Proof of Fermat's Last Theorem for specific exponents

Last updated

Fermat's Last Theorem is a theorem in number theory, originally stated by Pierre de Fermat in 1637 and proven by Andrew Wiles in 1995. The statement of the theorem involves an integer exponent n larger than 2. In the centuries following the initial statement of the result and before its general proof, various proofs were devised for particular values of the exponent n. Several of these proofs are described below, including Fermat's proof in the case n = 4, which is an early example of the method of infinite descent.

Contents

Mathematical preliminaries

Fermat's Last Theorem states that no three positive integers (a, b, c) can satisfy the equation an + bn = cn for any integer value of n greater than 2. (For n equal to 1, the equation is a linear equation and has a solution for every possible a and b. For n equal to 2, the equation has infinitely many solutions, the Pythagorean triples.)

Factors of exponents

A solution (a, b, c) for a given n leads to a solution for all the factors of n: if h is a factor of n then there is an integer g such that n = gh. Then (ag, bg, cg) is a solution for the exponent h:

(ag)h + (bg)h = (cg)h.

Therefore, to prove that Fermat's equation has no solutions for n > 2, it suffices to prove that it has no solutions for n = 4 and for all odd primes p.

For any such odd exponent p, every positive-integer solution of the equation ap + bp = cp corresponds to a general integer solution to the equation ap + bp + cp = 0. For example, if (3, 5, 8) solves the first equation, then (3, 5, −8) solves the second. Conversely, any solution of the second equation corresponds to a solution to the first. The second equation is sometimes useful because it makes the symmetry between the three variables a, b and c more apparent.

Primitive solutions

If two of the three numbers (a, b, c) can be divided by a fourth number d, then all three numbers are divisible by d. For example, if a and c are divisible by d = 13, then b is also divisible by 13. This follows from the equation

bn = cn − an

If the right-hand side of the equation is divisible by 13, then the left-hand side is also divisible by 13. Let g represent the greatest common divisor of a, b, and c. Then (a, b, c) may be written as a = gx, b = gy, and c = gz where the three numbers (x, y, z) are pairwise coprime. In other words, the greatest common divisor (GCD) of each pair equals one

GCD(x, y) = GCD(x, z) = GCD(y, z) = 1

If (a, b, c) is a solution of Fermat's equation, then so is (x, y, z), since the equation

an + bn = cn = gnxn + gnyn = gnzn

implies the equation

xn + yn = zn.

A pairwise coprime solution (x, y, z) is called a primitive solution. Since every solution to Fermat's equation can be reduced to a primitive solution by dividing by their greatest common divisor g, Fermat's Last Theorem can be proven by demonstrating that no primitive solutions exist.

Even and odd

Integers can be divided into even and odd, those that are evenly divisible by two and those that are not. The even integers are ...−4, −2, 0, 2, 4,... whereas the odd integers are ...−3, −1, 1, 3,.... The property of whether an integer is even (or not) is known as its parity. If two numbers are both even or both odd, they have the same parity. By contrast, if one is even and the other odd, they have different parity.

The addition, subtraction and multiplication of even and odd integers obey simple rules. The addition or subtraction of two even numbers or of two odd numbers always produces an even number, e.g., 4 + 6 = 10 and 3 + 5 = 8. Conversely, the addition or subtraction of an odd and even number is always odd, e.g., 3 + 8 = 11. The multiplication of two odd numbers is always odd, but the multiplication of an even number with any number is always even. An odd number raised to a power is always odd and an even number raised to power is always even, so for example xn has the same parity as x.

Consider any primitive solution (x, y, z) to the equation xn + yn = zn. The terms in (x, y, z) cannot all be even, for then they would not be coprime; they could all be divided by two. If xn and yn are both even, zn would be even, so at least one of xn and yn are odd. The remaining addend is either even or odd; thus, the parities of the values in the sum are either (odd + even = odd) or (odd + odd = even).

Prime factorization

The fundamental theorem of arithmetic states that any natural number can be written in only one way (uniquely) as the product of prime numbers. For example, 42 equals the product of prime numbers 2 × 3 × 7, and no other product of prime numbers equals 42, aside from trivial rearrangements such as 7 × 3 × 2. This unique factorization property is the basis on which much of number theory is built.

One consequence of this unique factorization property is that if a pth power of a number equals a product such as

xp = uv

and if u and v are coprime (share no prime factors), then u and v are themselves the pth power of two other numbers, u = rp and v = sp.

As described below, however, some number systems do not have unique factorization. This fact led to the failure of Lamé's 1847 general proof of Fermat's Last Theorem.

Two cases

Since the time of Sophie Germain, Fermat's Last Theorem has been separated into two cases that are proven separately. The first case (case I) is to show that there are no primitive solutions (x, y, z) to the equation xp + yp = zp under the condition that p does not divide the product xyz. The second case (case II) corresponds to the condition that p does divide the product xyz. Since x, y, and z are pairwise coprime, p divides only one of the three numbers.

n = 4

Portrait of Pierre de Fermat. Pierre de Fermat.png
Portrait of Pierre de Fermat.

Only one mathematical proof by Fermat has survived, in which Fermat uses the technique of infinite descent to show that the area of a right triangle with integer sides can never equal the square of an integer. [1] This result is known as Fermat's right triangle theorem. As shown below, his proof is equivalent to demonstrating that the equation

x4 − y4 = z2

has no primitive solutions in integers (no pairwise coprime solutions). In turn, this is sufficient to prove Fermat's Last Theorem for the case n = 4, since the equation a4 + b4 = c4 can be written as c4 − b4 = (a2)2. Alternative proofs of the case n = 4 were developed later [2] by Frénicle de Bessy, [3] Euler, [4] Kausler, [5] Barlow, [6] Legendre, [7] Schopis, [8] Terquem, [9] Bertrand, [10] Lebesgue, [11] Pepin, [12] Tafelmacher, [13] Hilbert, [14] Bendz, [15] Gambioli, [16] Kronecker, [17] Bang, [18] Sommer, [19] Bottari, [20] Rychlik, [21] Nutzhorn, [22] Carmichael, [23] Hancock, [24] Vrǎnceanu, [25] Grant and Perella, [26] Barbara, [27] and Dolan. [28] For one proof by infinite descent, see Infinite descent#Non-solvability of r2 + s4 = t4.

Application to right triangles

Fermat's proof demonstrates that no right triangle with integer sides can have an area that is a square. [29] Let the right triangle have sides (u, v, w), where the area equals uv/2 and, by the Pythagorean theorem, u2 + v2 = w2. If the area were equal to the square of an integer s

uv/2 = s2

then by algebraic manipulations it would also be the case that

2uv = 4s2 and −2uv = −4s2.

Adding u2 + v2 = w2 to these equations gives

u2 + 2uv + v2 = w2 + 4s2 and u2 − 2uv + v2 = w2 − 4s2,

which can be expressed as

(u + v)2 = w2 + 4s2 and (u − v)2 = w2 − 4s2.

Multiplying these equations together yields

(u2 − v2)2 = w4 − 16s4.

But as Fermat proved, there can be no integer solution to the equation x4 − y4 = z2, of which this is a special case with z = u2 − v2, x = w and y = 2s.

The first step of Fermat's proof is to factor the left-hand side [30]

(x2 + y2)(x2 − y2) = z2

Since x and y are coprime (this can be assumed because otherwise the factors could be cancelled), the greatest common divisor of x2 + y2 and x2 − y2 is either 2 (case A) or 1 (case B). The theorem is proven separately for these two cases.

Proof for case A

In this case, both x and y are odd and z is even. Since (y2, z, x2) form a primitive Pythagorean triple, they can be written

z = 2de
y2 = d2 − e2
x2 = d2 + e2

where d and e are coprime and d > e > 0. Thus,

x2y2 = d4 − e4

which produces another solution (d, e, xy) that is smaller (0 < d < x). As before, there must be a lower bound on the size of solutions, while this argument always produces a smaller solution than any given one, and thus the original solution is impossible.

Proof for case B

In this case, the two factors are coprime. Since their product is a square z2, they must each be a square

x2 + y2 = s2
x2 − y2 = t2

The numbers s and t are both odd, since s2 + t2 = 2x2, an even number, and since x and y cannot both be even. Therefore, the sum and difference of s and t are likewise even numbers, so we define integers u and v as

u = s + t/2
v = s − t/2

Since s and t are coprime, so are u and v; only one of them can be even. Since y2 = 2uv, exactly one of them is even. For illustration, let u be even; then the numbers may be written as u = 2m2 and v = k2. Since (u, v, x) form a primitive Pythagorean triple

s2 + t2/2 = u2 + v2 = x2

they can be expressed in terms of smaller integers d and e using Euclid's formula

u = 2de
v = d2 − e2
x = d2 + e2

Since u = 2m2 = 2de, and since d and e are coprime, they must be squares themselves, d = g2 and e = h2. This gives the equation

v = d2 − e2 = g4 − h4 = k2

The solution (g, h, k) is another solution to the original equation, but smaller (0 < g < d < x). Applying the same procedure to (g, h, k) would produce another solution, still smaller, and so on. But this is impossible, since natural numbers cannot be shrunk indefinitely. Therefore, the original solution (x, y, z) was impossible.

n = 3

Leonhard Euler by Jakob Emanuel Handmann. Leonhard Euler.jpg
Leonhard Euler by Jakob Emanuel Handmann.

Fermat sent the letters in which he mentioned the case in which n = 3 in 1636, 1640 and 1657. [31] Euler sent a letter to Goldbach on 4 August 1753 in which claimed to have a proof of the case in which n = 3. [32] Euler had the complete and pure elementary proof in 1760. [33] The case n = 3 was proven by Euler in 1770. [34] [35] [36] [37] Independent proofs were published by several other mathematicians, [38] including Kausler, [5] Legendre, [7] [39] Calzolari, [40] Lamé, [41] Tait, [42] Günther, [43] Gambioli, [16] Krey, [44] Rychlik, [21] Stockhaus, [45] Carmichael, [46] van der Corput, [47] Thue, [48] and Duarte. [49]

Chronological table of the proof of n = 3
dateresult/proofpublished/not publishedworkname
1621nonepublishedLatin version of Diophantus's Arithmetica Bachet
around 1630only resultnot publisheda marginal note in Arithmetica Fermat
1636, 1640, 1657only resultpublishedletters of n = 3Fermat [31]
1670only resultpublisheda marginal note in ArithmeticaFermat's son Samuel published the Arithmetica with Fermat's note.
4 August 1753only resultpublishedletter to Goldbach Euler [32]
1760proofnot publishedcomplete and pure elemental proofEuler [33]
1770proofpublishedincomplete but elegant proof in Elements of AlgebraEuler [32] [34] [37]

As Fermat did for the case n = 4, Euler used the technique of infinite descent. [50] The proof assumes a solution (x, y, z) to the equation x3 + y3 + z3 = 0, where the three non-zero integers x, y, and z are pairwise coprime and not all positive. One of the three must be even, whereas the other two are odd. Without loss of generality, z may be assumed to be even.

Since x and y are both odd, they cannot be equal. If x = y, then 2x3 = −z3, which implies that x is even, a contradiction.

Since x and y are both odd, their sum and difference are both even numbers

2u = x + y
2v = x − y

where the non-zero integers u and v are coprime and have different parity (one is even, the other odd). Since x = u + v and y = u − v, it follows that

−z3 = (u + v)3 + (u − v)3 = 2u(u2 + 3v2)

Since u and v have opposite parity, u2 + 3v2 is always an odd number. Therefore, since z is even, u is even and v is odd. Since u and v are coprime, the greatest common divisor of 2u and u2 + 3v2 is either 1 (case A) or 3 (case B).

Proof for case A

In this case, the two factors of −z3 are coprime. This implies that three does not divide u and that the two factors are cubes of two smaller numbers, r and s

2u = r3
u2 + 3v2 = s3

Since u2 + 3v2 is odd, so is s. A crucial lemma shows that if s is odd and if it satisfies an equation s3 = u2 + 3v2, then it can be written in terms of two integers e and f

s = e2 + 3f2

so that

u = e(e2 − 9f2)
v = 3f(e2 − f2)

u and v are coprime, so e and f must be coprime, too. Since u is even and v odd, e is even and f is odd. Since

r3 = 2u = 2e(e − 3f)(e + 3f)

The factors 2e, (e – 3f), and (e + 3f) are coprime since 3 cannot divide e: if e were divisible by 3, then 3 would divide u, violating the designation of u and v as coprime. Since the three factors on the right-hand side are coprime, they must individually equal cubes of smaller integers

−2e = k3
e − 3f = l3
e + 3f = m3

which yields a smaller solution k3 + l3 + m3 = 0. Therefore, by the argument of infinite descent, the original solution (x, y, z) was impossible.

Proof for case B

In this case, the greatest common divisor of 2u and u2 + 3v2 is 3. That implies that 3 divides u, and one may express u = 3w in terms of a smaller integer, w. Since u is divisible by 4, so is w; hence, w is also even. Since u and v are coprime, so are v and w. Therefore, neither 3 nor 4 divide v.

Substituting u by w in the equation for z3 yields

−z3 = 6w(9w2 + 3v2) = 18w(3w2 + v2)

Because v and w are coprime, and because 3 does not divide v, then 18w and 3w2 + v2 are also coprime. Therefore, since their product is a cube, they are each the cube of smaller integers, r and s

18w = r3
3w2 + v2 = s3

By the lemma above, since s is odd and its cube is equal to a number of the form 3w2 + v2, it too can be expressed in terms of smaller coprime numbers, e and f.

s = e2 + 3f2

A short calculation shows that

v = e(e2 − 9f2)
w = 3f(e2 − f2)

Thus, e is odd and f is even, because v is odd. The expression for 18w then becomes

r3 = 18w = 54f(e2 − f2) = 54f(e + f)(e − f) = 33 × 2f(e + f)(e − f).

Since 33 divides r3 we have that 3 divides r, so (r/3)3 is an integer that equals 2f(e + f)(e − f). Since e and f are coprime, so are the three factors 2f, e + f, and e − f; therefore, they are each the cube of smaller integers, k, l, and m.

−2f = k3
e + f = l3
f − e = m3

which yields a smaller solution k3 + l3 + m3 = 0. Therefore, by the argument of infinite descent, the original solution (x, y, z) was impossible.

n = 5

Portrait of Peter Gustav Lejeune Dirichlet. Peter Gustav Lejeune Dirichlet.jpg
Portrait of Peter Gustav Lejeune Dirichlet.
Caricature of Adrien-Marie Legendre (the only surviving portrait of him). Legendre.jpg
Caricature of Adrien-Marie Legendre (the only surviving portrait of him).

Fermat's Last Theorem for n = 5 states that no three coprime integers x, y and z can satisfy the equation

x5 + y5 + z5 = 0

This was proven [51] neither independently nor collaboratively by Dirichlet and Legendre around 1825. [32] [52] Alternative proofs were developed [53] by Gauss, [54] Lebesgue, [55] Lamé, [56] Gambioli, [16] [57] Werebrusow, [58] Rychlik, [59] van der Corput, [47] and Terjanian. [60]

Dirichlet's proof for n = 5 is divided into the two cases (cases I and II) defined by Sophie Germain. In case I, the exponent 5 does not divide the product xyz. In case II, 5 does divide xyz.

  1. Case I for n = 5 can be proven immediately by Sophie Germain's theorem(1823) if the auxiliary prime θ = 11.
  2. Case II is divided into the two cases (cases II(i) and II(ii)) by Dirichlet in 1825. Case II(i) is the case which one of x, y, z is divided by either 5 and 2. Case II(ii) is the case which one of x, y, z is divided by 5 and another one of x, y, z is divided by 2. In July 1825, Dirichlet proved the case II(i) for n = 5. In September 1825, Legendre proved the case II(ii) for n = 5. After Legendre's proof, Dirichlet completed the proof for the case II(ii) for n = 5 by the extended argument for the case II(i). [32]
Chronological table of the proof of n = 5
datecase I/IIcase II(i/ii)name
1823case I Germain
July 1825case IIcase II(i) Dirichlet
September 1825case II(ii) Legendre
after September 1825Dirichlet

Proof for case A

Case A for n = 5 can be proven immediately by Sophie Germain's theorem if the auxiliary prime θ = 11. A more methodical proof is as follows. By Fermat's little theorem,

x5 ≡ x (mod 5)
y5 ≡ y (mod 5)
z5 ≡ z (mod 5)

and therefore

x + y + z ≡ 0 (mod 5)

This equation forces two of the three numbers x, y, and z to be equivalent modulo 5, which can be seen as follows: Since they are indivisible by 5, x, y and z cannot equal 0 modulo 5, and must equal one of four possibilities: 1, −1, 2, or −2. If they were all different, two would be opposites and their sum modulo 5 would be zero (implying contrary to the assumption of this case that the other one would be 0 modulo 5).

Without loss of generality, x and y can be designated as the two equivalent numbers modulo 5. That equivalence implies that

x5 ≡ y5 (mod 25) (note change in modulus)
−z5 ≡ x5 + y5 ≡ 2x5 (mod 25)

However, the equation x ≡ y (mod 5) also implies that

−z ≡ x + y ≡ 2x (mod 5)
−z5 ≡ 25x5 ≡ 32x5 (mod 25)

Combining the two results and dividing both sides by x5 yields a contradiction

2 ≡ 32 (mod 25) ≡ 7

Thus, case A for n = 5 has been proven.

Proof for case B

n = 7

The case n = 7 was proven [61] by Gabriel Lamé in 1839. [62] His rather complicated proof was simplified in 1840 by Victor-Amédée Lebesgue, [63] and still simpler proofs [64] were published by Angelo Genocchi in 1864, 1874 and 1876. [65] Alternative proofs were developed by Théophile Pépin [66] and Edmond Maillet. [67]

n = 6, n = 10, and n = 14

Fermat's Last Theorem has also been proven for the exponents n = 6, n = 10, and n = 14. Proofs for n = 6 have been published by Kausler, [5] Thue, [68] Tafelmacher, [69] Lind, [70] Kapferer, [71] Swift, [72] and Breusch. [73] Similarly, Dirichlet [74] and Terjanian [75] each proved the case n = 14, while Kapferer [71] and Breusch [73] each proved the case n = 10. Strictly speaking, these proofs are unnecessary, since these cases follow from the proofs for n = 3, n = 5, n = 7, respectively. Nevertheless, the reasoning of these even-exponent proofs differs from their odd-exponent counterparts. Dirichlet's proof for n = 14 was published in 1832, before Lamé's 1839 proof for n = 7.

Notes

  1. ↑ Freeman L. "Fermat's One Proof" . Retrieved 2009-05-23.
  2. ↑ Ribenboim, pp. 15–24.
  3. ↑ Frénicle de Bessy, Traité des Triangles Rectangles en Nombres, vol. I, 1676, Paris. Reprinted in Mém. Acad. Roy. Sci., 5, 1666–1699 (1729).
  4. ↑ Euler L (1738). "Theorematum quorundam arithmeticorum demonstrationes". Comm. Acad. Sci. Petrop. 10: 125–146.. Reprinted Opera omnia, ser. I, "Commentationes Arithmeticae", vol. I, pp. 38–58, Leipzig:Teubner (1915).
  5. 1 2 3 Kausler CF (1802). "Nova demonstratio theorematis nec summam, nec differentiam duorum cuborum cubum esse posse". Novi Acta Acad. Petrop. 13: 245–253.
  6. ↑ Barlow P (1811). An Elementary Investigation of Theory of Numbers. St. Paul's Church-Yard, London: J. Johnson. pp. 144–145.
  7. 1 2 Legendre AM (1830). Théorie des Nombres (Volume II) (3rd ed.). Paris: Firmin Didot Frères. Reprinted in 1955 by A. Blanchard (Paris).
  8. ↑ Schopis (1825). Einige Sätze aus der unbestimmten Analytik. Gummbinnen: Programm.
  9. ↑ Terquem O (1846). "Théorèmes sur les puissances des nombres". Nouv. Ann. Math. 5: 70–87.
  10. ↑ Bertrand JLF (1851). Traité Élémentaire d'Algèbre. Paris: Hachette. pp.  217–230, 395.
  11. ↑ Lebesgue VA (1853). "Résolution des équations biquadratiques z2 = x4 ± 2my4, z2 = 2mx4 − y4, 2mz2 = x4 ± y4". J. Math. Pures Appl. 18: 73–86.
    Lebesgue VA (1859). Exercices d'Analyse Numérique. Paris: Leiber et Faraguet. pp. 83–84, 89.
    Lebesgue VA (1862). Introduction à la Théorie des Nombres. Paris: Mallet-Bachelier. pp. 71–73.
  12. ↑ Pepin T (1883). "Étude sur l'équation indéterminée ax4 + by4 = cz2". Atti Accad. Naz. Lincei. 36: 34–70.
  13. ↑ Tafelmacher WLA (1893). "Sobre la ecuación x4 + y4 = z4". Ann. Univ. Chile. 84: 307–320.
  14. ↑ Hilbert D (1897). "Die Theorie der algebraischen Zahlkörper". Jahresbericht der Deutschen Mathematiker-Vereinigung . 4: 175–546. Reprinted in 1965 in Gesammelte Abhandlungen, vol. I by New York:Chelsea.
  15. ↑ Bendz TR (1901). Öfver diophantiska ekvationen xn + yn = zn. Uppsala: Almqvist & Wiksells Boktrycken.
  16. 1 2 3 Gambioli D (1901). "Memoria bibliographica sull'ultimo teorema di Fermat". Period. Mat. 16: 145–192.
  17. ↑ Kronecker L (1901). Vorlesungen über Zahlentheorie, vol. I. Leipzig: Teubner. pp. 35–38. Reprinted by New York:Springer-Verlag in 1978.
  18. ↑ Bang A (1905). "Nyt Bevis for at Ligningen x4 − y4 = z4, ikke kan have rationale Løsinger". Nyt Tidsskrift Mat. 16B: 35–36.
  19. ↑ Sommer J (1907). Vorlesungen über Zahlentheorie. Leipzig: Teubner.
  20. ↑ Bottari A. "Soluzione intere dell'equazione pitagorica e applicazione alla dimostrazione di alcune teoremi dellla teoria dei numeri". Period. Mat. 23: 104–110.
  21. 1 2 Rychlík K (1910). "On Fermat's last theorem for n = 4 and n = 3". Časopis Pěst. Mat. (in Czech). 39: 65–86.
  22. ↑ Nutzhorn F (1912). "Den ubestemte Ligning x4 + y4 = z4". Nyt Tidsskrift Mat. 23B: 33–38.
  23. ↑ Carmichael RD (1913). "On the impossibility of certain Diophantine equations and systems of equations". Amer. Math. Monthly. 20 (7): 213–221. doi:10.2307/2974106. JSTOR   2974106.
  24. ↑ Hancock H (1931). Foundations of the Theory of Algebraic Numbers, vol. I. New York: Macmillan.
  25. ↑ Vrǎnceanu G (1966). "Asupra teorema lui Fermat pentru n=4". Gaz. Mat. Ser. A. 71: 334–335. Reprinted in 1977 in Opera matematica, vol. 4, pp. 202–205, București:Edit. Acad. Rep. Soc. Romana.
  26. ↑ Grant, Mike, and Perella, Malcolm, "Descending to the irrational", Mathematical Gazette 83, July 1999, pp.263-267.
  27. ↑ Barbara, Roy, "Fermat's last theorem in the case n = 4", Mathematical Gazette 91, July 2007, 260-262.
  28. ↑ Dolan, Stan, "Fermat's method of descente infinie", Mathematical Gazette 95, July 2011, 269-271.
  29. ↑ Fermat P. "Ad Problema XX commentarii in ultimam questionem Arithmeticorum Diophanti. Area trianguli rectanguli in numeris non potest esse quadratus", Œuvres, vol. I, p. 340 (Latin), vol. III, pp. 271–272 (French). Paris:Gauthier-Villars, 1891, 1896.
  30. ↑ Ribenboim, pp. 11–14.
  31. 1 2 Dickson (2005 , p. 546)
  32. 1 2 3 4 5 O'Connor & Robertson (1996)
  33. 1 2 Bergmann (1966)
  34. 1 2 Euler L (1770) Vollständige Anleitung zur Algebra, Roy.Acad. Sci., St. Petersburg.
  35. ↑ Freeman L. "Fermat's Last Theorem: Proof for n = 3" . Retrieved 2009-05-23.
  36. ↑ J. J. Mačys (2007). "On Euler's hypothetical proof". Mathematical Notes. 82 (3–4): 352–356. doi:10.1134/S0001434607090088. MR   2364600. S2CID   121798358.
  37. 1 2 Euler (1822 , pp. 399, 401–402)
  38. ↑ Ribenboim, pp. 33, 37–41.
  39. ↑ Legendre AM (1823). "Recherches sur quelques objets d'analyse indéterminée, et particulièrement sur le théorème de Fermat". Mém. Acad. Roy. Sci. Institut France. 6: 1–60. Reprinted in 1825 as the "Second Supplément" for a printing of the 2nd edition of Essai sur la Théorie des Nombres, Courcier (Paris). Also reprinted in 1909 in Sphinx-Oedipe, 4, 97–128.
  40. ↑ Calzolari L (1855). Tentativo per dimostrare il teorema di Fermat sull'equazione indeterminata xn + yn = zn. Ferrara.
  41. ↑ Lamé G (1865). "Étude des binômes cubiques x3 ± y3". C. R. Acad. Sci. Paris. 61: 921–924, 961–965.
  42. ↑ Tait PG (1872). "Mathematical Notes". Proc. R. Soc. Edinburgh. 7: 144. doi:10.1017/S0370164600041857.
  43. ↑ Günther S (1878). "Über die unbestimmte Gleichung x3 + y3 = z3". Sitzungsberichte Böhm. Ges. Wiss.: 112–120.
  44. ↑ Krey H (1909). "Neuer Beweis eines arithmetischen Satzes". Math. Naturwiss. Blätter. 6: 179–180.
  45. ↑ Stockhaus H (1910). Beitrag zum Beweis des Fermatschen Satzes. Leipzig: Brandstetter.
  46. ↑ Carmichael RD (1915). Diophantine Analysis. New York: Wiley.
  47. 1 2 Van der Corput JG (1915). "Quelques formes quadratiques et quelques équations indéterminées". Nieuw Archief Wisk. 11: 45–75.
  48. ↑ Thue A (1917). "Et bevis for at ligningen A3 + B3 = C3 er unmulig i hele tal fra nul forskjellige tal A, B og C". Arch. Mat. Naturv. 34 (15). Reprinted in Selected Mathematical Papers (1977), Oslo:Universitetsforlaget, pp. 555–559.
  49. ↑ Duarte FJ (1944). "Sobre la ecuación x3 + y3 + z3 = 0". Ciencias Fis. Mat. Naturales (Caracas). 8: 971–979.
  50. ↑ Ribenboim, pp. 24–49.
  51. ↑ Freeman L. "Fermat's Last Theorem: Proof for n = 5" . Retrieved 2009-05-23.
  52. ↑ Ribenboim, p. 49.
  53. ↑ Ribenboim, pp. 55–57.
  54. ↑ Gauss CF (1875). "Neue Theorie der Zerlegung der Cuben". Zur Theorie der complexen Zahlen, Werke, vol. II (2nd ed.). Königl. Ges. Wiss. Göttingen. pp. 387–391.
  55. ↑ Lebesgue VA (1843). "Théorèmes nouveaux sur l'équation indéterminée x5 + y5 = az5". J. Math. Pures Appl. 8: 49–70.
  56. ↑ Lamé G (1847). "Mémoire sur la résolution en nombres complexes de l'équation A5 + B5 + C5 = 0". J. Math. Pures Appl. 12: 137–171.
  57. ↑ Gambioli D (1903–1904). "Intorno all'ultimo teorema di Fermat". Il Pitagora. 10: 11–13, 41–42.
  58. ↑ Werebrusow AS (1905). "On the equation x5 + y5 = Az5(in Russian)". Moskov. Math. Samml. 25: 466–473.
  59. ↑ Rychlik K (1910). "On Fermat's last theorem for n = 5 (in Bohemian)". Časopis Pěst. Mat. 39: 185–195, 305–317.
  60. ↑ Terjanian G (1987). "Sur une question de V. A. Lebesgue". Annales de l'Institut Fourier. 37 (3): 19–37. doi: 10.5802/aif.1096 .
  61. ↑ Ribenboim, pp. 57–63.
  62. ↑ Lamé G (1839). "Mémoire sur le dernier théorème de Fermat". C. R. Acad. Sci. Paris. 9: 45–46.
    Lamé G (1840). "Mémoire d'analyse indéterminée démontrant que l'équation x7 + y7 = z7 est impossible en nombres entiers". J. Math. Pures Appl. 5: 195–211.
  63. ↑ Lebesgue VA (1840). "Démonstration de l'impossibilité de résoudre l'équation x7 + y7 + z7 = 0 en nombres entiers". J. Math. Pures Appl. 5: 276–279, 348–349.
  64. ↑ Freeman L. "Fermat's Last Theorem: Proof for n = 7" . Retrieved 2009-05-23.
  65. ↑ Genocchi A (1864). "Intorno all'equazioni x7 + y7 + z7 = 0". Ann. Mat. Pura Appl. 6: 287–288. doi:10.1007/BF03198884. S2CID   124916552.
    Genocchi A (1874). "Sur l'impossibilité de quelques égalités doubles". C. R. Acad. Sci. Paris. 78: 433–436.
    Genocchi A (1876). "Généralisation du théorème de Lamé sur l'impossibilité de l'équation x7 + y7 + z7 = 0". C. R. Acad. Sci. Paris. 82: 910–913.
  66. ↑ Pépin T (1876). "Impossibilité de l'équation x7 + y7 + z7 = 0". C. R. Acad. Sci. Paris. 82: 676–679, 743–747.
  67. ↑ Maillet E (1897). "Sur l'équation indéterminée axλt + byλt = czλt". Assoc. Française Avanc. Sci., Saint-Étienne. Série II. 26: 156–168.
  68. ↑ Thue A (1896). "Über die Auflösbarkeit einiger unbestimmter Gleichungen". Det Kongel. Norske Videnskabers Selskabs Skrifter. 7. Reprinted in Selected Mathematical Papers, pp. 19–30, Oslo:Universitetsforlaget (1977).
  69. ↑ Tafelmacher WLA (1897). "La ecuación x3 + y3 = z2: Una demonstración nueva del teorema de Fermat para el caso de las sestas potencias". Ann. Univ. Chile, Santiago. 97: 63–80.
  70. ↑ Lind B (1909). "Einige zahlentheoretische Sätze". Arch. Math. Phys. 15: 368–369.
  71. 1 2 Kapferer H (1913). "Beweis des Fermatschen Satzes für die Exponenten 6 und 10". Arch. Math. Phys. 21: 143–146.
  72. ↑ Swift E (1914). "Solution to Problem 206". Amer. Math. Monthly. 21: 238–239. doi:10.2307/2972379. JSTOR   2972379.
  73. 1 2 Breusch R (1960). "A simple proof of Fermat's last theorem for n = 6, n = 10". Math. Mag. 33 (5): 279–281. doi:10.2307/3029800. JSTOR   3029800.
  74. ↑ Dirichlet PGL (1832). "Démonstration du théorème de Fermat pour le cas des 14e puissances". J. Reine Angew. Math. 9: 390–393. Reprinted in Werke, vol. I, pp. 189–194, Berlin:G. Reimer (1889); reprinted New York:Chelsea (1969).
  75. ↑ Terjanian G (1974). "L'équation x14 + y14 = z14 en nombres entiers". Bull. Sci. Math. Série 2. 98: 91–95.

Related Research Articles

In number theory, two integers a and b are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides a does not divide b, and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also ais prime tob or ais coprime withb.

<span class="mw-page-title-main">Diophantine equation</span> Polynomial equation whose integer solutions are sought

In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, for which only integer solutions are of interest. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents.

<span class="mw-page-title-main">Modular arithmetic</span> Computation modulo a fixed integer

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

<span class="mw-page-title-main">Pythagorean triple</span> Integer side lengths of a right triangle

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime (that is, they have no common divisor larger than 1). For example, (3, 4, 5) is a primitive Pythagorean triple whereas (6, 8, 10) is not. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle.

<span class="mw-page-title-main">Pell's equation</span> Type of Diophantine equation

Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form where n is a given positive nonsquare integer, and integer solutions are sought for x and y. In Cartesian coordinates, the equation is represented by a hyperbola; solutions occur wherever the curve passes through a point whose x and y coordinates are both integers, such as the trivial solution with x = 1 and y = 0. Joseph Louis Lagrange proved that, as long as n is not a perfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately approximate the square root of n by rational numbers of the form x/y.

<span class="mw-page-title-main">Gaussian integer</span> Complex number whose real and imaginary parts are both integers

In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as or

In number theory, Fermat's little theorem states that if p is a prime number, then for any integer a, the number ap − a is an integer multiple of p. In the notation of modular arithmetic, this is expressed as

Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 are two perfect powers (that is, powers of exponent higher than one) of natural numbers whose values (8 and 9, respectively) are consecutive. The theorem states that this is the only case of two consecutive perfect powers. That is to say, that

In number theory, Euler's theorem states that, if n and a are coprime positive integers, and is Euler's totient function, then a raised to the power is congruent to 1 modulo n; that is

In mathematics, a Fermat number, named after Pierre de Fermat, the first known to have studied them, is a positive integer of the form

In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.

In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli numbers.

In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. It is a method which relies on the well-ordering principle, and is often used to show that a given equation, such as a Diophantine equation, has no solutions.

In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as:

The Beal conjecture is the following conjecture in number theory:

In mathematics and statistics, sums of powers occur in a number of contexts:

<span class="mw-page-title-main">Fermat's Last Theorem</span> 17th-century conjecture proved by Andrew Wiles in 1994

In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions.

A pseudoprime is a probable prime that is not actually prime. Pseudoprimes are classified according to which property of primes they satisfy.

<span class="mw-page-title-main">Fermat's right triangle theorem</span> Rational right triangles cannot have square area

Fermat's right triangle theorem is a non-existence proof in number theory, published in 1670 among the works of Pierre de Fermat, soon after his death. It is the only complete proof given by Fermat. It has several equivalent formulations, one of which was stated in 1225 by Fibonacci. In its geometric forms, it states:

In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to Q, the field of rational numbers.

References

Further reading

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.