertion that for any natural number $n>2$ the equation $x^n+y^n=z^n$ (the Fermat equation) has no solution in non-zero integers $x,y,z$. It was stated by P. Fermat in about 1630 in the margins of his copy of the book Aritmetika [1] by Diophantus as follows: "It is impossible to partition a cube into two cubes, or a biquadrate into two biquadrates, and in general any power greater than the second into two powers with the same exponent" . And he then added: "I have discovered a truly marvellous proof of this, which this margin is too narrow to contain" . A proof of the theorem for $n=4$ was found in Fermat's papers. No general proof has so far been obtained (1984), despite the efforts of many mathematicians (both professional and amateur). An unhealthy interest in proving this theorem was stimulated at one time by a large international prize, which was abolished at the end of the First World War.

It has been conjectured that there is no proof of Fermat's last theorem at all.

For $n=3$ the theorem was proved by L. Euler (1770), for $n=5$ by P.G.L. Dirichlet and A. Legendre (1825), and for $n=7$ by G. Lamé (1839) (see [2]). It is sufficient to prove the theorem for $n=4$ and for every prime exponent $n=p>2$, that is, it is enough to prove that the equation

$$ x^p+y^p=z^p $$

(1)

has no solutions in non-zero relatively-prime integers $x,y,z$. One can also assume that $x$ and $y$ are relatively prime to $p$. For a proof of Fermat's theorem one considers two cases: case 1 when $(xyz,p)=1$, and case 2 when $p|z$. The proof of the second case is more difficult and is usually carried out by the method of infinite descent. An important contribution to proving Fermat's last theorem was made by E. Kummer, who invented a fundamentally new method based on his development of the arithmetic theory of a cyclotomic field. It makes use of the fact that in the field $\Q(\zeta)$, $\zeta=e^{2\pi i/p}$, the left-hand side of equation (1) splits into linear factors $x^p+y^p=\prod_{i=0}^{p-1} (x+y\zeta^i)$, which are $p$-th powers of ideal numbers (cf. Ideal number) in $\Q(\zeta)$ in case 1 and differ from $p$-th powers by a factor $(1-\zeta)$, $i>0$, in case 2. If $p$ divides the numerators of the Bernoulli numbers $B_{2n}$ ($n=1,\ldots, (p-3)/2$), then by the regularity criterion $p$ does not divide the class number $h$ of $\Q(\zeta)$ and these ideal numbers are principal. Kummer proved Fermat's theorem in this case. It is not known whether the number of regular numbers $p$ is infinite or finite (by Jensen's theorem the number of irregular prime numbers is infinite [4]). Kummer [5] proved the theorem for some irregular prime numbers and also established its validity for all $p<100$. In case 1 he showed that (1) implies the congruences

$$ B_n \left[ \frac{d^{p-n}}{dv^{p-n}} \ln(x+e^vy)\right]_{v=0} \equiv 0 \pmod{p}, \qquad n=2,4,\ldots,p-3, $$

which are valid for any permutation of $x,y,-z$. Hence he obtained that if equation (1) has a solution in case 1, then for $n=3,5$,

$$ B_{p-n} \equiv 0 \pmod{p}. $$

(2)

In case 2 Kummer proved Fermat's theorem under the following conditions: 1) $p|h_1$, $p^2 \nmid h_1$, where $h_1$ is the first factor of the class number of $\Q(\zeta)$ (this is equivalent to the requirement that only one of the numerators of the numbers $B_{2n}$, where $2n = 2,4,\ldots,p-3$, is divisible by $p$); 2) $B_{2np} \not\equiv 0 \pmod{p^3}$; and 3) there is an ideal modulo which the unit

$$ E_n = \sum_{i=1}^{(p-3)/2} e_i^{g^{-2ni}} $$

is not congruent to the $p$-th power of an integer in $\Q(\zeta)$, where $g$ is a primitive root modulo $p$ and

$$ e_i = \frac{\zeta^{g^{i+1}/2} - \zeta^{-g^{i+1}/2}}{\zeta^{g^i/2} - \zeta^{-g^i/2}} $$

Kummer's method has been widely developed in several articles on Fermat's last theorem (see [6], [7]). It has been established that (2) holds if (1) does in case 1 for n=7, 9, 11, 13, 15, 17, 19. Under the same conditions M. Krasner [8] showed that there is a number $p_0$ such that for $p > p_0$ (2) is true for all numbers $n=2k+1$, where $1 \le k \le [(\ln p)^{1/3}]$.

H. Brückner [9] showed that the amount of numbers $B_n$, $n=2,4,\ldots,p-3$, with numerators divisible by $p$ is greater than $p^{1/2}-2$. Suppose that $p^k|h_1$, $p^{k+1}\nmid h_1$. P. Remorov [10] showed that there are constants $N_k$ and $M_k$, $N_k < M_k$, such that for all $p < N_k$, $p > M_k$, case 1 of the Fermat theorem is true. M. Eichler [11] established that case 1 is true for $H < [p^{1/2}] - 1$, where $H$ is the index of irregularity of $\Q(\zeta)$, $p^H|h_1$. H. Vandiver [12] proved case 1 for $p\nmid h_2$, where $h_2$ is the second factor of the class number of $\Q(\zeta)$. He obtained interesting results on case 2 in [13] and [6]. For example, he showed that the Fermat theorem is true under the following conditions: 1) $(h_2,p)=1$; and 2) $B_{np} \not\equiv 0 \pmod{p^3}$, $n=2,4,\ldots,p-3$. The following theorem is most important: Let $p$ be an irregular prime number and let $2a_1, \ldots, 2a_s$ be the indices of the Bernoulli numbers among $B_2, B_4, \ldots, B_{p-3}$ with numerators divisible by $p$; if none of the units $e_a$ ($a = a_1, \ldots, a_s$) is congruent to the $p$-th power of an integer in $\Q(\zeta)$ modulo $\mathfrak{P}$, where $\mathfrak{P}$ is the prime ideal dividing a prime number $q < p^2-p$ with $q \equiv 1 \pmod{p}$, then Fermat's theorem is true. From this Vandiver [14] obtained an effectively-verifiable criterion for irregular prime numbers by means of which the Fermat theorem has been proved on a computer for all $p < 125000$ (see [15]).

There are various results on case 1 of Fermat's last theorem. As early as 1823 Legendre published a result of S. Germain: If there is a prime number $q$ such that the congruence $\xi^p + \eta^p \zeta^p \equiv 0 \pmod{q}$ has no integer solutions $\xi,\eta,\zeta$ not divisible by $q$, and $p$ is not a $p$-th power residue modulo $q$, then case 1 of the Fermat theorem holds (see [2]). Hence he showed that if at least one of the numbers $2kp+1$, $k\not\equiv 0\pmod{3}$ , $k\le8$, is prime, then case 1 holds. This proposition has been extended to all $k \le 55$. A. Wieferich [16] discovered the following criterion: If $p \nmid q(2)$, where $q(m) = (m^{p-1}-1)/p$ is the Fermat quotient, then case 1 is true. D. Mirimanoff [17] proved this for $p\nmid q(3)$. Subsequently, case 1 was established by a number of other authors for all $p$ for which $p\nmid q(m)$, where $m$ is any prime number $\le 43$. From this the first case of Fermat's theorem follows for $p = a\pm b$, where $a,b\in\N$ contain only prime numbers $\le 43$ in their prime factorizations. Calculations on a computer showed [18] that among the numbers $p<6.10^9$ only two: $p=1093$ and $p=3511$ satisfy the condition $p| q(2)$, but for these $p\nmid q(3)$. This proves case 1 for all $p < 6.10^9$. P. Furtwängler [19] gave fairly simple new proofs of the results of Wieferich and Mirimanoff based on Eisenstein's reciprocity law. He also proved that if $x,y,z$ is a solution of (1) and $(x,y)=1$, then $p|q(r)$, where $r|x$ but $p\nmid x$, or $r|y$ but $p \nmid y$, or $r|(x\pm y)$ but $p\nmid(x^2-y^2)$.

A great variety of other criteria are known for case 1 of the Fermat theorem. They are connected with the solvability of certain congruences or with the existence of prime numbers of a certain form. The equation $x^{2p}+y^{2p}=z^{2p}$ is not valid if $2p$ divides neither $x$ nor $y$ (see [20]). It is impossible in practice to produce a counterexample to Fermat's last theorem. K. Inkeri [21] showed that if the integers $x,y,z$, $0<x<y<z$, satisfy (1), then $x>p^{3p-4}/2$, and in case 1: $x> ((2p^3+p)/\ln 3p)^p$.

Fermat's last theorem can be stated as follows: For every natural number $n>2$ there are no rational points on the Fermat curve $x^n+y^n=1$ except the trivial ones, $(0, \pm1)$ and $(\pm1, 0)$. Rational points on the Fermat curve have been studied by methods of algebraic geometry. By these methods it has been proved (1983) that the number of rational points on the Fermat curve is finite in every case. This follows from the Mordell conjecture, which was proved by G. Faltings [23]. D.R. Heath-Brown has shown, using the Mordell conjecture, that Fermat's last theorem holds for almost-all primes $p$, cf. [24]. Also, by methods of analytic number theory, L.M. Adleman, Foury and Heath-Brown have shown that case 1 holds for infinitely many primes $p$, cf. [25].

One can look at Fermat's equation in algebraic integers, entire functions, matrices, etc. There is a generalization of Fermat's theorem for equations of the form $x^n + y^n = Dz^n$.

References

[1]	Diophantus of Alexandria, "Aritmetika and the book on polygonal numbers" , Moscow (1974) (In Russian; translated from Greek)
[2]	H.M. Edwards, "Fermat's last theorem. A genetic introduction to algebraic number theory" , Springer (1977)
[3a]	E. Kummer, "Bestimmung der Anzahl nicht äquivalenter Classen für die aus $\lambda$ten Wurzeln der Einheit gebildeten complexen Zahlen" J. Reine Angew. Math. , 40 (1850) pp. 93–116
[3b]	E. Kummer, "Zwei besondere Untersuchungen über die Classen-Anzahl und über die Einheiten der aus $\lambda$ten Wurzeln der Einheit gebildeten complexen Zahlen" J. Reine Angew. Math. , 40 (1850) pp. 117–129
[3c]	E. Kummer, "Allgemeiner Beweis des Fermatschen Satzes, dass die Gleichung $x^\lambda+y^\lambda=z^\lambda$ durch ganze Zahlen unlösbar ist, für alle diejenigen Potenz-Exponenten $\lambda$, welche ungerade Primzahlen sind und in den Zählern der ersten $\frac12(\lambda-3)$ Bernoullischen Zahlen als Factoren nicht vorkommen" J. Reine Angew. Math. , 40 (1950) pp. 130–138
[4]	Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) MR0195803 Zbl 0145.04902
[5]	E. Kummer, "Einige Sätze über die aus den Wurzeln der Gleichung $a^\lambda=1$ gebildeten complexen Zahlen, für den Fall, dass die Klassenanzahl durch $\lambda$ teilbar ist, nebst Anwendung derselben auf einen weiteren Beweis des letzten Fermat'schen Lehrsatzes" Abh. Akad. Wiss. Berlin, Math. Kl. (1857) pp. 41–74
[6]	H. Vandiver, "Fermat's last theorem" Amer. Math. Monthly , 53 (1946) pp. 555–578 Zbl 52.0161.13
[7]	P. Ribenboim, "Thirteen lectures on Fermat's last theorem" , Springer (1979)
[8]	M. Krasner, "Sur le premier cas du théorème de Fermat" C.R. Acad. Sci. Paris , 199 (1934) pp. 256–258 Zbl 0010.00702 Zbl 60.0129.01
[9]	H. Brückner, "Zum Beweis des ersten Falles der Fermatschen Vermutung für pseudoreguläre Primzahlen $l$" J. Reine Angew. Math. , 253 (1972) pp. 15–18
[10]	P.N. Remorov, "On Kummer's theorem" Uchen. Zap. Leningrad. Gosudarstv. Univ. Ser. Mat. Nauk , 144 : 23 (1952) pp. 26–34 (In Russian) MR81310
[11]	M. Eichler, "Eine Bemerkung zur Fermatschen Vermutung" Acta Arith. , 11 (1965) pp. 129–131 MR0182607 Zbl 0135.09401
[12]	H. Vandiver, "Fermat's last theorem and the second factor in the cyclotomic class number" Bull. Amer. Math. Soc. , 40 (1934) pp. 118–126 MR1562807
[13]	H. Vandiver, "On Fermat's last theorem" Trans. Amer. Math. Soc. , 31 (1929) pp. 613–642 MR1501503
[14]	H.S. Vandiver, "Examination of methods of attack on the second case of Fermat's last theorem" Proc. Nat. Acad. Sci. USA , 40 : 8 (1954) pp. 732–735 MR62758
[15]	S. Wagstaff, "The irregular primes to 125.000" Math. Comp. , 32 (1978) pp. 583–591 MR491465
[16]	A. Wieferich, "Zum letzten Fermatschen Theorem" J. Reine Angew. Math. , 136 (1909) pp. 293–302
[17]	D. Mirimanoff, "Zum letzten Fermatschen Theorem" J. Reine Angew. Math. , 139 (1911) pp. 309–324
[18]	D.H. Lehmer, "On Fermat's quotient, base 2" Math. Comp. , 36 (1981) pp. 289–290 MR595064
[19]	P. Furtwängler, "Letzter Fermat'scher Satz und Eisenstein'sches Reziprozitätsprinzip" Sitzungsber. Akad. Wiss. Wien Math.-Naturwiss. Kl. IIa , 121 (1912) pp. 589–592
[20]	G. Terjanian, "Sur l'equation $x^{2p} + y^{2p} = z^{2p}$" C.R. Acad. Sci. Paris , A285-B285 : 16 (1977) pp. 973A-975A (English abstract) MR0498370
[21]	K. Inkeri, "Abschätzungen für eventuelle Lösungen der Gleichung im Fermatschen Problem" Ann. Univ. Turku Ser. A , 16 : 1 (1953) pp. 3–9 MR0058629 Zbl 0051.28003
[22]	M.M. Postnikov, "An introduction to algebraic number theory" , Moscow (1982) (In Russian)
[23]	G. Faltings, "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" Invent. Math. , 73 (1983) pp. 349–366
[24]	D.R. Heath-Brown, "Fermat's last theorem for "almost all" exponents" Bull. London Math. Soc. , 17 (1985) pp. 15–16
[25]	L.M. Adleman, D.R. Heath-Brown, "The first case of Fermat's last theorem" Invent. Math. , 79 (1985) pp. 409–416 MR778135

Comments

In fact, Heath-Brown and, independently, A. Granville, cf. [a1], have proved that the density of the exponents $n$ for which Fermat's last theorem holds is one.

It is now (1988) known that Fermat's last theorem holds for all $n < 150000$, and that case 1 holds for all primes up to $714591416091389$, cf. [a2].

Recently (1987), K. Ribet, using ideas of G. Frey and J.-P. Serre, showed that Fermat's last theorem is implied by the Weil–Taniyama conjecture in the theory of elliptic curves (cf. Elliptic curve).

References

[a1]	S. Wagon, "Fermat's last theorem" Math. Intelligencer , 8 : 1 (1986) pp. 59–61 MR823221
[a2]	P. Ribenboim, "Recent results about Fermat's last theorem" Cand. Math. Bull. , 20 (1977) pp. 229–242 MR463088